Potential at a Certain Location

1. Add up the contribution of all point charges at this point

q1 r1

q2

r2

A

2. Travel along a path from point very far away to the location of interest and add up at each step:

q1

q2

AE

dl

Potential at a Certain Location

A

Example: E = 0 inside a charged metal sphere, but V is not!

Common PitfallAssume that the potential V at a location is defined by the electric field at this location.

A negative test charge Q = -0.6C was moved from point A to point B In a uniform electric field E=5N/C. The test charge is at rest before and after the move. The distance between A and B is 0.5m and the line connecting A and B is perpendicular to the electric field. How much work was done by the net external force while moving the test charge from A to B?

A

B

0.5mE = 5 N/C

A. 1.5JB. 0JC. –1.5JD. 3.0JE. –3.0J

After moving the -0.6C test charge from A to B, it was then moved from B to C along the electric field line. The test charge is at rest before and after the move. The distance between B and C also is 0.5m. How much work was done by the net external force while moving the test charge from A to C?

A. 1.5JB. 0JC. –1.5JD. 3.0JE. -3.0J

A

CB

0.5mE = 5 N/C

Instead of moving the test charge from A to B then to C, it is moved from A to D and then back to C. The test charge is at rest before and after the move. How much work was done by the net external force while moving the test charge this time?

A. 1.5JB. 0JC. –1.5JD. Infinitely bigE. Do not know at this time.

E = 5 N/C

A

B

0.5m

C0.5m

D

+1.5 J of work was done by the net external force while moving the -0.6 C test charge from B to C. The test charge is at rest before and after the move. What is the voltage difference between B and C, and at which point is the voltage larger?

A. 2.5 V, voltage higher at B.B. 2.5 V, voltage higher at C.C. 0.9 V, voltage higher at B.D. 1.5 V, voltage higher at B.E. 1.5 V, voltage higher at C.

A

E = 5 N/C

B

0.5m

C0.5m

D

In a multiparticle system we can either consider a change in potential energy or a change in field energy (but not both); the quantities are equal.

The idea of energy stored in fields is a general one:Magnetic and gravitational fields can also carry energy.

The concept of energy stored in the field is very useful:

- electromagnetic waves

Potential Energy and Field Energy

e+ e-

SystemSurroundings

Release electron and positron – the electron (system) will gain kinetic energy

Conservation of energy surrounding energy must decrease

Does the energy of the positron decrease? - No, it increases

Where is the decrease of the energy in the surroundings?

- Energy stored in the fields must decrease

An Electron and a Positron

e+ e-

SystemSurroundings

Single charge:

Dipole:

(far)

Energy:

Energy stored in the E fields decreases as e+ and e- get closer!

An Electron and a Positron

e+ e-

SystemSurroundings

(Field energy) + Kpositron + Kelectron = 0

(Field energy) = -2(Kelectron )

Principle of conservation of energy:

Alternative way: e+ and e- are both in the system:

Uel = -2(Kelectron )

Change in potential energy for the two-particle system is the same as the change in the field energy

An Electron and a Positron

Chapter 18

Magnetic Field

A compass needle turns and points in a particular direction

there is something which interacts with it

Magnetic field (B): whatever it is that is detected by a compass

Compass: similar to electric dipole

Magnetic Field

Magnetic fields are produced by moving chargesCurrent in a wire: convenient source of magnetic fieldStatic equilibrium: net motion of electrons is zeroCan make electric circuit with continuous motion of electrons

The electron current (i) is the number of electrons per second that enter a section of a conductor.

Counting electrons: complicated

Indirect methods:measure magnetic fieldmeasure heating effect

Electron Current

Both are proportional to the electron current

If 1.81016 electrons enter a light bulb in 3 ms – what is the magnitude of electron current at that point in the circuit?

Exercise

We will use a magnetic compass as a detector of B.

How can we be sure that it does not simply respond to electric fields?

Interacts with iron, steel – even if they are neutral

Unaffected by aluminum, plastic etc., though charged tapes interact with these materials

Points toward North pole – electric dipole does not do that

Detecting Magnetic Fields

Compass needle:

Make electric circuit:

What is the effect on the compass needle?

What if we switch polarity?What if we run wire under compass?

What if we change the current or there is no current in the wire?

The Magnetic Effects of Currents

Experimental results:• The magnitude of B depends on the amount of current• A wire with no current produces no B• B is perpendicular to the direction of current• B under the wire is opposite to B over the wire

Oersted effect:discovered in 1820 by H. Ch. Ørsted

How does the field around a wire look?

The Magnetic Effects of Currents

Hans Christian Ørsted (1777 - 1851)

Magnetic Field Due to Long Current-Carrying Wire

Principle of superposition:

What can you say about the magnitudes of BEarth and Bwire?

What if BEarth were much larger than Bwire?

The Magnetic Effects of Currents

The moving electrons in a wire create a magnetic field

A current-carrying wire is oriented N-S and laid on top of a compass. The compass needle points 27o west. What is the magnitude and direction of the magnetic field created by the wire Bwire if the magnetic field of Earth is BEarth= 210-5 T (tesla).

Exercise

Biot-Savart Law for a Single Charge

Electric field of a point charge:

Moving charge makes a curly magnetic field:

B units: T (tesla) = kg s-2A-1

Jean-Baptiste Biot(1774-1862)

Felix Savart(1791-1841) Nikola Tesla

(1856-1943)

Calculate magnitude:

Right-hand rule

The Cross Product

Calculate direction:

Question

What is the direction of

< 0, 0, 3> x < 0, 4, 0>?

A) +xB) –xC) +yD) –yE) zero magnitude

|�̂� �̂� �̂�0 0 30 4 0|¿ (0 −12) �̂�

𝑥

𝑦h𝑟𝑖𝑔 𝑡 h𝑎𝑛𝑑𝑟𝑢𝑙𝑒

Ä a vector (arrow) is facing into the screen a vector (arrow) is facing out of the screen

v

rB BB

B

B

Two-dimensional Projections

Why must the field change direction above and below the dashed line?

What is B straight ahead?

What if the charge is negative?

Exercise

v

rB1

B2

B3

Which is larger, B1 or B3 ?

Which is larger, B1 or B2 ?

Distance Dependence

v

rB1

v

r B

v

rB1

Magnetic field depends on qv: Positive and negative charges produce the same B if moving in opposite directions at the same speed

For the purpose of predicting B we can describe current flow in terms of ‘conventional current’ – positive moving charges.

Moving Charge Sign Dependence

+

-

-

QuestionAn electron passing through the origin is traveling at a constant velocity in the negative y direction. What is the direction of the magnetic field at a point on the positive z axis?

A) -xB) +xC) -zD) +zE) No magnetic field

x

z

y

v

A current-carrying wire lies on top of a compass. What is the direction of the electron current in this wire?

What would the direction of conventional current have to be?

Exercise

𝑒−𝑒−𝑒−𝑒−

�⃗�

Electric fields: produced by charges

Magnetic fields: produced by moving charges

charged tape

Any magnetic field?

Frame of Reference

Must use the velocities of the charges as you observe them in your reference frame!

There is a deep connection between electric field and magnetic fields (Einstein’s special theory of relativity)

Frame of Reference

If we suddenly change the current in a wire: Magnetic field will not change instantaneously.

Electron and positron collide: Produce both electric and magnetic field, these fields exist even after annihilation.Changes propagate at speed of light

There is no time in Biot-Savart law:Speed of moving charges must be small

Retardation

A steady flow of charges in one direction will create a magnetic field. How can we cause charges to flow steadily?

Need to find a way to produce and sustain E in a wire.

Use battery

Electron Current

Electron current:

mobile electrondensity

wireCross sectionalarea

Averagedrift speed

Electron Current

Typical electron current in a circuit is ~ 1018 electrons/s.What is the drift speed of an electron in a 1 mm thick copper wire of circular cross section?

Typical Mobile Electron Drift Speed

How much time would it take for a particular electron to movethrough a piece of wire 30 cm long?

How can a lamp light up as soon as you turn it on?

Typical Mobile Electron Drift Speed

Observing magnetic field around copper wire: Can we tell whether the current consists of electrons or positive ‘holes’?

In some materials current moving charges are positive:Ionic solution“Holes” in some materials (same charge as electron but +)

Conventional Current

The prediction of the Biot-Savart law is exactly the same in either case.

Metals: current consists of electronsSemiconductors:

n-type – electronsp-type – positive holes

Most effects are insensitive to the sign of mobile charges:introduce conventional current:

Conventional Current

Units: C/s A (Ampere)

André Marie Ampère(1775 - 1836)

A typical electron current in a circuit is 1018 electrons/s.What is the conventional current?

Exercise

Superposition principle is valid

The Biot-Savart law for a short length of thin wire

The Biot-Savart Law for Currents

Moving charge produces a curly magnetic field

B units: T (Tesla) = kg s-2A-1

Single Charge:

Biot-Savart Law

The Biot-Savart law for a short length of thin wire

Current: