Sources of Magnetic Field What are some sources of Magnetic Field?

Current Elements Moving Point Charges:



B

   4  4 0  0 

q q v v r r

2   3

r r

 ˆ

(CH 27, #23)

r

  4 0   10  7 T  m / A  10  7 N / A 2

d



(Biot-Savart Law):

B

  4 0 

Id

     4 0 

Id r

 

r

2 3 

r

ˆ

r



points from source to field point



A

 

B

 

C

| 

C

|  | 

A

|| 

B

| sin  

A

, 

B

Direction from RHR I

Sources of Magnetic Field Magnets (magnetic dipole moment



):



B

N S Electric dipole

 E

Magnetic dipole moment



N S



B

Magnetic field along axis, distance

x

away from center:

B

  0

2 4 x

 3

Exercise: Compass and B-Field

Take your compass and place it flat on your table.

The compass needle will lie along the direction of the total local magnetic field.

Hold your cylindrical bar magnet so its axis is perpendicular to the compass needle and it is in the plane of the compass (see figure) .

Hold magnet far from the compass to start. Slowly bring it closer to the compass and observe the needle deflection.

Biot-Savart and RHR The Biot-Savart Law is

d



B

  4 0 

Id r

  2 

r

ˆ   4 0 

Id

 

r

3 

r



A small piece of a current carrying wire lying along the

y

-axis is indicated on the sketches.

P

3 (x 3 ,0) a) What is the direction of the magnetic field at the point P 1 ?

Id

 

●

P

1 (0,

y

1 )

( 0,y )

P

2 (x 1 ,0) b) What is the direction of the magnetic field at the point P 2 ?

c) What is the direction of the magnetic field at the point P 3 ?

More Biot-Savart A long straight current carrying wire, length 2

L

, is oriented along the

y

-axis as shown. A small piece of the wire at (0,

y

) is indicated. A field point

P

is on the x-axis at a distance

x

P .

r



What is ?



d B

  4 0 

Id

 

r

2 

r

ˆ   4 0 

Id

 

r

3 

r



I d

 

L P

(x p ,0)

Id

 

-L

( 0,y )

Example: Biot-Savart Applied to Long Straight Wire

d



B

  4 0 

Id r

  2 

Id

   (

Idy

)

j

ˆ

r

ˆ   4 0 

r

 

Id

 

r

3 

x P i

ˆ 

y j

ˆ

r

3

r



x

2

P



y

2  (

x

2

P



y

2 ) 3 / 2

Id

  

r

  (

Idy

)

j

ˆ  (

x P i

ˆ 

y j

ˆ )  (

Idy

)(

x p

)

j

ˆ 

i

ˆ  (

Idy

)( 

y

)

j

ˆ 

j

ˆ  (

Idy

)(

x p

)( 

k

ˆ )  (

Idy

)( 

y

)( 0 )   (

Ix p

)(

dy

)

k

ˆ 

d B

  4 0 

Id

 

r

2 

rˆ

  4 0 

Id

 

r

3 

r

    4 0  (

x

2

P Ix



P dy y

2 ) 3 / 2

k

ˆ 

B

  

d B

   4 0 

( Ix P ) kˆ L

 

L ( x

2

P



dy y

2

)

3

/

2 Limits of integration from sketch; smallest

y

-value for wire is –

L

, highest

y

-value for wire is

L

.

L

Id

 

-L

( 0,y )  (

b

2 

du u

2 ) 3 / 2 

b

2

u b

2 

u

2

L

 

L

(

x

2

P



x

2

P



dy y

2 ) 3 / 2

x L

2

p



L

2     

x

2

P x

2

P y x

2

p



y

2   

y



L y

 

L



L x

2

p



L

2 

x

2

P

2

L x

2

p



L

2 

B

     4 0  (

Ix P

 2 0 

x P

)

k

ˆ

IL x

2

P x

2

P



L

2 2

x

2

P k

ˆ

L



L

2

P

(x p ,0)

Magnetic Field from Straight Wire

On bisector:

B

 

B

  2 0 

R IL R

2 

L

2 “Infinite” wire: take limit as

L

 

B

  2 0 

I R

Direction: Magnetic field makes closed circles around wire.

RHR applied to straight wires: Point thumb in current direction; fingers curl in direction of magnetic field.

R

Exercise: Magnetic Field from Wires

1) Take your compass and place it flat on your desk or notebook.

2) The compass needle will lie along the direction of the total local magnetic field.

3) Take out your batteries/pack and an alligator clip lead (or several).

4) Lay the a length of wire on top of the compass so that the wire is parallel to the needle (don’t complete circuit yet.

5) Arrange so that the current in the wire will flow from south to north (north should point towards the front of the room). Don’t complete circuit yet.

6) Use RHR to PREDICT the direction the needle should deflect.

7) Check your prediction.

8) Repeat, but this time with wire beneath compass.

9) Repeat, but this time with wire perpendicular to compass needle. Try both above and below the compass.

ConcepTest #15: Two long straight wires each carry the same current and are the same distance from the origin. For the wire on the

y

-axis, the current goes into the page, and for the wire on the

x

-axis, the current comes out of the page, as shown.

Consider the following directions. Hold up as many cards as you need to specify the direction.

What is the direction of the net magnetic field at the origin?

1. Up 2. Down 4. Left 5. Into the page 3. Right 6. Out of the page (back of card) No Direction Use formula to get magnitude of B Use RHR to get direction of B DRAW vectors Add VECTORS!

ConcepTest #16: Wire 1 carries current

I

distance

d

going up as shown. Wire 2 is a away from wire 1, and carries current

I

going down as shown.

Consider the following directions. Hold up as many cards as you need to specify the direction.

1. Up 2. Down 4. Left 5. Into the page 3. Right 6. Out of the page (back of card) No Direction 1

d



d F



Id

   

B

2 a) What is the direction of the magnetic field at the location of wire 2 due to the current in wire 1?

b) What is the direction of the magnetic force acting on wire 2 due to the magnetic field from wire 1?

Example: Biot-Savart Applied to Circular Arc (at center of arc)



d B

  4 0 

Id

 

r

2 

r

ˆ

r



Id

 

Id

 

RHR applied to arcs or loops of current at center of arc or loop: Curl fingers in current direction; thumb points in magnetic field direction.

Biot-Savart Applied to Circular Arc (at center of arc) Direction: Magnetic field perpendicular to plane of loop.

B

 

dB

   0

Id

4 R 2 

d

   4 0 

IRd



R

2   4 0 

I R



f d



i

 

Rd



B

  4 0 

I

 

R

RHR applied to arcs or loops: Curl fingers in current direction, thumb points in direction of magnetic field.

Circular Loop at center of loop

B

  4 0 

I

 

R

  4 0 

I

 2 

R B

  2 0

I R

Example: Biot-Savart Applied to Circular Loops (on axis of loop) Direction: Magnetic field along axis of loop.

RHR applied to loops: Curl fingers in current direction, thumb points in direction of magnetic field.

I

 

R

2 

I



A

 

B

  4 0  (

x

2 2

I

  

R

2

R

2 ) 3 / 2

Note resemblance to field from magnetic moment!

Equivalence of Current Loops and Bar Magnets: MAGNETIC MOMENTS



B B axis

  4 0  (

x

2 2

I

  

R

2

R

2 ) 3 / 2

N S



B B axis

  0

2 4 x

 3

Magnetic Field of Special Current Configurations

Axis of Loop of radius

R

, distance

x

from center:

B

  4 0  (

x

2 2

I

  

R

2

R

) 2 3 / 2 Axis of bar magnet with moment  , distance

x

from center:

B

  0 2 4 x  3

Direction: fingers along current; thumb along magnetic moment Direction: magnetic moment from South pole to North pole

Center of Loop of radius

R

, so that

x

= 0 :

B

  4 0  2 

R I

  2 0

R I

Direction: fingers along current; thumb in field direction

Axis of long solenoid, away from edges, with

n

turns per meter:

B

  0

nI

Direction: fingers along current; thumb in field direction

Center of Arc of radius

R

, angle  :

B

  4 0 

I

 

R

  2 0

R I

  2 

Direction: fingers along current; thumb in field direction

Long (“infinite”) Straight Wire, distance

R

away from wire:

B

  4 0  2

R I

Direction: fingers along current; thumb in field direction