How do most objects interact with each other? Electric Charge (q or Q) –An intrinsic property of matter (as is mass) –S.I. unit: coulombs (C), (also: μC)

How do most objects interact with each other?

•Electric Charge (q or Q)–An intrinsic property of matter (as is mass)–S.I. unit: coulombs (C), (also: μC)–Two types (+) or (-): add algebraically to give a net charge.–Like charges repel; unlike charges attract.–Coulomb’s Law: The force between two charges (q1) and q2) separated by a distance r is: Fe=k q1 q2 / r2.– k is Coulomb’s constant: k = 9 × 109 Nm2/C2

–Force by q1 on q2 is equal and opposite to the force by q2 on q1.

More about Charge.• Quantization of charge (e): charge comes in discrete

amounts of e = 1.6×10-19C• e is the “elementary charge”; we always treat e as

positive.• Electrons have a charge of –e• Protons have a charge of +e and about 2000 times the

mass of electrons.• Neutrons have no charge and are about the same mass

as a proton.• Conservation of charge: In any reaction, net charge

remains the same. (Is this different from conservation of mass?)

Using Coulomb’s Law

1. A charge (Q) of +2µC is fixed in the ground.

A charge (q) of +3µC having a mass of 0.24 kg is held 0.15m directly above the fixed charge and released.

How does the charge q move?

Give the magnitude and direction of its

acceleration.

Use (g = 10 m/s2 and k = 9 × 109 Nm2/C2)

Using Coulomb’s Law (cont’d)

2. Two protons are separated by 5×10-15m.

a. Calculate the force on one proton due to the other. (qp= + e = 1.6×10-19 C). Does the force seem small or large as felt by:

a person? a fly? a proton?

b. Calculate the acceleration of the proton. Is this a large number? (mp= 1.67×10-27 kg)

Examples• Book sitting on a table• Pushing off

– A hard wall– A soft wall– Electric Force is a measure of hardness in repulsion.

• Sodium Metal vs. Aluminum Metal.• Potassium Metal vs. Sodium Metal.

– Electric Force is a measure of hardness in attraction.

• What happens if we hold a– Negatively charged rod held near a neutral atom?– Positively charged rod held near a neutral atom?

How can charge be transferred between objects?

1. Friction• Electrons transferred by rubbing.• Rubbing plastic with fur; fur loses electrons.• Rubbing glass with silk; silk gains electrons.

2. Induction (In Insulators)• A charged object causes electrons to move in a

neutral object.• Atoms become polarized; they turn into dipoles.• Charged rod (+) or (-) near a neutral object; what

happens?

How can charge be transferred between objects? (cont’d)

3. Conduction• Electrons can move freely through some

materials, called conductors.

• Conductors can be charged directly through contact with charged objects (not friction).

• Conductors can be charged through induction Electrons move freely to one side or another;

object as a whole becomes polarized.

How is Charge shared between Conductors?

• Electrons (- charges) move freely through conducting materials.

• Mutual repulsion forces them to spread apart as much as possible.

• Result is charge being evenly spread through touching conductors.

• If conductors are not touching, induction between them occurs.

Van de Graf Generator• Device to concentrate positive

charge on a conducting sphere.• Operation:

– Lower roller (G) rubs e- off belt.

– Belt becomes (+)

– (+) belt attracts e- from brush (B), bleeding them off the conducting sphere (A) and neutralizing belt.

– Neutralized Belt travels back to lower roller.

– Sphere (A) becomes (+) more and more.

Electrical Properties of Materials

• Conductors: Outer electrons can move nearly freely through the material. (What is it about metallic bonding that allows this?)

• Insulators: Outer electrons are bound tightly to the atoms. Cannot move freely.

Electrical Properties of Materials (cont’d)

• Semiconductors: Outer electrons are bound, but not so tightly. (e.g. silicon, germanium)

– Can be made to conduct electricity in very controlled ways.

– Doping: impurities to increase or decrease electrons in crystal structure (e.g. transistors)

– Photoelectric Effect: electrons can be energized by absorbing light ot leave their atoms and travel through the material. (e.g. photocells)

Using Coulomb’s Law

1. A charge (Q) of +2µC is fixed in the ground.

A charge (q) of +3µC having a mass of 0.24 kg is held 0.15m directly above the fixed charge and released.

How does the charge q move?

Give the magnitude and direction of its

acceleration.

Use (g = 10 m/s2 and k = 9 × 109 Nm2/C2)

Using Coulomb’s Law (cont’d)

2. Two protons are separated by 5×10-15m.

a. Calculate the force on one proton due to the other. (qp= + e = 1.6×10-19 C). Does the force seem small or large as felt by:

a person? a fly? a proton?

b. Calculate the acceleration of the proton. Is this a large number? (mp= 1.67×10-27 kg)

Using Coulomb’s Law (cont’d)

3. Three charges (q1 and q3 are fixed):

q1 = -86 µC at (52 cm, 0)

q2 = +50 µC at (0, 0)

q3 = +65 µC at (0,30 cm)

What is the net force on q2? (magnitude and direction)

What is its acceleration if it has a mass of 0.5 kg?

• How does one charge “know” that another charge is there? (“Action at a distance”).

• A charge actually changes the space around it; a force field is caused.

• Other charges interact with this “Electric Field”.• E is a VECTOR:

The direction of E at a point is in the same direction that a small positive point charge (qT) would move if placed at that point. (BY DEFINITION)

The magnitude of E at a point equals the Force on qT divided by qT. E = Fe/qT (BY DEFINITION)

Electric Field (E)

Advantages of the E-Field

• The Electric fields caused by several charges add up as vectors.

• For a system of charges or objects:– Find the resultant (net) E-Field– Now you can easily find the force on any charge (q) you

place in the system: Fe= q E– If q is (+), then Fe and E are in the same direction,

otherwise they are opposite to each other.

• If you change q, you don’t need to recalculate the E-field.

• The E-field is physical; energy is actually stored in space.

Using the E-Field

1. A charge of q = +4.0 µC feels a force of 2.0N to the right. What are the magnitude and direction of the E-Field?

2. A charge of q = -8.0µC feels a force of 2.0N to the right. What are the magnitude and direction of the E-Field?

Using the E-field

3. A charge of (q= +6µC) and mass (m=2×10-3kg) is at the origin and moving in the +x-direction with an initial speed (vi=2×102m/s). A uniform E-field (with magnitude: E=3×105N/C) acts in the negative x-direction.

a. Find the force on the charge.

b. Find the acceleration of the charge.

c. Find the maximum value of +x reached by the charge.

d. Find the time for the charge to return to its start point.

Using the E-field (cont’d)

4. A pendulum with: charge (+Q) and mass (m) is in equilibrium in a uniform E-field applied in the +x-direction. In terms of Q, m, and θ,

a. Find the magnitude of the E-field.

b. Find the tension in the string.c. Describe the motion of the ball if

the string is cut. (Be as specific as you can.)

Using the E-field (cont’d)

5. How can we find the E-field due to a point charge Q at a point (P), a distance (r) from Q?

Find magnitude of E at the point P.Show direction of E at P if Q is (+)Show direction of E at P if Q is (-)

Electric Field Lines

• Used to visualize an electric field’s – Direction: Directed

path along which a positive test charge would move.

– Magnitude: Proportional to the number of field lines coming through an area (line density).

Drawing Electric Field Lines

To draw the electric field lines for a single charge

1. Draw lines directed away from (+) charges.

2. Draw lines directed towards (-) charges.

3. The number of lines entering or leaving the charge is proportional to the amount of charge.

4. Examples to do: Using 4 lines for a (+1 C) charge, draw the field lines for:

a. +1C charge, +2C charge, -1C charge, -2C charge.

Drawing Electric Field Lines (cont’d)

To draw electric field lines for a system of charges:

1. Draw field lines near each charge in the system.

2. Connect field lines smoothly between (+) and (-) charges.

3. Field lines between like charges can go to infinity.

4. Draw field lines perpendicular (normal) to conducting surfaces (why?).

Examples

1. Dipole (+ -)

2. Two (+) charges (+ +)

3. Quadrupole +

- -

+

4. Oppositely charged plates that are close together.(*Important*)

Check answers at Electric Field Applet

How does a Conductor behave in an E-Field?

• In electrostatics, charges don’t move. (Fields due to stationary charges are called electrostatic fields.)

• Recall what happens when we put a charged rod near a conductor:– Transient current– Polarization of entire object– Electrons moved so as to cancel the E-field inside the

conductor. (Why must this be so?)

• Excess charges move to surface and distribute themselves out of mutual repulsion.

• And so ....

How does an E-field behave in a Conductor?

• No electrostatic field can exist in the material of a conductor.

• At the surface of a charged conductor: – E-field is perpendicular to the surface.– Component tangential to the surface is zero.– Higher concentrations of charge occur at surfaces

of higher curvature.

• In an empty cavity within a conductor– E-field is zero– Region is shielded from any external charges.

Charge densities at curved surfaces

On flat surfaces of low curvature, repulsive forces are directed mostly parallel to surface, keeping charges further apart.

On highly curved surfaces, parallel components of the repulsive forces are smaller, allowing charges to be closer together.

Result: Strong E-fields near highly curved surfaces. (Ex: Lightning Rods)

Electric Field Lines near a Conductor

Conductor in an E-field(summary)

Field is normal to surface Charge concentrates at high curvature.

Field is ZERO in an empty cavity.

Field is NOT zero in a cavity containing charge.

How do we describe the ability of an Electric Field to do Work?

• We want to discuss the ability of a field to do work without regard to the material being worked on.

• Approach:– Develop an intuitive gravitational analog.

– Extend the concept to the electrical case.

– Apply the concept to different combinations of positive/negative source/test charges.

What is the Gravitational Equivalent to an E-field?

• For uniform fields:Fg = mg Fe = q E

So E-field is to charge as ____________ is to mass.

Gravitational Potential Energy

b _______________

a _______________

• Mass falling from b to a loses P.E.: ΔPEg = -mgh

• More mass will lose more P.E.

• It takes work to lift mass from a to b:

Wexternal = mgh

• More mass will require more work to lift.

Gravitational Potential• Define Gravitational Potential: Vg = Peg/m

(Gravitational P.E. per unit mass).

• Gravitational Potential Difference:

Vg = ΔPeg/m = W/m = Work per unit mass.

ΔVg = gh in going from a to b.

• ΔVg is a measure of How difficult it is to lift material from a to b ORHow fast and how far material will fall from b to a.

• ΔVg describes the terrain through which we can climb or fall without regard to our mass.

Topographical Maps

• Topo map: maps contours or lines of equal gravitational potential.

• No work is done to move objects along these lines.

• Moving across lines requires work– Closely spaced lines indicates steepness.– Steepness indicates the strength of the field in the

direction of your movement.

(e.g. “field strength” here would be analogous to an an inclined plane: g sinθ; direction of fall would be down the plane.)

Topo map

Electrical Potential EnergyLet’s extend the concept to the E-field:• A (+q) charge “falls” from high (P.E.)b to low

(P.E.)a : ΔPEe = PEea - PEeb < 0 • What work is done by the field on the charge?

WbyField = F d = qE d• Change in P.E. :

ΔPEe = -qEd• Or if we “lift” (+q) from a to b, we do work:

W = Fd = qEd = ΔPEe • More charged material means

– More work done by the field in falling– More P.E. lost in falling– More external work to push the charge “uphill” against

the field.

+ - | | | | (b) (a) E

Electric Potential• Define Electric Potential: V= PEe/q

(Electric potential energy per unit charge). (Units: joule/coulomb or “volt”; J/C or V)• Potential Difference or (“Voltage”): (going from a to b)

ΔV = Vb-Va=ΔPEe/q = Wext /q • ΔV is a measure of

How fast and how far charged material will fall from b to a.How difficult it is to lift charged material from a to b.ΔPEe = q ΔV (the signs of q and ΔV are important)

• ΔV describes the electrical terrain through which we can climb or fall without regard to our charge.

• BUT: The direction of your fall depends on the sign of your charge.

Electric Potential and Charged Material

• E-field direction specifies region of high V to region of low V.

• (+) charges fall to:___________ potential_________ potential energy

• (-) charges fall to:___________ potential_________ potential energy

+ - | | | | (b) (a)High V Low V E

ΔPEe = q ΔV

Some Examples

1. An electron is accelerated across a voltage of 5000 volts between two plates.

a. What is the change in potential energy of the electron?

b. What is the work done on the electron by the E-field between the two plates?

c. What speed does the electron attain?

d. Who works with these kinds of problems?

e. What is an “electron-volt (eV)”?

The Electron Volt• The amount of work needed to push an electron

across a potential difference of 1 volt.

• Allows MUCH easier calculations of energies.

• Allows MUCH easier grading of the calculations.

• Use them when charges are given in units of “e”.

• 1 eV = (1.60×10-19C)(1 J/C) = 1.60×10-19J• Notice: instead of multiplying voltage by the electronic

charge in coulombs, you just put an “e” in front of it.• If you’re asked for speeds; you’re back to S.I. units.

Examples (cont’d)

2. Two plates are charged to a voltage 50V. If their separation is 5.0 cm, what is the E-field between them?

+ - | | | | | | | | +50 V 0V

| d = 5 cm |

The E-field is uniform between the plates.

Visualizing Electric Potential• Draw Equipotentials: Lines along which

the electric potential is constant.• Similar to contours on a topo map.• Questions

– How would you calculate the work done in moving a charge along an equipotential?

– How are equipotentials related to electric field lines?

They are perpendicular to each

other at every point.

Mapping Equipotentials

• What do equipotentials look like for– A point charge?

– A dipole?

– Two like charges?

– Applet as before: Electric Field Applet

• For a charge distribution, how could you experimentally map– Equipotentials?

– Electric field lines?

STOP HERE

• The slides after this are still at AP level.

• I’ll remove this slide when I’ve amended the later slides.