magnetic interaction

++

v

B

F

O

There is interaction between a particle and other bodies which depends on the charge of the particle, its position and its velocity (and its spin). We call this interaction a magnetic interaction. Moving charged particles in the body cause the magnetic interaction.

The region of space in which magnetic forces can be exerted is called a magnetic field.

magnetic field vector

The magnetic field vector at position is a vector such that, at this position, the magnetic force exerted on a particle with charge q, moving with velocity , would be

B

r

v

BvFB

q

B

F ++ v

the cyclotron frequency

++

v

B

F

O

am

BF

Bv

mq vB

mq

Recall that in a uniform circular motion:

v =aIn a uniform magnetic field, the particle moves with constant angular velocity along the magnetic field

qmB

work due to magnetic interaction

BdWv

B

F

ds

sdF B dtB vF

0

The magnetic work performed on the particle is zero.

(A magnetic field cannot change the speed of a particle.)

+

magnetic force "on a current"

I

IdsB

dF

Fd

Bs

ˆqvd BsdAqnvd Bsd

I

The (differential) force , exerted on a (differential) segment, depends on the current I in the wire, the size ds and orientation of the segment, and the magnetic field vector at the location of the segment:

BsdFd

I

Fd

sB

nAds

magnetic moment

B

The magnetic moment of an object is a vector such that the magnetic torque exerted on the object (about its center of the mass) placed in the magnetic field is

B

Puzzle. What is the direction of the magnetic moment of a compass needle?

BB

BB

N

S

magnetic moment of a current loop

nB

1F

2F

4321 00BnBn

ˆI2

ˆI2

baba BA

I

a

b

The magnetic moment of a wire loop carrying current depends on the current I in the loop and the area A of the loop.

IA

N

S

NS

potential energy

The potential energy of an object in a magnetic field depends on the magnetic moment of the object and the magnetic field at its location

U B

The Lorentz force

When the particle moves in the presence of both a magnetic field and an electric field, the net force depends on both fields:

F E v B q q

Example. The Hall effect

FB

FE

vdd I

+ ++ ++ ++

_ __ __ __

Vqv B qEd H

dEV HH Bdvd

BInqA

dVH +

Maxwell's equations

. . . and God said:

Let . . .

0Surface

Gaussian

Q

AdE

0

SurfaceGaussian

AdB

B

AmperianLoop

dsdt

E d E

0 0Amperian

Loop

ds Idt

B d

. . . and there was light.

Gauss‘s law for magnetic fields

The net electric flux through any closed (Gaussian) surface is proportional to the net charge inside the surface:

0

in

S

q AdE

The net magnetic flux through any closed (Gaussian) surface is equal to zero:

0

SurfaceGausian

AdB NN

Faraday's law of induction

N

The line integral of the electric field vector around any closed path equals the rate of change in the magnetic flux through any surface bounded by that path.

dtdd B

loopclosed

sE

BE

Ampere-Maxwell law

The circulation of the magnetic field vector around any amperian loop proportional the sum of the total conduction current and the displacement current through any surface bounded by that path.

dt

dId E00

loopclosed

sB

The proportionality coefficient is called the permeability of free space.

I

E

E

B

displacement current

The rate of change in the electric field multiplied by the permittivity of free space is called the displacement current

I ddtd

E0

I

I

Q

-Q

Edt

dI E0d

Example:

dt

Qd0

0

dtdQ I

+

Example: Magnetic field of a long straight wire with current

loop

sdB

vB

F

I

ds

R2

0Bds

R2

0dsB BR2 I0

B IR

0

2R

the Biot-Savart law

I

Ids

dB

rP

The (differential) magnetic field at a certain position P produced by a differential element carrying electric current I depends on the value of the current and the size and orientation of the segment.

dB ds r

0

24I

r

Example: infinite straight wire with current

dB

I

R

Ids

s r

P

ds

sR

RsR

14

I2222

0

2

0y r

ˆI4

dB rsd

2

0

rdssinI

4

2322

0

sR

ds4

IR

23

22

0y

sR

dsIR4

BB-

222

0

sRR

sIR4 R2

I0

x

y

z

Interaction between two parallel current

I1 I2

F21

B1

l

21a

21F

12I B

l

aII2 ˆ2 1

0

al

aIIIaI2 ˆˆ

2 2110

al

aII ˆ2 21

0

al

Parallel "currents" attract and antiparallel repel.

The magnitude of the magnetic force exerted on segment l of a wire by the other wire (infinite) is

F I I210

1 22

la

magnetic field of a solenoid

IN

I

S

L

The magnetic field outside the solenoid is zero

Bout 0

The magnetic field inside is uniform, its direction is parallel to the axis, and the magnitude depends on the current and the number of loops per length of the solenoid

Bin 0nI

Magnetic properties of matter

When a substance is placed in a (external) magnetic field, its molecules acquire a magnetic moment related to the external field. This creates an additional magnetic field (internal).

B0

Bm

0m0 BBBB

1

paramagnetics: > 1

diamagnetics: < 1