The Ultimate Arciaga

Physics 72 Arciaga CHAPTER 21. ELECTRIC CHARGE AND ELECTRIC FIELD

Review vector addition, geometry and calculus (differentiation and integration) Four fundamental forces:

o Gravitational force involves mass o Electromagnetic force involves electric charge o Strong force (or Nuclear force) o Weak force

A. Electric charge Electric charge is a scalar; it has no direction. SI unit of electric charge is coulomb (C) Three properties of electric charge:

1. Dichotomy property The electric charge is either positive (+) or negative (). Like charges repel; opposite charges attract.

2. Conservation property The algebraic sum of all the electric charges in any closed system is constant. In charging, charge is not created nor destroyed; it is only transferred from one

body to another. This is a universal conservation law.

3. Quantization property The magnitude of charge of the electron or proton is a natural unit of charge. Basic unit of charge e = 1.60210-19 C

a. Charge of 1 proton = +e = 1.60210-19 C b. Charge of 1 electron = e = 1.60210-19 C

Every observable amount of electric charge is always an integer multiple of this basic unit.

Other keywords: o Electrostatics involves electric charges that are at rest (i.e. speed is zero) in the

observers reference frame o Atom composed of electron, proton, and neutron o Neutral atom atom with zero net charge (# of electrons = # of protons) o Positive ion (cation) atom with positive net charge (lost one or more electrons) o Negative ion (anion) atom with negative net charge (gained one or more electrons) o Ionization gaining or losing of electrons

B. Types of materials in terms of electric conduction Conductors

o Objects that permit the easy movement of electrons through them o Ex: most metals, copper wire, earth o In metals, the mobile charges are always negative electrons sea of free electrons o The earth can act as an infinite source or sink of electrons grounding

Insulators o Objects that does NOT permit the easy movement of electrons through them o Ex: most nonmetals, ceramic, wood, plastic, rubber, air o The charges within the molecules of an insulator can shift slightly polarization

Physics 72 Arciaga Semiconductors

o Objects with properties between conductors and insulators o Ex: silicon, diodes, transistors

Superconductors o Objects with zero resistance against the movement of electrons o Ex: some compounds at very low temperatures

C. Ways of charging a material Charging by rubbing charge of charger changes; electrons transfer Charging by contact charge of charger changes; electrons transfer Charging by induction (without grounding) charge of charger does NOT change Charging by induction (with grounding)

charge of charger does NOT change negative charger induces a positive charge (positive charger induces a negative charge)

Charging by polarization charge of charger does NOT change charged object can still attract a neutral object by polarization

D. Coulombs law The amplitude of the electric force between two point charges is directly proportional to the

product of the charges and inversely proportional to the square of the distance between them.

Mathematically: 1 2 1 2e 2 2o

q q q q1F k4r r

= =

pi

; where Fe = magnitude of the electric force between two point charges q1 and q2 = electric charges of the two point charges r = distance between the 2 point charges k = proportionality constant = 1/4pio o = permittivity of free space (permittivity of vacuum)

NOTES: 1. The direction of eF

is along the line joining the two point charges. 2. The electric force on q1 by q2 is equal in magnitude but opposite in direction to the electric force on q2 by q1. [Recall: Newtons third law of motion] 3. It is an inverse square law. [Compare: Newtons law of gravity] 4. k = 1/4pio = 8.988109 Nm2/C2 5. o = 8.85410-12 C2/Nm2 6. If there are more than two point charges, use the principle of superposition of forces. Use vector addition (not scalar addition). 7. For atomic particles, the electric force is much greater than the gravitational force.

E. Electric field and electric forces Electric field:

e

test

FEq

=

; where E

= electric field at a particular position qtest = charge of a test charge placed at the particular position eF

= net electric force experienced by the test charge at the particular position

NOTES: 1. Electric field is a vector. 2. Electric field is an intermediary for the electric force; an aura of electric charges.

Physics 72 Arciaga 3. A charged body experiences an electric force when it feels an electric field created by other charged bodies. 4. Compare it with the gravitational field. 5. SI unit of electric field is newton per coulomb (N/C).

Electric force experienced by a point charge due to a given electric field: eF qE=

; where eF

= net electric force experienced by a point charge at a particular position q = charge of a point charge placed at the particular position E

= electric field at the particular position

NOTES: 1. eF

and E

are in the same direction if q is positive. 2. eF

and E

are in the opposite direction if q is negative.

Electric field created by a point charge:

2o

1 qE r

4 r=

pi

; where E

= electric field created by a point charge q = charge of the point charge r = distance from the point charge r = unit vector pointing away from the point charge (i.e. radially outward) NOTES: 1. E

points away from a positive charge.

2. E

points toward a negative charge. 3. The electric field by a point charge is an inverse-square relation. 4. If there are more than one point charge, use the principle of superposition of electric fields. Use vector addition (not scalar addition). 5. Other keywords:

o Source point location of the point charge that creates the electric field o Field points locations at which the electric field are being determined o Vector field infinite set of vectors drawn in a region of space o Uniform field constant vector field (i.e. magnitude and direction are constant)

Electric field created by a continuous distribution of charge: 1. Use principle of superposition of electric fields; perform an integration! 2. Imagine the continuous distribution of charge as composed of many point charges. 3. Sometimes symmetry analysis makes the solution easier. 4. Other keywords:

o linear charge density [] charge per unit length (C/m) o surface charge density [] charge per unit area (C/m2) o volume charge density [] charge per unit volume (C/m3)

F. Electric field lines (also called lines of force) Electric field lines

o imaginary line or curve drawn so that its tangent at any point is in the same direction of the electric field vector at that point

o tangent at an electric field line determines direction of the electric field o spacing of electric field lines determines magnitude of the electric field

electric field lines are closer together indicates strong electric field electric field lines are farther apart indicates weak electric field

o electric field lines never intersect o electric field magnitude can vary along one electric field line

Physics 72 Arciaga

G. Electric dipoles Electric dipole

o a pair of point charges with equal magnitude and opposite sign separated by a particular distance

o ex: water molecule, polar molecules, TV antenna Electric dipole moment

pp qd r=

; where p = electric dipole moment of a dipole q = magnitude of the electric charge (of a charge) in the dipole d = separation distance between the two charges

pr = unit vector pointing from the negative to the positive charge NOTE: Electric dipole moment is a vector:

a. magnitude = |qd| b. direction = from the negative to the positive charge

Torque of an electric dipole in a uniform electric field p E =

; where = torque experienced by an electric dipole in an electric field

p = electric dipole moment of a dipole E

= electric field NOTE: Torque is a vector [recall Physics 71]:

a. magnitude = pE sin ; where = small (tail-to-tail) angle between p and E b. direction = use right-hand rule [recall Physics 71]

Potential energy of an electric dipole in a uniform electric field U p E=

; where U = potential energy experienced by an electric dipole

in an electric field p = electric dipole moment of a dipole E

= electric field NOTE: Potential energy is a scalar [recall Physics 71]:

a. magnitude = pE cos ; where = small (tail-to-tail) angle between p and E

Equilibrium concepts [recall Physics 71] If both the net force and the net torque on an object are ZERO, then that object is in

EQUILIBRIUM; otherwise, that object is NOT in equilibrium. If the potential energy of an object is a MINIMUM, then that object is in STABLE

equilibrium. But if the potential energy of an object is a MAXIMUM, then that object is in UNSTABLE equilibrium.

Physics 72 Arciaga CHAPTER 22. GAUSSS LAW

A. Electric flux Electric flux like a flow of the electric field through an imaginary surface For a uniform electric field through a flat surface:

E E A E n A E A cos = = =

; where E = electric flux E

= electric field

A n A=

= vector area

n = unit vector perpendicular to the area (unit normal vector) = tail-to-tail angle between E and A

(or equivalently, E and n ) NOTES: 1. Electric flux is a scalar. 2. Electric flux is zero if E

is parallel to the surface.

3. The vector area has: a. magnitude equal to the area of the surface; and b. direction perpendicular to the surface.

General definition: For any electric field through any surface E E dA E n dA E cos dA = = =

NOTES: 1. This is called a surface integral of E dA

.

2. For a closed surface: a. unit vector n points outward (by convention) b. electric flux is positive if flowing outward the closed surface c. electric flux is negative if flowing inward the closed surface

B. Gausss law Qualitative statements of Gausss law:

1. The net electric flux through a closed surface is outward (/inward) if the net enclosed charge is positive (/negative).

2. The net electric flux through a closed surface is zero if the net enclosed charge is zero. 3. The net electric flux through a closed surface is unaffected by charges outside the closed

surface. 4. The net electric flux through a closed surface is directly proportional to the net amount of

enclosed charge. 5. The net electric flux through a closed surface is independent of the size and shape of the

closed surface (if the net amount of enclosed charge is constant). Mathematically:

encEo

QE dA E n dA = = =

NOTES: 1. The symbol means surface integral for a closed surface. 2. The closed surface to be used is imaginary !!! called a Gaussian surface 3. Two possible uses: a. Given a charge distribution, enclose it with a proper Gaussian (imaginary) surface that utilizes the symmetry of the situation, then determine the electric field. b. Given an electric field, construct a Gaussian (imaginary) surface, then determine the charge distribution inside it.

Physics 72 Arciaga C. Conductors in electrostatics

The electric field is zero ( E 0= ) in the bulk material of a conductor. Any excess charge resides entirely on the surface of the conductor; no charge can be found in the

bulk material. The electric field at the surface of the conductor is always perpendicular to the surface; there is no

tangential or parallel component. The electric field at the surface of the conductor has a magnitude equal to /o. The magnitudes of electric field and surface charge density on the surface of the conductor are

higher at the sharper locations. The electric field is discontinuous (in magnitude and/or direction) wherever there is a sheet of

charge.

Physics 72 Arciaga CHAPTER 23. ELECTRIC POTENTIAL

A. Electric potential energy Review of some important remarks [recall Physics 71]:

1. Work done by a force F

on a particle that moves from position a to position b.

b

a ba

W F ds =

2. Electric force is a conservative force. A conservative force has the following properties: a. The work it does on a particle is independent of the path taken by the particle and

depends only on the initial and final positions. b. The total work it does on a particle is zero when the particle moves around any

closed path, in which the initial and final positions are the same. c. The work it does on a particle is reversible, i.e. energy can always be recovered

without loss. d. The work it does on a particle can be expressed as the difference between the initial

and final values of a potential-energy function. Wab = U = (Ub Ua) = Ua Ub

; where Wab = work done by a conservative force when a particle moves from position a to position b

U = change in the potential energy Ua and Ub = potential energies at positions a and b, respectively

3. Conservation of mechanical energy can be applied when only internal force and conservative force do work on the system.

4. A system tends to attain the lowest possible potential energy (i.e. it tends to attain a state of stable equilibrium).

A charged particle in a uniform electric field: U = Uo + qEh ; where U = electric potential energy of a charged particle

in a uniform electric field q = electric charge of the charged particle E = magnitude of the uniform electric field h = position of the charged particle against E

Uo = reference potential energy (i.e. value of U at h = 0) Wab = U = qE(ha hb)

; where ha and hb = positions at a and b, respectively NOTE: Compare with the gravitational potential energy (i.e. UGPE = Uo + mgh).

Two point charges:

1 2oo

q q1U U4 r

= +pi

; where U = electric potential energy of two point charges q1 and q2 = electric charges of the two point charges r = separation distance of the two point charges Uo = reference potential energy

NOTES: 1. Commonly, Uo = 0. Meaning, U = 0 at r = . 2. U is negative if the two charges have opposite signs. 3. U is positive if the two charges have the same sign. 4. The above formula can also be used if one or both of the point charges is/are replaced

by any spherically symmetric charge distribution (in that case, r is the distance between the centers).

Physics 72 Arciaga

a b 1 2o a b

1 1 1W U q q4 r r

= =

pi

; where ra and rb = separation distance of the two point charges at positions a and b, respectively

A point charge with other point charges:

32 io 1 o 1io 12 13 o 1i

qq q1 1U U q ... U q4 r r 4 r

= + + + = +

pi pi

; where U = electric potential energy of a point charge with other point charges q1 = electric charge of the point charge

q2, q3, = electric charges of the other point charges r12, r13, = separation distance of q1 from q2, q1 from q3, Uo = reference potential energy

NOTES: 1. Commonly, Uo = 0. Meaning, U = 0 if q1 is very far away from the other charges. 2. The above formula can also be used if any of the point charges is replaced

by any spherically symmetric charge distribution (in that case, r is the distance between the centers).

3. The above formula is just scalar addition of electric potential energies. a b 1 i

io 1i,a 1i,b

1 1 1W U q q4 r r

= = pi

Interpretations of the electric potential energy 1. The work done by the electric force when a charged particle moves from position a to

position b is equal to (Ua Ub). Wby electric force = U = Uinitial Ufinal

2. The work that must be done by other external force to move the charged particle slowly from position a to position b is equal to (Ub Ua).

Wby other external force = U = Ufinal Uinitial

B. Electric potential Electric potential electric potential energy per unit charge

often called simply as potential

test

UVq

= ; where V = potential

U = electric potential energy qtest = electric charge of a test charge

NOTES: 1. Compare with electric field ( e testE F q= ).

2. Potential is a scalar. 3. SI unit of potential is volt (V) : 1 V = 1 J/C

By a uniform electric field: V = Vo + Eh ; where V = potential in a uniform electric field

E = magnitude of the uniform electric field h = position against E

Vo = reference potential (i.e. value of V at h = 0)

Physics 72 Arciaga By a point charge:

oo

1 qV V4 r

= +pi

; where V = potential by a point charge

q = electric charges of the point charge r = distance from the point charge Vo = reference potential

NOTES: 1. Commonly, Vo = 0. Meaning, V = 0 at r = . 2. The above formula can also be used if the point charge is replaced by any spherically symmetric charge distribution (in that case, r is the distance from the center).

By a collection of point charges:

32 io oio 2 3 o i

qq q1 1V V ... V4 r r 4 r

= + + + = +

pi pi

; where U = potential by several point charges q2, q3, = electric charges of the point charges

r2, r3, = separation distance from q2, from q3, Vo = reference potential

NOTES: 1. Commonly, Vo = 0. Meaning, V = 0 somewhere very far away from the point charges. 2. The above formula can also be used if any of the point charges is replaced

by any spherically symmetric charge distribution (in that case, r is the distance from the centers).

3. The above formula is just scalar addition of potentials. By a continuous distribution of charge:

o

1 dqV4 r

=

pi NOTES: 1. The integration is done over the entire distribution of charge (length, area, or volume). 2. For finite distribution of charge, you can set V = 0 at r = . 3. For infinite distribution of charge, you cannot set V = 0 at r = . What you can do is to set V = 0 somewhere else.

C. Potential difference (or Voltage) Some important relations:

Wab = U = (Ub Ua) = Ua Ub Wab = qV = q(Vb Va) = q(Va Vb) = qVab

b b

a b ea a

W F ds qE ds = =

b

ab a ba

V V V E ds= =

; where Wab = work done by the electric force in moving a charged particle from position a to position b

q = electric charge of the charged particle U = change in the electric potential energy Ua and Ub = electric potential energies at positions a and b, respectively Va and Vb = potential at positions a and b, respectively Vab = Va Vb = potential at a with respect to b (or voltage between a and b) eF

= electric force

E

= electric field

Physics 72 Arciaga NOTES: 1. The E

points toward decreasing V.

2. If E 0=

in a certain region, V is constant in that region (e.g. body of a conductor). 3. If E 0=

at a certain location, it does not necessarily mean that V = 0 at that location.

4. If V = 0 at a certain location, it does not necessarily mean that E 0=

at that location. Some common units:

Units of electric field newton per coulomb (N/C) volt per meter (V/m) : 1 V/m = 1 N/C

Units of energy joule (J) electron volt (eV) : 1 eV = 1.60210-19 J

D. Equipotential surfaces Equipotential surface 3D surface on which the potential is the same at every point (V = constant) Some notes:

1. Contour lines on a topographic map curves of constant grav. potential energy per test mass Equipotential surfaces curved surfaces of constant elec. potential energy per test charge

2. Electric field lines curved lines (arrows) to represent E ; E

is not necessarily constant in an electric field line

3. Equipotential surfaces curved surfaces to represent V; V is constant in an equipotential surface 4. Electric field line is perpendicular to equipotential surfaces. 5. Electric field points toward decreasing potential. 6. Magnitude of electric field is large in regions where equipotential surfaces are close to each

other.

E. Potential gradient Gradient operator

i j kx y z = + +

NOTES: 1.

= gradient operator (also called as grad or del operator) 2. A mathematical operation that can convert a scalar to a vector. 3. Utilizes partial differentiation.

Potential gradient

V V V

E V i j kx y z

= = + +

E

= electric field V = potential

= gradient operator

NOTES: 1. V

= gradient of V (also called as the potential gradient) 2. From the scalar V, a vector E

can be obtained.

3. V

is directed toward the rapid decrease of V. 4. If V depends only on the radial distance [i.e. V = V(r)], then

V

E V rr

= =

; where r = unit radial vector

Physics 72 Arciaga CHAPTER 24. CAPACITANCE AND DIELECTRICS

A. Capacitors Capacitor composed of two conductors separated by an insulator or vacuum can store electric potential energy and electric charge Capacitance characteristic property of a capacitor

measure of the ability of a capacitor to store energy

ab

QCV

=

; where C = capacitance of a capacitor Q = charge of the capacitor (i.e. charge on one conductor is +Q;

and charge on the other is Q) Vab = potential difference between the two conductors

NOTES: 1. SI unit of capacitance is farad (F) : 1 F = 1 C/V 2. The capacitance depends on the insulator between the two conductors. [see Section D] 3. In vacuum, the capacitance depends only on the shape, configuration, and size of the capacitor. 4. In vacuum, the capacitance does NOT depend on the charge and potential difference of the capacitor.

Parallel-plate capacitor (in vacuum): o

ACd

=

; where C = capacitance of a parallel-plate capacitor in vacuum A = area of the parallel plates d = distance separation between the two parallel plates

B. Connections of capacitors Key idea: A connection of several capacitors can be replaced by a single capacitor with a certain

equivalent capacitance (also called effective capacitance in other textbooks). Capacitors in series connection

1

eq1 2 3

1 1 1C ...C C C

= + + +

; where Ceq = equivalent capacitance of a series connection C1, C2, C3, = capacitances of the capacitors in the series connection

NOTES: 1. Ceq is less than any of C1, C2, C3, 2. Qseries = Q1 = Q2 = Q3 = [i.e. equal charges] 3. Vseries = V1 + V2 + V3 + [i.e. sum of potential differences]

Capacitors in parallel connection eq 1 2 3C C C C ...= + + +

; where Ceq = equivalent capacitance of a parallel connection C1, C2, C3, = capacitances of the capacitors in the parallel connection

NOTES: 1. Ceq is greater than any of C1, C2, C3, 2. Qparallel = Q1 + Q2 + Q3 + [i.e. sum of charges] 3. Vparallel = V1 = V2 = V3 = [i.e. equal potential differences]

Physics 72 Arciaga C. Energy stored in capacitors

Capacitors can store electric potential energy and electric charge. Two equivalent interpretations of energy storage in capacitors:

1. Energy stored is a property of the charge in the capacitor

2

21 Q 1 1U CV QV2 C 2 2

= = =

; where U = electric potential energy stored in a capacitor Q = charge of the capacitor V = potential difference across the capacitor C = capacitance of the capacitor

NOTES: 1. These assign U = 0 if the capacitor is uncharged (Q = 0). 2. Work needed to charge the capacitor:

2

ch arge1 QW U2 C

= =

2. Energy stored is a property of the electric field produced by the capacitor 2o

1u E

2=

; where u = electric energy density stored in a capacitor (in a vacuum) E = electric field in the capacitor

NOTES: 1. Electric energy density is electric potential energy per unit volume: Uuvolume

=

2. Total electric potential energy: volume

U u dv=

D. Dielectrics Dielectric a nonconducting material (i.e. insulator) usually inserted between the plates of a capacitor Characteristic properties associated with a dielectric:

1. Dielectric constant: o symbol: K o pure number; dimensionless; no units o in general, K 1 o for vacuum, K = 1 o for air (at 1 atm), K = 1.00059 1 o for Mylar, K = 3.1

2. Permittivity: o symbol: o = Ko o SI unit is C2/Nm2 or F/m o o = permittivity of free space (permittivity of vacuum) o in general, o o for air (at 1 atm), o

3. Dielectric strength: o dielectric strength maximum electric field (magnitude) that a dielectric can withstand

without the occurrence of dielectric breakdown o dielectric breakdown phenomenon at which the dielectric becomes partially ionized

and becomes a conductor

Physics 72 Arciaga Effects of inserting a dielectric in the capacitor:

1. Separates the two plates even at very small distances 2. Increases the maximum possible potential difference between the plates (because some

dielectrics have higher dielectric strength than air) 3. Increases the capacitance of the capacitor

Cw = KCwo 4. Decreases the potential difference between the plates when Q is kept constant

Vw = Vwo / K 5. Decreases the electric field when Q is kept constant (because of polarization and induced

charges in the dielectric) Ew = Ewo / K

6. Decreases the electric potential energy stored when Q is kept constant (because the electric field fringes do work on the dielectric)

Uw = Uwo / K uw = uwo / K = Ew2

; where K = dielectric strength of the inserted dielectric = permittivity of the inserted dielectric Cw, Cwo = capacitances with and without the inserted dielectric Vw, Vwo = potential differences with and without the inserted dielectric Ew, Ewo = electric fields with and without the inserted dielectric Uw, Uwo = electric potential energies with and without the inserted dielectric uw, uwo = electric energy densities with and without the inserted dielectric

Remark: In solving problems about capacitors, you must determine whether the voltage or the charge is constant. Here are some common situations:

1. capacitor is directly connected to a battery (or emf source) implies constant voltage 2. charged capacitor is isolated (i.e. not connected to anything) implies constant charge

Physics 72 Arciaga CHAPTER 25. CURRENT, RESISTANCE, AND ELECTROMOTIVE FORCE

A. Current Remarks about conductors (particularly metals):

1. In electrostatics, a. electric field is zero within the material of the conductor. b. the free electrons move randomly in all directions within the material of the conductor;

comparable with the motion of gas molecules. c. there is no net current in the material of the conductor.

2. In electrodynamics, a. electric field is nonzero within the material of the conductor. b. the free electrons move with a drift velocity in the opposite direction of the electric field

(aside from the random motion described in 1b). c. there is a net current in the material of the conductor.

Current o any motion of charge from one region to another o rate of flow of charge (i.e. charge flowing per unit time) o moving charges:

a. metals electrons b. ionized gas (plasma) electrons, positive ions, negative ions c. ionic solution electrons, positive ions, negative ions d. semiconductors electrons, holes (sites of missing electrons)

o direction of current flow = same direction as the electric field in the conductor = same direction as the flow of positive charge = opposite direction to the flow of negative charge o mathematically:

ddQI n q v Adt

= = ; where I = current flowing through an area

dQ = net charge flowing through the area dt = unit time n = concentration of the charged particles (i.e. number of particles per unit volume) q = charge of the individual particles vd = drift speed of the particles A = cross-sectional area

NOTES: 1. Current is a scalar; not a vector. 2. SI unit of current is ampere (A) : 1 A = 1 C/s 3. If there are different kinds of moving charges, the total current is the sum of the currents due to each kind of moving charge.

Current density o current per unit area o mathematically:

dJ nqv=

; where J

= current density n = concentration of charged particles q = charge of the individual particles

dv

= drift velocity of the particles

Physics 72 Arciaga NOTES: 1. Current density is a vector. 2. Magnitude: dJ I / A n q v= = 3. Direction: same direction as the electric field in the conductor (see direction of current flow described above) 4. SI unit of current density is ampere per meter squared (A/m2) 5. If there are different kinds of moving charges, the total current density is the sum of the current densities due to each kind of moving charge.

Two classifications of current: 1. Direct current direction of current is always the same (i.e. does not change) 2. Alternating current direction of current continuously changes

B. Resistivity

EJ

! = ; where ! = resistivity of a material

E = magnitude of electric field in the material J = magnitude of current density in the material

NOTES: 1. Resistivity is a scalar; not a vector. 2. Summary: a. perfect conductors: ! = 0 b. (nonperfect) conductors: low ! c. insulators: high ! d. semiconductors: ! between conductor and insulator e. superconductos: ! = 0 (at temperatures below a critical temperature Tc) 3. Conductivity reciprocal of resistivity (i.e. " = 1/! ) 4. A material with high resistivity has low conductivity.

( )o o1 T T! = ! +# $ ; where ! = resistivity of a conductor at a temperature T To = reference temperature (usually To = 20 oC or 0 oC) ! o = resistivity of the conductor at the reference temperature To # = temperature coefficient of resistivity

NOTES: 1. The above equation is an equation of a line. 2. The above equation is only an approximation valid for small temperature range (usually up to %100 oC). 3. Summary: a. most conductors (especially metals): # > 0 [i.e. ! increases if T increases] b. manganin: # = 0 [i.e. ! does not change with T] c. graphite: # < 0 [i.e. ! decreases if T increases] d. semiconductors: # < 0 [i.e. ! decreases if T increases]

C. Resistance

V LRI A

= = ! ; where R = resistance of a conductor

V = potential difference between the ends of the conductor I = current flowing through the conductor ! = resistivity of the conductor L = length of the conductor A = (cross-sectional) area of the conductor

Physics 72 Arciaga NOTES: 1. R = V/I is a definition of resistance for any conductor. 2. SI unit of resistance is ohm (): 1 = 1 V/A 3. SI unit of resistivity is ohmmeter (m): 1 m = 1 Vm/A

( )o oR R 1 T T= + ; where R = resistance of a conductor at a temperature T To = reference temperature (usually To = 20 oC or 0 oC) o = resistance of the conductor at the reference temperature To = temperature coefficient of resistance

NOTES: 1. The above equation is an equation of a line. 2. The above equation is only an approximation valid for small temperature range (usually up to 100 oC). 3. In most conductors, the temp. coeff. of resistivity is equal to the temp. coeff. of resistance (especially if the length and area do not change much with temp.).

Resistor a circuit element or device that is fabricated with a specific value of resistance between its ends

D. Ohms law Ohms law:

o At a given temperature, the current density flowing through a material is nearly directly proportional to the electric field in that material.

o Mathematically: J E

(or equivalently, I V ) NOTE: This is not actually a law because it is obeyed only by some materials (i.e. not all).

Two classifications of materials: 1. Ohmic material (or linear material)

o material that obeys Ohms law o ex: resistors, metals, conductors o at constant temperature, its and R are constant (i.e. do not depend on E or V) o its I-V curve (i.e. current vs. voltage plot) is a straight line passing through the origin

2. Nonohmic material (or nonlinear material) o material that does not obey Ohms law o ex: semiconductors, diodes, transistors o at constant temperature, its and R vary (i.e. depends on E or V) o its I-V curve (i.e. current vs. voltage plot) is not a straight line, or a straight line but

does not pass though the origin

E. Circuits Circuit a path for current Two classifications:

1. Incomplete circuit o also called open loop or open circuit o no steady current will flow through it (i.e. current eventually stops or dies)

2. Complete circuit o also called closed loop or closed circuit o a steady current will flow through it (i.e. current does not stop or die) o needs a source of emf

Physics 72 Arciaga F. Electromotive force

Electromotive force o something that can make the current flow from lower to higher potential energy o abbreviation: emf o symbol: o its not a force; its a potential (i.e. potential energy per unit charge) o SI unit of emf is volt (V)

Source of emf o any device that can provide emf (i.e. potential or voltage) o ex: battery, electric generator, solar cell, fuel cell, etc. o can transform a particular for of energy (ex: chemical, mechanical, thermal, etc.) into

electric potential energy o two classifications:

1. ideal source of emf no internal resistance provides a constant voltage across its terminals (called terminal voltage) Vab = ; where Vab = terminal voltage provided by the source of emf = emf in the source of emf

2. real (or nonideal) source of emf has an internal resistance provides a terminal voltage that depends on the current and resistance Vab = Ir ; where Vab = terminal voltage provided by the source of emf = emf in the source of emf I = current through the source of emf r = internal resistance in the source of emf

NOTES: 1. For an ideal source of emf, the terminal voltage is always equal to . 2. For a real source of emf, the terminal voltage becomes equal to only when there is no current flowing (i.e. open circuit).

Some keywords: 1. Ammeter a device that measures the current passing through it must be connected in series to a circuit element or device

ideal ammeter = has zero resistance inside (so that there is no potential difference across its terminals)

2. Voltmeter a device that measures the potential difference (or voltage) across its terminals must be connected in parallel to a circuit element or device ideal voltmeter = has infinitely large resistance inside (so that there is no current passing through it) 3. Short circuit a closed circuit in which the terminals of a source of emf are connected directly

to each other creates very large current that can damage the devices in the circuit !!!

Physics 72 Arciaga G. Energy and power in electric circuits

Recall: Power = energy per time = rate of energy change or flow P = IV ; where P = power delivered to or extracted from a circuit element or device I = current passing through the device V = voltage across the terminals (or ends) or the device

NOTES: 1. Power is delivered to a resistor. A resistor dissipates energy (transforms electric potential energy into thermal energy or heat).

P = IV = I2R = V2/R 2. Power can be extracted from a source of emf. A source of emf provides energy (transforms chemical energy, mechanical energy, fuel energy, etc. into electric potential energy).

P = IV = I( Ir) = I I2r 3. Power can be delivered to a source of emf (ex: charging of batteries).

P = IV = I( + Ir) = I + I2r

Physics 72 Arciaga CHAPTER 26. DIRECT-CURRENT CIRCUITS

A. Connections of resistors Key idea: A connection of several resistors can be replaced by a single resistor with a certain

equivalent resistance (also called effective resistance in other textbooks).

Resistors in series connection eq 1 2 3R R R R ...= + + +

; where Req = equivalent resistance of a series connection R1, R2, R3, = resistances of the resistors in the series connection

NOTES: 1. Req is greater than any of R1, R2, R3, 2. Iseries = I1 = I2 = I3 = [i.e. equal currents] 3. Vseries = V1 + V2 + V3 + [i.e. sum of potential differences]

Resistors in parallel connection

1

eq1 2 3

1 1 1R ...R R R

= + + +

; where Req = equivalent resistance of a parallel connection R1, R2, R3, = resistances of the resistors in the parallel connection

NOTES: 1. Req is less than any of R1, R2, R3, 2. Iparallel = I1 + I2 + I3 + [i.e. sum of currents] 3. Vparallel = V1 = V2 = V3 = [i.e. equal potential differences]

B. Kirchhoffs rules Keywords:

o Junction (or node) any point in a circuit where three or more conductors meet o Loop any closed conducting path in a circuit

Kirchhoffs junction rule (or Kirchhoffs current law): o The algebraic sum of the currents into any junction is zero. o Mathematically: I = 0 (at any junction)

NOTES: 1. This is a consequence of conservation of electric charge. 2. At any junction, Iin = Iout.

Kirchhoffs loop rule (or Kirchhoffs voltage law): o The algebraic sum of the potential differences in any closed loop is zero. o Mathematically: V = 0 (for any closed loop)

NOTES: 1. This is a consequence of conservation of energy. 2. Consider voltage rise and voltage fall carefully.

Problem-solving tips: 1. Usually, you first have to assume the direction of the current in each branch of the circuit. If

the calculated current in the end is positive, then the assumed direction is correct (but if the calculated current is negative, then the assumed direction is opposite to the correct one).

2. Recall that current flows from high to low potential across a resistor. 3. Using Kirchhoffs rules, setup a number of independent equations equal to the number of

unknowns. Usually, you first apply the junction rule to all the junctions; then, complete the number of equations by applying the loop rule.

Physics 72 Arciaga C. R-C circuits

Charging a capacitor: ( ) ( )t / RC t /FQ C 1 e Q 1 e = = t / RC t /o

dQI e I edt R

= = =

; where Q and I = charge on and current through the capacitor, respectively t = time R and C = resistance and capacitance, respectively = terminal voltage (of the ideal emf source) QF = final charge on the capacitor = C Io = initial current = /R = time constant (or relaxation time) = RC

NOTES: 1. Charge in the capacitor exponentially increases with time: a. t = 0: Q = 0 b. t = : Q = QF(1 1/e) = 0.63 QF c. t = : Q = QF 2. Current (magnitude) through the capacitor exponentially decreases with time: a. t = 0: I = Io b. t = : I = Io/e = 0.37 Io c. t = : I = 0 3. Recall the voltages across the resistor and capacitor: VR = IR and VC = Q/C 4. Rule of thumb

a. Transient voltage across a charging capacitor is zero if it has no initial charge like a short circuit element b. At steady-state of a fully-charged capacitor, current is zero like an open circuit element

Discharging a capacitor: t /oQ Q e = t / RC t /o o

QdQI e I edt RC

= = =

; where Qo = initial charge on the capacitor Io = initial current = Qo/RC = time constant (or relaxation time) = RC

NOTES: 1. Charge in the capacitor exponentially decreases with time: a. t = 0: Q = Qo b. t = : Q = Qo/e = 0.37 Qo c. t = : Q = 0 2. Current (magnitude) through the capacitor exponentially decreases with time: a. t = 0: I = Io b. t = : I = Io/e = 0.37 Io c. t = : I = 0 3. Recall the voltages across the resistor and capacitor: VR = IR and VC = Q/C 4. Rule of thumb

a. At steady-state of a fully-discharged capacitor, current is zero like an open circuit element

Physics 72 Arciaga CHAPTER 27. MAGNETIC FIELD AND MAGNETIC FORCES

In understanding the concepts of magnetism, I strongly suggest that you compare, contrast, or find analogies with the concepts of electricity (Chaps. 21 and 22).

Please review the cross product (vector product) that you learned from Physics 71 !!!

A. Magnetic pole Key ideas:

1. A permanent magnet has a north pole and a south pole. 2. North pole repels north pole, but attracts south pole.

South pole repels south pole, but attracts north pole. 3. No experimental evidence of a magnetic monopole. Poles always appear in pairs. 4. A bar magnet sets up a magnetic field.

The earth is a magnet: North geographic pole it is actually (near) a south magnetic pole South geographic pole it is actually (near) a north magnetic pole

B. Magnetic field Analogy:

1. Electric field ( E ) produced by electric charges that may be at rest or moving exerts an electric force ( F qE= ) on another electric charge that may be at rest or

moving 2. Magnetic field ( B )

produced by moving electric charges (i.e. current) exerts a magnetic force ( F qv B= ) on another electric charge that must be moving

Direction of magnetic field: o same direction where the north pole of the compass needle points to o for a permanent magnet, the magnetic field points out of its north pole and into its south

pole (but inside the magnet, the field points from the south to the north) NOTES: 1. SI unit of magnetic field is tesla (T): 1 T = 1 N/Am 2. Another common unit of magnetic field is gauss (G): 1 G = 10-4 T

C. Magnetic force on a moving charged particle Mathematically:

F qv B=

; where F

= magnetic force on a moving charged particle q = electric charge of the moving charged particle v

= velocity of the charged particle B

= (external) magnetic field acting on the charged particle NOTES: 1. Magnetic force is a vector.

Magnitude: F = |q|vB = |q|vB perpendicular components !!! Direction: use right-hand rule perpendicular to both v and B

2. The direction of the magnetic force depends on the sign of q and the directions of both v and B

.

3. Compare with the electric force ( F qE= ).

Physics 72 Arciaga Implications:

o The magnetic force can never do work on a charged particle. o The magnetic force can only change the direction but not the magnitude (i.e. speed) of the

velocity of a charged particle. Remark: When a situation involves both the electric force and the magnetic force, be careful on

how you use the principle of superposition. o You can add together all electric fields (vector addition) o You can add together all magnetic fields (vector addition) o You can add together electric forces and magnetic forces (vector addition) o Never add electric fields with magnetic fields !!!

D. Magnetic field lines Key ideas:

Magnetic field lines represent magnetic field in space. Direction: The magnetic field is tangent to the magnetic field line at a particular point. Magnitude: The closer (i.e. denser) the magnetic field lines are, the stronger the magnetic

field is at that region. Different magnetic field lines do not intersect.

Compare with the concept of electric field lines. Recall: The north pole of a compass needle points toward the same direction as the magnetic field

at that position.

E. Magnetic flux Similar idea as the electric flux (E). It is like a flow of magnetic field though a surface. Mathematically:

B B dA B n dA B dA B dA = = = =

NOTES: 1. Magnetic flux ( B ) is a scalar. 2. Magnetic flux is zero if B

is parallel to the surface.

3. Recall the unit normal vector ( n ) and the vector area ( A ) (Chap. 22) 4. SI unit of magnetic flux is weber (Wb): 1 Wb = 1 Tm2 5. Sometimes the magnetic field is also called magnetic flux density (i.e. flux per unit area).

F. Gausss law for magnetism Similar idea as the Gausss law for electrostatics (Chap. 22) Mathematically:

B B dA B n dA 0 = = =

NOTES: 1. Magnetic flux through any closed surface is zero !!! 2. This is because of the absence (at least experimentally) of magnetic monopole. 3. This implies that magnetic flux lines always form closed loops (but not necessarily circular loops). A magnetic field line has no end points.

Physics 72 Arciaga G. Motion of charged particles

Recall the following (from Physics 71): 1. Newtons 2nd law: F ma=

2. Circular motion: 2

Cv

ar

= ; v = r ; 2 2 fTpi

= = pi

Examples: 1. Circular motion

uniform magnetic field; velocity has perpendicular component only

cyclotron radius (or Larmor radius or gyroradius) : mvrq B

=

angular speed : q Bv

r m = =

cyclotron frequency (or Larmor frequency or gyrofrequency): q B1f2 2 m

= =

pi pi

2. Helical motion uniform magnetic field; velocity has perpendicular and parallel components

cyclotron radius (or Larmor radius or gyroradius) : mvrq B

= [v matters!]

angular speed : q Bv

r m = = [v matters!]

cyclotron frequency (or Larmor frequency or gyrofrequency): q B1f2 2 m

= =

pi pi

pitch : Pm

x v T v 2q B

= = pi [v|| matters!]

3. Mirror motion non-uniform magnetic field; magnetic mirror (or magnetic bottle) configuration

Some applications: 1. Velocity selector (or velocity filter)

purpose: to select ions moving with the prescribed velocity how: balance the electric force and the magnetic force

example: selectE

vB

=

2. Thomsons e/m experiment purpose: to determine the value of e/m how: velocity selector with speed determined from conservation of mechanical energy

example: 2

2e Em 2VB

=

3. Mass spectrometer purpose: to determine the mass (or the species) of ions assuming |q| is known how: cyclotron radius due to a uniform magnetic field

example: r q B

mv

=

Remark: Do NOT memorize the above formulas !!! Just start thinking from the fundamentals and learn to derive the above formulas.

Physics 72 Arciaga H. Magnetic force on a current-carrying conductor

Straight wire: F IL B=

; where F

= magnetic force on a current-carrying straight wire

I = current flowing through the straight wire L

= vector length (see NOTE 2 below) B

= magnetic field acting on the straight wire NOTES: 1. This comes from adding the magnetic force ( F qv B= ) acting on all the charged particles in the conductor. 2. L

let us call it the vector length:

a. direction: along the wire, same direction as the flow of current b. magnitude: equal to the length of the straight wire

Any shape: F I dL B=

NOTES: 1. This is a line integral. 2. The integration is done throughout the length of the wire (not necessarily straight).

I. Current loop Current loop a conductor that forms a loop and has a current flowing through it Magnetic dipole any object that experiences a magnetic torque most common example is a current loop analogy: electric dipole Magnetic dipole moment property of a magnetic dipole also called magnetic moment analogy: electric dipole moment

IA =

; where = magnetic dipole moment I = current flowing through the current loop (i.e. magnetic dipole) A

= vector area NOTES: 1. Magnetic dipole moment is a vector; same direction as the vector area (see Chap. 22) 2. Direction: use the right-hand rule curl fingers to the direction of the current 3. Its arrow head is the north pole; while its arrow tail is the south pole.

Torque on a current loop (in a uniform magnetic field) B =

; where = torque acting on a current loop in a uniform magnetic field

= magnetic dipole moment of the current loop B

= magnetic field acting on the current loop NOTES: 1. Compare with torque on an electric dipole ( p E = ). 2. Torque on a magnetic dipole is NOT always zero, but the magnetic force on the current loop in a uniform magnetic field is always zero. 3. Recall: If both the net force and the net torque on an object are ZERO, then that object is in EQUILIBRIUM; otherwise, that object is NOT in equilibrium.

Physics 72 Arciaga Potential energy of a current loop (in a uniform magnetic field)

U B=

; where U = potential energy of a current loop in a uniform magnetic field = magnetic dipole moment of the current loop B

= magnetic field acting on the current loop NOTES: 1. Compare with potential energy of an electric dipole ( U p E= ).

2. Recall: If the potential energy of an object is a MINIMUM, then that object is in STABLE equilibrium. But if the potential energy of an object is a MAXIMUM, then that object is in UNSTABLE equilibrium.

Remark: For multiple loops or conducting coils consisting of several plane loops that are close together (e.g. solenoid), all the magnetic force, magnetic dipole moment, torque, and potential energy increase by a factor of N (i.e. number of loops).

Physics 72 Arciaga CHAPTER 28. SOURCES OF MAGNETIC FIELD

Recall: Electric field produced by electric charges that may be at rest or moving Magnetic field produced by moving electric charges (including current)

Please practice your right-hand rules.

A. Magnetic field of a moving point charge with constant velocity

o

2qv rB

4 r

=

pi

; where B

= magnetic field of a moving point charge with constant velocity

q = electric charge of the moving point charge v

= velocity of the point charge r = distance from the point charge r = unit vector (indicates direction) o = permeability of free space (permeability of vacuum)

NOTES: 1. This expression is valid only for constant velocity (or approximately constant). 2. This is an inverse square law. 3. o = 4pi10-7 Tm/A

4. If there are more than two moving point charges, use the principle of superposition of magnetic fields. Use vector addition (not scalar addition).

B. Magnetic field of an infinitesimal current element

o

2I dL rdB

4 r

=

pi

; where B

= magnetic field of an infinitesimal current element

I = current through the current element dL

= infinitesimal vector length NOTES: 1. This is called the Biot-Savart law. 2. This is used for infinitesimal current element only. 3. To find the total magnetic field of a current element of any shape:

o 2I dL rB dB

4 r

= =

pi

The integration is done over the entire length of the current element.

C. Amperes law o encB dL I =

; where Ienc = net current enclosed by the integration path NOTES: 1. This is a line integral for a closed path. 2. The sign of the current is determined by the right-hand rule. 3. Only enclosed current matters.

4. Compare with Gausss law: enco

QE dA =

How to use Amperes law: 1. Very useful only for highly symmetrical situations. 2. Create a closed path for integration; this path is usually imaginary. 3. Assign a direction for the integration along the path. 4. Determine the net enclosed current; be careful with the proper signs. 5. Use Amperes law to determine the magnetic field.

Physics 72 Arciaga CHAPTER 29. ELECTROMAGNETIC INDUCTION

A. Electromagnetic induction Key idea: When the magnetic flux through a circuit or loop changes, then an emf and current are

induced in the circuit or loop. Keywords: electromagnetic induction, induced emf, induced current

B. Faradays law The induced emf in a closed loop equals the negative of the time rate of change of magnetic flux

through the loop.

Mathematically: Bddt

=

; where = induced emf in the circuit or loop B = magnetic flux through the circuit or loop dB/dt = rate of change of the magnetic flux

Remarks: 1. depends on the change of B only. [independent of the material of the circuit] 2. Induced current depends on (hence B) and resistance (since I = /R). [depends on the

material of the circuit] 3. Recall: B ABB A BA cos = =

It can be possibly changed by the following:

a) Changing magnitude of B b) Changing magnitude of A c) Changing angle between B and A (i.e. orientation)

4. is larger if the rate of change of B is faster. 5. The sign is related to the polarity of related to right-hand rule and Lenzs law 6. To know the polarity of , it is important to know whether B is increasing or decreasing. 7. For a coil with N identical loops or turns under the same change of B, the = N(dB/dt).

C. Lenzs law The direction of any magnetic induction effect is such as to oppose the cause of the effect. Remarks:

1. If B increases dB/dt is positive is negative If B increases must create an induced magnetic field to decrease B !!!

2. If B decreases dB/dt is negative is positive If B decreases must create an induced magnetic field to increase B !!!

3. Right-hand rule must be utilized.

D. Motional electromotive force Key idea: When a conductor (either loop or not) moves through a region of magnetic field, then

an emf can be induced on the conductor depending on the orientation of the magnetic field, conductor, and its motion.

Keyword: motional emf d ( )v B dL= ; where d = motional emf produced on the conductor

v

= velocity of the conductor B

= (external) magnetic field dL

= infinitesimal vector length of the conductor

Physics 72 Arciaga Remarks:

1. d may be zero or nonzero depending on the orientation of v, B, and dL

.

2. For a closed conducting loop (i.e. conductor is part of closed circuit): ( )v B dL= [closed line integral over the entire loop]

3. This is actually an alternate form of Faradays law for the case of moving conductors. 4. Motional emf is just a special case of induced emf for the case of moving conductors.

E. Induced electric fields Key idea: When the magnetic flux through a stationary loop changes, then an electric field is

induced in that loop. Keyword: induced electric field

Bd E dL

dt

= =

; where dB/dt = rate of change of the magnetic flux through a stationary loop E

= induced electric field produced in the loop

dL

= infinitesimal vector length along the loop = induced emf on the loop

Remarks: 1. The above expression is actually an alternate form of Faradays law for the case of changing

magnetic flux through stationary conductors. 2. The induced emf has an associated induced electric field. 3. Changing magnetic field creates an electric field !!! 4. Two classifications of electric field:

a) Electrostatic field (also called conservative electric field) - electric field produced by stationary charge distributions - conservative - causes an electric force qE

b) Nonelectrostatic field (also called nonconservative electric field) - induced electric field produced by changing magnetic flux

- nonconservative - causes an electric force qE

F. Generalized Amperes law Displacement current:

ED odI

dt

= ; where ID = displacement current through a region

E = electric flux through the region dE/dt = rate of change of E through the region

NOTES: 1. Fictitious current invented by Maxwell to correct or complete Amperes law and to satisfy Kirchhoffs junction rule. 2. No actual flow of charged particles in a displacement current. 3. For distinction, current with flow of charged particles is called conduction current. 4. Changing electric flux has an associated displacement current. 5. Example: in the region between the plates of a charging capacitor 6. Displacement current can produce a magnetic field just like the conduction current. 7. Changing electric field creates a magnetic field !!!

Physics 72 Arciaga Generalized Amperes law:

( )o C,enc D,encB dL I I = + ; where IC,enc = enclosed conduction current ID,enc = enclosed displacement current

G. Maxwells equations of electromagnetism These are not new equations. They were just summarized neatly by Maxwell to emphasize their

significance, particularly in building the idea of electromagnetic wave. Maxwells four equations for electromagnetism:

1. Gausss law for electric fields:

enc

o

QE dA =

Implications: o static charges create an electric field (i.e. conservative electric field) o electric field lines start from positive charges and end at negative charges o Coulombs law can be derived from the above expression

2. Gausss law for magnetic fields: B dA 0 =

Implications: o magnetic monopoles do not exist o magnetic field lines have no start and end (i.e. they are closed loops)

3. Amperes law with Maxwells correction:

( ) Eo C,enc D,enc o C,enc oenc

dB dL I I Idt

= + = +

Implications:

o moving charges (i.e. conducton currents) create a magnetic field o varying electric fields create a magnetic field o Biot-Savart law can be derived from the above expression

4. Faradays law:

BdE dL

dt

=

Implication:

o varying magnetic fields create an electric field (i.e. nonconservative electric field)

Amazing remark: Equations 1 and 2 look similar!!! Equations 3 and 4 look similar!!!

Physics 72 Arciaga CHAPTER 30. INDUCTANCE

Recall the concept of electromagnetic induction (Chap. 29).

A. Mutual inductance Key idea: A time-varying current in a coil (or circuit) causes an induced emf and induced current

in another coil (or circuit), depending on their mutual inductance. Keywords: mutual inductance, mutually-induced emf Mutual inductance:

B2 B12 11 2

M N NI I

= =

; where M = mutual inductance (between coils 1 and 2) N1, N2 = number of turns of coils 1 and 2, respectively B1, B2 = magnetic flux through each turn of coils 1 and 2, respectively

I1, I2 = current in coils 1 and 2, respectively NOTES: 1. Mutual inductance is scalar. 2. It is a shared property of two separated and independent coils (i.e. M = M12 = M21). 3. Depends on the geometry of the 2 coils (i.e. size, shape, number of turns, orientation, and separation) and the core material enclosed by the coils (vacuum, air, iron, etc.) 4. Independent of the current. 5. SI unit of mutual inductance is henry (H): 1 H = 1 Wb/A = 1 Vs/A = 1 s = 1 J/A2 6. High mutual inductance means that the 2 coils highly affect each other.

Mutually-induced emf:

2 1dIMdt

=

1 2dIMdt

=

; where 1, 2 = mutually-induced emf in coils 1 and 2, respectively M = mutual inductance (between coils 1 and 2)

I1, I2 = current in coils 1 and 2, respectively

B. Self-inductance and inductors Key idea: A time-varying current in a coil (or circuit) causes an induced emf and induced current

in itself, depending on its self-inductance. Keywords: self-inductance, self-induced emf Self-inductance:

BL NI

=

; where L = self-inductance (of a coil) N = number of turns of the coil B = magnetic flux through each turn of the coil I = current in the coil

NOTES: 1. Self-inductance is scalar. 2. It is a self property of a single coil. 3. Depends on the geometry of the coil (i.e. size, shape, number of turns) and the core. 4. Independent of the current. 5. SI unit of self-inductance is the same as that of mutual inductance.

Physics 72 Arciaga Self-induced emf:

dILdt

=

; where = self-induced emf in a coil L = self-inductance (of the coil)

I = current in the coil Inductor:

o a circuit device that is designed to have a particular inductance (i.e. self-inductance) o also called choke o opposes any variation of current through the circuit o voltage across an inductor depends on the time-rate of change of the current

dIV Ldt

=

; where V = voltage across an inductor (i.e. Ventry Vexit) L = inductance of the inductor dI/dt = time-rate of change of the current through the inductor

NOTES: 1. V is zero if dI/dt is zero (i.e. constant current). 2. V is either a rise or a drop depending on the sign of dI/dt (i.e. whether the current is increasing or decreasing).

C. Magnetic-field energy Inductors can store magnetic-field energy (or simply magnetic energy). Magnetic-field energy:

21U LI2

= ; where U = magnetic energy stored in an inductor

L = inductance of the inductor I = current through the inductor

NOTES: 1. Energy in the inductor is constant if the current is constant. 2. The inductor is storing energy while the current is increasing. 3. The inductor is releasing energy while the current is decreasing. 4. Compare with a capacitor that can store or release electric-field energy (U = Q2/C). 5. Compare with a resistor that always dissipates energy.

Magnetic energy density:

2

o

U 1 Bu

volume 2= =

; where u = magnetic energy density in an inductor

B = magnetic field produced by the inductor NOTES: 1. Energy stored in an inductor is proportional to the square of the magnetic field. 2. Compare with the electric energy density of a capacitor (u = oE2).

Physics 72 Arciaga D. The R-L circuit

Time constant for an R-L circuit

LR

= ; where = time constant of an R-L circuit

L = inductance of the inductor R = resistance of the resistor

High slow growth or decay of current Low fast growth or decay of current

Current growth in an RL circuit (if connected with an emf source): Exponential increase

o t = 0 I = 0 and dI/dt = /L o t = I = 0.63 Imax o t = I = Imax and dI/dt = 0

Rule of thumb o Transient current through a charging inductor is zero if it has no initial current

like an open circuit element o At steady-state of a fully-charged inductor, current is constant and voltage is zero

like a short circuit element Current decay in an RL circuit (if disconnected from an emf source):

Exponential decrease o t = 0 I = Imax and dI/dt = ImaxR/L o t = I = 0.37 Imax o t = I = 0 and dI/dt = 0

Rule of thumb o At steady-state of a fully-discharged inductor, current and voltage is zero

like a short circuit element but no current Remark: Compare the above situations with that of the R-C circuit (Chap. 26).

E. The L-C circuit Key ideas:

a. Electrical oscillation happens in an L-C circuit. i. Charge and current oscillate back and forth the oscillation is sinusoidal in time

ii. Total energy is conserved; but energy transforms from electric-field energy to magnetic-field energy back and forth the oscillation is sinusoidal-squared

iii. When charge is full, current is zero. When current is full, charge is zero. b. Analogous to a mechanical oscillation (review your Physics 71).

i. Position and velocity oscillate sinusoidally simple harmonic motion ii. Total energy is conserved; but energy transforms from potential energy to kinetic

energy back and forth iii. For more interesting analogies, see Table 30.1 of Young (but this is OPTIONAL).

Angular frequency of the electrical oscillation.

1

LC = ; where = angular frequency of an L-C electrical oscillation

L = inductance of the inductor C = capacitance of the capacitor

NOTES: 1. Recall from Physics 71: T = 2pi/ and f = /2pi. 2. High L slow oscillation [and Low L fast oscillation] 3. High C slow oscillation [and Low C fast oscillation]

Physics 72 Arciaga

F. The L-R-C circuit Key ideas:

a. Resistance is analogous to friction; they both dissipate energy. b. Damped electrical oscillation happens in an L-R-C circuit. c. Analogous to the damped mechanical oscillation (review Physics 71).

Three cases: 1. Underdamped

happens for low R (i.e. R 2 L C< ) charge and current still oscillate but they die with an exponential decay envelope increasing R causes slower oscillation but quicker death

2. Critically damped happens at moderate R (i.e. R 2 L C= ) charge and current do not oscillate they die the quickest possible way

3. Overdamped happens for high R (i.e. R 2 L C> ) charge and current do not oscillate they die slower compared to the critically damped case increasing R causes slower death

Physics 72 Arciaga CHAPTER 31. ALTERNATING CURRENT

Start thinking or visualizing sinusoidal graphs. This skill will help you a lot!

A. Alternating current Alternating current (ac) direction of the current continuously changes AC source any device that supplies a sinusoidally varying voltage or current

V = Vmax cost I = Imax cost

; where V and I = instantaneous voltage and current, respectively Vmax and Imax = voltage and current amplitudes, respectively = angular frequency t = time Example: In the Philippines, f = 60 Hz (i.e. = 377 rad/s).

Phasors o rotating vectors that can be used to represent sinusoidally carrying voltages and currents o they are only geometric tools for easier analysis of ac circuits o characteristics:

a phasor rotates counterclockwise with a constant angular speed () length of a phasor is equal to the amplitude value (Vmax or Imax) projection of a phasor onto the horizontal axis is the instantaneous value (V or I) angular displacement of the phasor is t after an elapsed time t

o key idea: When using phasor diagram for ac circuit analysis, it is just like performing vector addition but taking the x-component of the final answer.

Average values: 1. Root-mean-square value (rms value)

maxrmsVV

2=

maxrmsII

2=

; where Vrms and Irms = rms values of voltage and current, respectively Vmax and Imax = voltage and current amplitudes, respectively

Example: In the Philippines, Vrms = 110 V or 220 V (i.e. Vmax = 156 V or 311 V). 2. Rectified average value (rav)

rav max2V V=pi

rav max2I I=pi

; where Vrav and Irav = rav of voltage and current, respectively Vmax and Imax = voltage and current amplitudes, respectively

NOTES: 1. rms value is more commonly used instead of rav. 2. Some quick guides:

a. ave. value of (A sint) or (A cost) over a period is zero. b. ave. value of (A sint) or (A cost) over a quarter-period is 2A/pi. c. ave. value of (A2 sin2t) or (A2 cos2t) over a period is A2. d. ave. value of (A cost sint) over a period is zero.

Physics 72 Arciaga B. Resistance and reactance

General forms: I = Imax cos(t) V = Vmax cos(t + ) Vmax = ImaxX Vrms = IrmsX

; where = phase angle X = resistance or reactance (SI unit is ohm)

NOTE: The phase angle is the angle of the voltage phasor with respect to the current phasor. Resistor:

I = Imax cos(t) V = IR = ImaxR cos(t) Vmax = ImaxR

NOTES: 1. = 0o voltage is in phase with the current 2. XR = R resistance 3. does not affect R

Inductor: I = Imax cos(t) omax

dIV L I Lcos( t 90 )dt

= = +

Vmax = ImaxL NOTES: 1. = 90o voltage is out of phase with the current voltage leads the current by 90o 2. XL = L inductive reactance 3. XL Imax inductor hates high and loves low

Capacitor: I = Imax cos(t) omax

IQV cos( t 90 )C C

= =

maxmaxIV

C=

NOTES: 1. = 90o voltage is out of phase with the current voltage lags the current by 90o

2. C1XC

=

capacitive reactance

3. XC Imax capacitor loves high and hates low

Physics 72 Arciaga C. The L-R-C series circuit

Recall the properties of a series connection: a. IL = IR = IC = Isource instantaneous currents are equal (not amplitude) b. VL + VR + VC = Vsource instantaneous voltages are added (not amplitude)

Apply the phasor diagram analysis. Some important relations:

Vmax = ImaxZ 2 2L CZ R (X X )= + XL = L

C1XC

=

1 L CX X

tanR

=

; where Z = impedance of the ac circuit (SI unit is ohm) = phase angle

NOTES: 1. R, XL, XC, and Z are analogous. They are all measures of resistance to current flow. 2. If XL > XC, then > 0 (i.e. the voltage leads the current) 3. If XL < XC, then < 0 (i.e. the voltage lags the current) 4. If XL = XC, then = 0 (i.e. the voltage is in phase with the current) and resonance occurs ( see Section E).

D. Power in ac circuits Instantaneous power

P = IV ; where P, I, and V = instantaneous power, current, and voltage, respectively Average power

Pave = ImaxVmaxcos = IrmsVrmscos Pave = average power

Vmax and Imax = voltage and current amplitudes, respectively Vrms and Irms = rms values of voltage and current, respectively = phase angle

NOTES: 1. cos is called the power factor of the ac circuit. 2. For a pure resistor R connected to an ac source:

cos = 1

22 rms

ave max max rms rms rms

V1P I V I V = I R2 R

= = =

3. For a pure inductor L or a pure capacitor C connected to an ac source: cos = 0 Pave = 0

4. For a series L-R-C circuit connected to an ac source: cos = R/Z

2 2ave max max rms rms rms rms 2

1 R R RP I V I V = I R V2 Z Z Z

= = =

Physics 72 Arciaga E. Resonance in a series L-R-C circuit

Key ideas: 1. If a series L-R-C circuit is connected to an ac source, then there will be an electrical driven

oscillation. (Analogous to the mechanical driven oscillation in Physics 71.) 2. If a series L-R-C circuit is connected to an ac source, electrical resonance can occur if the

frequency of the source is the same as the natural frequency of the series L-R-C circuit. (Analogous to the mechanical resonance in Physics 71.)

Conditions for resonance in a series L-R-C circuit 1. XL = XC

2. source natural1LC

= =

What happens at resonance 1. Z is minimum (i.e. Z = R). 2. Imax is largest (i.e. Imax = Vmax/R).

F. Transformers Transformer device that employs the idea of electromagnetic induction to step-up or step-down

the voltage amplitudes from a primary coil to a secondary coil. Important parts of a transformer:

1. Primary coil or winding connects to an ac source 2. Secondary coil or winding connects to a circuit or device 3. Core (usually iron) ensures that almost all magnetic field lines from primary coil pass

through the secondary coil Important relations:

max,2 2

max,1 1

V NV N

=

Vmax,2Imax,2 = Vmax,1Imax,1 ; where Vmax,1 and Vmax,2 = voltage amplitudes in the primary and secondary coils,

respectively Imax,1 and Imax,2 = current amplitudes in the primary and secondary coils,

respectively N1 and N2 = number of turns in the primary and secondary coils,

respectively NOTES: 1. The first relation comes from Faradays law.

For constant flux change, turns induced emf 2. The second relation comes from conservation of energy.

For constant power, voltage current 3. Step-up transformer:

Vmax,2 > Vmax,1 N2 > N1 and Imax,2 < Imax,1 4. Step-down transformer:

Vmax,2 < Vmax,1 N2 < N1 and Imax,2 > Imax,1

Physics 72 Arciaga CHAPTER 32. ELECTROMAGNETIC WAVES

Review the key concepts about mechanical waves (Physics 71) Important: Review the Maxwells equations and its implications (Chap. 29)

A. Electromagnetic waves Wave

o transports disturbance, energy, and momentum from one region to another o speed of the wave: v = f ; where v = speed of a wave

= wavelength of the wave f = frequency of the wave

Electromagnetic wave (EM wave) o also called electromagnetic radiation o a wave that can propagate even when there is no matter (i.e. vacuum) or no medium o predicted by the four Maxwells equations o consists of time-varying electric and magnetic fields (i.e. waving electric and magnetic

fields) o produced by accelerating charges (e.g. transmitter antenna)

General characteristics of electromagnetic wave (as predicted by Maxwells equations) o speed in vacuum

o o

1c =

= 3.00108 m/s ; where c = speed of EM wave in vacuum

o = permittivity of free space or vacuum o = permeability of free space or vacuum

o speed in matter (i.e. not vacuum) or medium

1 cv

n= =

; where v = speed of EM wave in a medium

c = speed of EM wave in vacuum = permittivity of the medium = permeability of the medium n = index of refraction of the medium

Remarks: v c nothing is faster than c n 1 index of refraction is a property of matter EM waves slow down when moving in a medium commonly, for EM waves in a medium, replace o, o, and c

by , , and v, respectively o transverse wave

the electric field, magnetic field, and direction of propagation of the EM wave are all perpendicular to each other

E B

points the direction of propagation of the EM wave o definite ratio of amplitude

E = cB ; where E = magnitude of the electric field B = magnitude of the magnetic field c = speed of EM wave in vacuum

Remark: E >> B magnitude of the magnetic field is usually small

Physics 72 Arciaga B. Energy and momentum in electromagnetic waves

Energy o Poynting vector in vacuum

o

1S E B=

; where S

= Poynting vector of the EM wave in vacuum

E and B

= electric and magnetic fields, respectively Remarks:

Poynting vector points toward the direction of propagation of the EM wave S = EB/o since the E and B

are perpendicular in an EM wave

significance of Poynting vector: o energy flowing per unit time per unit area o power transfer per unit area

o 1 dUSA dt

=

o Intensity of sinusoidal EM wave in vacuum

ave max max maxo

1 1I S S E B2 2

= = =

; where I = intensity of the sinusoidal EM wave in vacuum Save = average Poynting vector Smax = maximum Poynting vector Emax = electric field amplitude Bmax = magnetic field amplitude

Remark: significance of intensity:

o average energy flowing per unit time per unit area o average power transfer per unit area

o ave

1 dUIA dt

=

Momentum o Radiation pressure of EM wave if totally absorbed

averadS Ip

c c= = ; where prad = radiation pressure by an absorbed EM wave

o Radiation pressure of EM wave if totally reflected

averad2S 2Ip

c c= = ; where prad = radiation pressure by a reflected EM wave

Remarks: significance of radiation pressure:

o average rate of momentum transfer per unit area o average force per unit area

o rad aveave

1 dp 1p FA dt A

= =

larger force is imparted by the EM wave to a surface it hits when it is reflected than when it is absorbed by the surface

Physics 72 Arciaga C. Electromagnetic spectrum

Categories Frequency (Hz) Wavelength (m) Applications Radiowave 3108 1 radio (AM, FM), TV (UHF, VHF) Microwave ~3108-1012 ~10-4-1 cellphone, oven, radar, wi-fi Infrared ~31011-1015 ~10-7-10-3 camera focusing, stove, heat sensor Visible light

Red ~405-4801012 ~625-74010-9 Orange ~480-5101012 ~590-62510-9 Yellow ~510-5301012 ~565-59010-9 Green ~530-6001012 ~500-56510-9 Blue ~600-7001012 ~430-50010-9 Violet ~700-7901012 ~380-43010-9

Ultraviolet ~31015-1017 ~10-9-10-7 high-precision apps, eye surgery X ray ~31016-1021 ~10-13-10-8 x-ray imaging, crystal structure analysis Gamma ray 31018 10-10 cancer treatment, sterilization

Remarks: o for any EM wave, v = f always o in general, EM waves with higher frequency have shorter wavelengths o v = c = 3108 m/s if EM wave moves in vacuum o some categories overlap in the spectrum o in general, EM waves with higher frequency have higher energy Physics 73 o the range of values in the table above are just approximate values

Physics 72 Arciaga CHAPTER 33. THE NATURE AND PROPAGATION OF LIGHT

Optics branch of physics that deals with the behavior of light and other EM waves Geometric optics focuses on ray analysis of light Physical optics focuses on wave behavior of light

A. The nature of light Light

o usually refers to the visible portion of the electromagnetic spectrum o has a wave-particle duality (i.e. possesses both wave-like and particle-like properties) o particle-like appropriate when discussing the emission and absorption of light

Plato, Socrates, Euclid, Newton, Einstein o wave-like appropriate when discussing the propagation of light

Huygens, Maxwell, Hertz, Young, Fresnel, Fraunhofer Some keywords to describe wave propagation

a. wave front locus of all adjacent points at which the phase of the wave is the same distance between two adjacent wave fronts is equal to the wavelength

b. spherical wave produced by a point source represented by spherical wave fronts centered at the point source c. plane wave produced by a very far point source represented by plane wave fronts d. ray an imaginary line that indicates the direction of propagation of the wave perpendicular to the wave fronts

Huygens principle o Geometrical method to determine the succeeding wave front from the preceding wave front o Every point of a wave front may be considered the source of secondary wavelets that

spread out in all directions with a speed equal to the speed of propagation of the wave.

B. Reflection and refraction Key idea: light can be partially reflected and partially transmitted (refracted) at an interface

between two media (i.e. materials) with different indexes of refraction Some keywords:

a. incident ray ray that describes the wave coming to the interface b. reflected ray ray that describes the wave reflected from the interface c. refracted ray ray that describes the (bent) wave transmitted through the interface

Types of reflection 1. specular reflection well-directed reflection from a smooth surface 2. diffused reflection scattered reflection from a rough surface

Index of refraction (or refractive index) o recall: n = c/v see Chap. 32 o dimensionless number that describes the medium o for vacuum, n =1 o for air, n = 1.0003 (i.e. n 1) o in general, n 1 o affects the speed, direction, and wavelength of the wave (but not the frequency)

high n, low speed [v = c/n] high n, short wavelength [ = vacuum/n]

Physics 72 Arciaga Law of reflection

o i = r ; where i and r = angle of incidence and angle of reflection, respectively o Remarks

angles are measured between the ray and the normal (i.e. perpendicular to surface) reflected ray is at the same angle as the incident ray also, the incident ray, reflected ray, and normal all lie in the same plane

Law of refraction o ni sini = nt sint

; where i and t = angle of incidence and angle of refraction, respectively ni = index of refraction of primary medium (incident side)

nt = index of refraction of secondary medium (refracted side) o Remarks

angles are measured between the ray and the normal (i.e. perpendicular to surface) refracted ray is bent with respect to the incident ray

a. bends toward the normal if light moves from low n to high n b. bends away from the normal if light moves from high n to low n

also, the incident ray, refracted ray, and normal all lie in the same plane

C. Total internal reflection (TIR) Key idea: light can be totally reflected at an interface between two materials even if the second

medium is transparent Critical angle

o tcriti

nsin

n = ; where crit = critical angle for TIR

o Remarks Two necessary conditions for TIR to happen

a. n of primary medium is greater than n of second medium [ni > nt] b. angle of incidence is greater than or equal to the critical angle [i crit]

If TIR happens, then there is no refracted (transmitted) ray

D. Dispersion Dispersion dependence of the index of refraction on the wavelength (in vacuum) of the wave Dispersion curve plot showing the dependence of n on vacuum [n vs. vacuum curve] Key idea: ordinary white light, which is composed of EM waves with different wavelengths, can

be separated into its different colors by dispersion (e.g. prism dispersion) Remarks

o vacuum determines n, and n determines v o If n decreases as vacuum increases, then

long are faster than short [recall: = vacuum/n] long have smaller deviation than short

o If n increases as vacuum increases, then long are slower than short [recall: = vacuum/n] long have larger deviation than short

Physics 72 Arciaga Rainbow formation

o dispersion + refraction + reflection o primary rainbow

single reflection inside the water droplet bright, but thin red has larger radius than violet

o secondary rainbow double reflection inside the water droplet thick, but faint violet has larger radius than red (i.e. reverse order of primary rainbow)

E. Polarization Tip: When thinking about polarization, it helps a lot if you will imagine about components of an

oscillating or rotating vector (in this case, the vector is the electric field) Polarization characteristic of all transverse waves for EM waves, it describes the direction of oscillation of the electric field this is different from the polarization you learned in Chap. 21 Unpolarized light (or natural light) light with no polarization (i.e. random direction) Types of polarization

1. linearly polarized electric field oscillates along a line can be composed of two perpendicular wave components with phase difference

equal to 0 or pi 2. circularly polarized

tip of electric field traces a circle [looks like a rotating helix] a. right circularly polarized clockwise rotation (as viewed opposite to

direction of propagation) b. left circularly polarized counterclockwise rotation (as viewed opposite to

direction of propagation) can be composed of two perpendicular wave components with same amplitude and

phase difference equal to pi/2 [i.e. quarter-cycle or quarter-wave difference] 3. elliptically polarized

tip of electric field traces an ellipse [looks like a rotating distorted helix] can be composed of two perpendicular wave components with same amplitude and

phase difference NOT equal to 0, pi, or pi/2 can be composed of two perpendicular wave components with different amplitudes

and phase difference NOT equal to 0 or pi Methods of polarization

1. radiowave: a straight antenna creates a linearly polarized radiowave 2. radiowave: two perpendicular straight antennas with a phase-shifting network can create

circularly or elliptically polarized radiowave 3. microwave: a grill-like array of conducting wires can transform any microwave into

linearly polarized microwave 4. light: a quarter-wave plate birefringent material can transform a linearly polarized light to

a circularly polarized light, and vice-versa birefringent a material with different indexes of refraction for different directions

of polarization (e.g. calcite) birefringence behavior of birefringent materials

Physics 72 Arciaga 5. light: a polarizing filter (or polarizer) composed of a dichroic material can transform any

light into linearly polarized light dichroic a material which absorbs a particular direction of polarization (e.g.

Polaroids in sunglasses and cameras) dichroism behavior of dichroic materials polarizing axis the orientation of the transmitted linearly polarized light

6. light: reflection can cause partial or total polarization of light key idea: the component of the electric field parallel to the interface is reflected

MORE than the non-parallel component Brewsters law for the polarizing angle

t

poli

ntan

n =

; where pol = polarizing angle ni = index of refraction of primary medium (incident side) nt = index of refraction of secondary medium (refracted side)

If incident angle is equal to the polarizing angle, then a. reflected ray is completely linearly polarized parallel to the interface b. refracted ray is partially linearly polarized non-parallel to the interface c. reflected and refracted rays are perpendicular to each other

Intensity after polarization o When an unpolarized light passes through a single polarizer, the intensity of the

transmitted linearly polarized light is halved Iline = Iunpol

o When a circularly polarized light passes through a single polarizer, the intensity of the transmitted linearly polarized light is halved

Iline = Icirc o When a linearly polarized light passes through another polarizer, the intensity of the

transmitted linearly polarized light depends on the orientation of the polarizing axis Iout = Iin cos2 ; where = angle between the directions of the incident

linearly polarized light and the polarizing axis Iin = intensity of the incident light Iout = intensity of the transmitted light

Remarks The above equation is called Maluss law. It can be used successively for a series of 2 or more polarizers. A series of 2 polarizers is usually called a polarizer-analyzer setup. If the 2 polarizing axes are aligned, then Iout = Iin. If the 2 polarizing axes are perpendicular, then Iout = 0.

F. Scattering of light Scattering when light is absorbed and then re-radiated to different directions Key idea: long wavelength is less scattered

o scattered 41I