The graded exams will be returned next Tuesday, Nov 7. You will have until the next class on Thursday, Oct 6 to rework the problems you got wrong and receive 50% added credit. Make sure you are in class as you will no have another opportunity to rework the exam. I will be going over the answers in class on Thursday. This will also be your only opportunity to ask for corrections/clarifications on any grading mistakes.

The homework assignment will be on line this afternoon but will not be due until Tuesday, Nov 14. This will give you the opportunity to start work on the problems so that you will not be overloaded with homework and the exam rework next week.

Forces and Torques on the Current Loop

' sin(90 ) cos

F IaB

F IbB IbB

Net force is zero

The Torque is not zero

2 ( / 2)sin ( )( sin )

sin

F b IBa b

IBA

Magnetic Dipole Moment

In general:

Magnetic (dipole) moment

(valid for any orientation and shape)

Potential energy of the loop

Torque wants to rotate the loop so

that its moment is oriented along

I IA

I

U

τ A B n B μ B

μ A

μ B

B

Loops and Coils

sinNIAB

For solenoid

Example: Force and Torque on a Circular Current Loop

( sin ) (cos)d l Rd i Rd j

d F Id l B

( sin ) (cos ) 0

0 0x

i j k

d l B Rd Rd

B

d r dF cos sinr R i R j

j IBA B

€

rr × d

r F =

i j k

Rcosθ Rsinθ 0

0 0 − IBxRdθ cosθ

Example: Force on a Current Loop in Non-Uniform Field

0 0 (only z and y components)B z B y

B j kL L

(1)

(2)

(3)

(4)

d F Id l B

01Component to dl is B=

B yk

L

0 01

02

L B y IB LF i I dy i

L

0 0

0 0

0

i j k

d l B dx

B z B y

L L

2F I dl B 2 0F j IB L

03 2

IB LF i

0

4 0B y

F j I dxL

Magnetic Dipole in a Non-uniform Magnetic Field

( )F B

y

; 0

0 ( )

yB B

Fy y

F along y axis

0

0 ( )

B

y

F opposite to y axis

Magnetic Dipoles and How Magnets Work

The Direct-Current Motor

Magnetic Field of a Moving Charge

02

02

sin

4

4

qvB

r

q v rB

r

Magnetic field of a point charge moving with constant velocity

700 0 2

110 T m/A;

4 c

Example: Force between two moving protonsFind the ratio of electric and magnetic forces on the protons

2

20

1

4E

qF

r

024

qvB k

r

Magnetic field of the lower proton at the position of the top one

( )BF q v B

2 20

24B

q vF j

r

22

0 0 2B

E

F vv

F c

Magnetic Field of Current Element

The Biot-Savart law.

02

02

0 1212

12

12

For element of a (fine) wire:

ˆ

4constant permeability of free space:

For the whole "circuit":

ˆ

4For arbitrary distribution of charge flow:

ˆ(1)(2)

4

ˆ( is fro

I dld

r

I dl

r

dVr

rB

rB

j rB

r m point 1 to point 2)

dflow with velocity v

dQ nqAdl

Magnetic field around a straight wire

wire) thefrom distance theis (where

2sin

4

]sin

;cot ;sin

[

sin

4

:magnitude field For the

0

0

0

2

20

a

a

Id

a

I

dadxax

ar

r

dxIB

Magnetic Field of Two Wires

Field at points on the x-axisto the right of point (3)

0 01 2

02 1 2 2

; ;2 ( ) 2 ( )

( )total

I IB B

x d x d

IdB B B

x d

Magnetic field outside of a conductor pair falls off more rapidly

Magnetic field of a circular arc

€

For the field magnitude at O :

B =μ0I

4πR2ds =∫ μ0I

4πR2Rθ

=μ0Iθ

4πR

Magnetic Field of a Circular Current Loop

30

2/3220

2/322

20

2/3220

220

2

][

)(2)(2

2)(4

cos

4)()(

:axis on the fieldFor

x

Rx

RxRx

IR

RRx

RIRx

dsIxBxB x

Falls off just as the electric field of the electric dipole

Magnetic Field on the Axis of a Coil2

02 2 3/ 2

02 2 3/ 2

2( )

;2 ( )

x

x

NIRB

x R

B NIAx R

The magnetic field of a (small) loop behaves “on the outside” like the electric field of the electric dipole of the same orientation – that’s why “magnetic dipole”.

Magnetic force between two parallel conductors with currents

02

'' 0

1 2

1 2

'0

Magnetic field from conductor 2:

2Magnetic force on conductor 1:

2Absolutely the same magnitude is

for the magnetic force on conductor 2

but

2Currents in the same

B

IB

r

IIF I LB L

r

IIF

L r

F F

direction attract

Currents in opposite directions repel

N/m 102 is

meter 1per force when the

Ampere 1 is

m 1by separated

wiresin twocurrent Identical

:Ampere 1 of Definition

7

Example: Two straight, parallel, superconducting wires 4.5 mm apart carry 15,000 A current each in opposite directions

Should we carry about the mechanical strength of the wires?

'40 10 /

2

IIFN m

L r