fundamental laws for calculating b-field biot-savart law...

Springr2004 OSU Sources of Magnetic Field

Chapter 28

1

Magnetic Field of a Wire

• Fundamental Laws for Calculating B-field

– Biot-Savart Law (long method, but works always)

– Ampere’s Law (high symmetry)

• B-Field of a Straight Wire

– For a thin straight conductor carrying current the

magnetic field is: We expect B from wire to be

proportional to I/R.


Chapter 32

2


Chapter 32

3

Magnetic Field of a Wire

• B-Field of a Straight Wire

• Where

• is the Permeability of free space

• Use the right hand rule to determine the

direction. (Place thumb in direction of

current and B Field is in direction of fingers

grabbing the wire.)

7

0 4 10 .T mA

0

2

IB

r

dl

r I


Chapter 32

4

Ampere’s Law

I

r dl


Chapter 32

5

Today

Example B-field Calculations (Application of

Ampere’s Law)

– Inside a Long Straight Wire

– Infinite Current Sheet

– Solenoid

– Toroid

– Biot-Savart Law

– Circular Loop


Chapter 32

6

Ampere’s Law

A) Evaluate the integral dl around each loop

1) 0 3) ab

2) 2(a + b) 4) Not enough info

R

dl

1) 0 3) 2R

2) R2 4) not enough information

b

a

dl


Chapter 32

7

Ampere’s Law - Examples

Two identical loops are placed in

proximity to two identical current

carrying wires.

1) A 2) B 3) Same

3) For which loop is B • dl the greatest?

A B


Chapter 32

8

Two identical loops are

placed in proximity to

two identical current

carrying wires.

1) B 2) C 3) Same

D) Now compare loops B and C. For which loop is B • dl

the greatest?

C B


Chapter 28

9

B Field Inside a Long Wire

Suppose a total current I flows

through the wire of radius a

into the screen as shown.

To calculate B field as a function

of r, from center of the wire:

Take an amperian loop of radius r outside the wire,

using Ampere’s Law:

The enclosed current is all of current through wire:

The B-field diminishes as 1/r outside the wire

r

R

r

0 encB dl I 02 encB rd rB I

0

2

IB

r


Chapter 28

10


Now B-field inside the wire:

We choose an amperean loop

of radius r inside the wire:

But the enclosed current is a fraction of total; since

current is uniform:

Inside the wire the B-field is linear with r.

r

R

r 0

02

enc

enc

B dl I

B rd rB I

2

2enc

rI I

R

0

22

IrBtherefo e

Rr


Chapter 32

11


Inside the wire: (r < R)

B

r x

b1(x);b2(x)

0 4

0

1

x =

y =

• Outside the wire:( r>R )

B = 0 I

2 r

2

0

π 2

μ

R

r I B

R


Chapter 32

12

B Field of Current Sheet

Consider an sheet of current described

by n wires/length each carrying current I I

into the screen as shown. Calculate the B field.

What is the direction of the field?

From the Symmetry +/ y direction w

square of side w

• BwBwBwldB 200

• nwiI

therefore, IμldB 0

2

0niμB

x

x

x x

x

x

x x

x

x x

x

x

y

constant constant

Calculate using Ampere's law for a


Chapter 32

13

B-Field of A Solenoid

A uniform magnetic field can

be produced by a solenoid A solenoid is defined by a current

I flowing through a wire that is

wrapped n turns per unit length

on a cylinder of radius a and

length L.

dl l

0B dl Bl NI 0

0

NIB nI

l


Chapter 32

14

B-Field of A Toroid

• Toroid defined by a solenoid of N total turns with

current I connected at both ends. It becomes a donut

shape with a coil wrapped around it.

outside Toroid B = 0

integrating B on circle outside

toroid, enclosed current = I +(- I) = 0

Inside the

Toroid

Get a concentrated field inside the toroid.

dl

r

r

NIB

2

0

02B dl B r NI


Chapter 32

15

Examples

Two cylindrical conductors, one solid and the other hallow in middle, each carry current I into the screen as shown. The conductor has radius R=4a. The conductor on the right has a hole in the middle and carries current only between R=a and R=4a.

At R = 5a which conductor produces stronger B-field?

1) Left conductor 3) Both are the same

2) Right Conductor 4) both are zero

a

4a

4a

I I


Chapter 32

16

Examples

Two cylindrical conductors, one solid and the other hallow in middle, each carry current I into the screen as shown. The conductor has radius R=4a. The conductor on the right has a hole in the middle and carries current only between R=a and R=4a.

At R = 2a which conductor produces stronger B-field?

1) Left conductor 3) Both are the same

2) Right Conductor 4) both are zero

a

4a

4a

I I


Chapter 32

17

Examples

Use Ampere’s Law in both cases by drawing a loop in the plane of the screen at R = 5a and R = 2a. Both fields have cylindrical symmetry, so the fields are tangent to the loop at all points, thus the field at R=5a only depends on current enclosed

Ienc = I in both cases

Field only depends on

enclosed current

a

4a

4a

I I


Chapter 32

18

Examples

A current carrying wire is wrapped around an iron core,

forming an electro-magnet.

Which direction does the

magnetic field point inside

the iron core?

1) left 4) right 2) up 5) down

3) out of the screen 6) into the screen

Which side of the solenoid should be labeled as the

magnetic north pole?

1) left 3) right

2) up 4) down


Chapter 32

19


Inside the wire: (r < R)

R

r x

b1(x);b2(x)

0 4

0

1

x =

y =

• Outside the wire:( r>R )

B = 0 I

2 r

2

0

π 2

μ

R

r I B


Chapter 28

20

B Field of Current Sheet

Consider an sheet of current described

by n wires/length each carrying current I I

into the screen as shown. Calculate the B field.

What is the direction of the field?

From the Symmetry +/ y direction w

square of side w

• BwBwBwldB 200

• nwiI

therefore, IμldB 0

2

0niμB

x

x

x x

x

x

x x

x

x x

x

x

y

constant constant

Calculate using Ampere's law for a

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