Potential at a certain location
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Potential at a Certain Location. q 2. r 2. q 1. A. r 1. 2. Travel along a path from point very far away to the location of interest and add up at each step:. q 2. q 1. dl. A. E. 1. Add up the contribution of all point charges at this point. Common Pitfall.

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Potential at a certain location

Potential at a Certain Location

q2

r2

q1

A

r1

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

q2

q1

dl

A

E

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


Common pitfall

Common Pitfall

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

A

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


Potential at a certain 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 atrestbefore 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

E = 5 N/C

0.5m

B

  • 1.5J

  • 0J

  • –1.5J

  • 3.0J

  • –3.0J


Potential at a certain location

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 restbefore 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?

E = 5 N/C

0.5m

C

B

  • 1.5J

  • 0J

  • –1.5J

  • 3.0J

  • -3.0J

A


Potential at a certain location

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 restbefore and after the move. How much work was done by the net external force while moving the test charge this time?

A

D

E = 5 N/C

0.5m

0.5m

B

C

  • 1.5J

  • 0J

  • –1.5J

  • Infinitely big

  • Do not know at this time.


Potential at a certain location

+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 restbefore and after the move. What is the voltage difference between B and C, and at which point is the voltage larger?

A

D

E = 5 N/C

0.5m

0.5m

B

C

  • 2.5 V, voltage higher at B.

  • 2.5 V, voltage higher at C.

  • 0.9 V, voltage higher at B.

  • 1.5 V, voltage higher at B.

  • 1.5 V, voltage higher at C.


Chapter 18

Chapter 18

Magnetic Field


Magnetic field

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


Electron current

Electron Current

Magnetic fields are produced by moving charges

Current in a wire: convenient source of magnetic field

Static equilibrium: net motion of electrons is zero

Can 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 field

measure heating effect

Both are proportional to the electron current


Detecting magnetic fields

Detecting Magnetic Fields

We use a magnetic compass as a detector of B.

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

Compass needle:

Interacts with iron, steel – even if they are neutral

Unaffected by aluminum, plastic etc., though charged objects polarize and interact with these materials

Points toward North pole – electric dipole does not do that


The magnetic effects of currents

The Magnetic Effects of Currents

Imagine anelectric 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 currents1

The Magnetic Effects of Currents

Hans Christian Ørsted

(1777 - 1851)

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 currents2

The Magnetic Effects of Currents

Principle of superposition:

The moving electrons in a wire create a magnetic field

What can you say about the magnitudes of BEarthand Bwire?

What if BEarth were much larger than Bwire?


Exercise

Exercise

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= 210-5 T (tesla).


Biot savart law for a single charge

Biot-Savart Law for a Single Charge

Jean-Baptiste Biot

(1774-1862)

Felix Savart

(1791-1841)

Nikola Tesla

(1856-1943)

Electric field of a point charge:

Moving charge makes a curly magnetic field:

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


Nikola tesla 1856 1943

Nikola Tesla (1856-1943)

High tension coil demonstration


The cross product

The Cross Product

Calculate magnitude:

Calculate direction:

Right-hand rule


Question

Question

A

) +x

W

h

a

t i

s

t

he

d

ir

e

ct

i

o

n

o

f

B

) –

x

C

) +y

D

) –

y

<

0

,

0

,

3

> x < 0

,

4

,

0

>

?

E

)

z

er

o m

ag

n

it

u

d

e


Two dimensional projections

Two-dimensional Projections

B

B

B

B

B

  • a vector (arrow) is facing into the screen

     a vector (arrow) is facing out of the screen

r

v

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


Exercise1

Exercise

What is B straight ahead?

What if the charge is negative?


Distance dependence

Distance Dependence

B1

B2

B3

r

v

Which is larger, B1or B3?

Which is larger, B1or B2?


Moving charge sign dependence

Moving Charge Sign Dependence

r

v

B1

B1

r

v

r

v

B

+

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.

-


Question1

Question

An 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?

y

-x

+x

-z

+z

No magnetic field

x

v

z


Exercise2

Exercise

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?


Frame of reference

Frame of Reference

Any magnetic field?

charged tape

Electric fields: produced by charges

Magnetic fields: produced by moving charges


Frame of reference1

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)


Retardation

Retardation

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


Electron current1

Electron Current

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 current2

Electron Current

mobile

electron

density

wire

Cross sectional

area

Average

drift

speed

Electron current:


Typical mobile electron drift speed

Typical Mobile Electron Drift Speed

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 speed1

Typical Mobile Electron Drift Speed

How much time would it take for a particular electron to move

through a piece of wire 30 cm long?

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


Conventional current

Conventional Current

In some materials current moving charges are positive:

Ionic solution

“Holes” in some materials (same charge as electron but +)

Observing magnetic field around copper wire:

Can we tell whether the current consists of electrons or positive ‘holes’?

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


Conventional current1

Conventional Current

André Marie Ampère

(1775 - 1836)

Metals: current consists of electrons

Semiconductors:

n-type – electrons

p-type – positive holes

Most effects are insensitive to the sign of mobile charges:

introduceconventional current:

Units: C/s  A (Ampere)


Exercise3

Exercise

A typical electron current in a circuit is 1018 electrons/s.

What is the conventional current?


The biot savart law for currents

The Biot-Savart Law for Currents

Superposition principle is valid

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


Biot savart law

Biot-Savart Law

Single Charge:

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

Current:

Moving charge produces a curly magnetic field

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


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