Magnetic Field of a Wire Fundamental Laws for Calculating B-field - - PowerPoint PPT Presentation

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.

Springr2004 OSU Sources of Magnetic Field Chapter 32 2

Springr2004 OSU Sources of Magnetic Field 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

4 10

. T m A

 



 

2 I B r   

dl

 r I

Springr2004 OSU Sources of Magnetic Field Chapter 32 4

Ampere’s Law

I r dl

Springr2004 OSU Sources of Magnetic Field 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

Springr2004 OSU Sources of Magnetic Field 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) 2R 2) R2 4) not enough information

b a dl

Springr2004 OSU Sources of Magnetic Field 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

Springr2004 OSU Sources of Magnetic Field 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

Springr2004 OSU Sources of Magnetic Field 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

• f 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 enc

B dl I   



2

enc

B rd rB I     



2 I B r   

Springr2004 OSU Sources of Magnetic Field Chapter 28 10

B Field Inside a Long Wire

Now B-field inside the wire: We choose an amperean loop

• f 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

2

enc enc

B dl I B rd rB I        

 

2 2 enc

r I I R   

2

2 Ir B therefo e R r    

Springr2004 OSU Sources of Magnetic Field Chapter 32 11

B Field Inside a Long Wire

Inside the wire: (r < R)

B

r

1

• Outside the wire:( r>R )

B =



0 I

2  r

2

π 2 μ R r I B 

R

Springr2004 OSU Sources of Magnetic Field 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

• Bw

Bw Bw l d B 2      



 

• nwi

I 

therefore,

I μ l d B  



 



2

0ni

μ B 

x x x x x x x x x x x x x y

constant constant

Calculate using Ampere's law for a

Springr2004 OSU Sources of Magnetic Field 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

• n a cylinder of radius a and

length L.

dl l

B dl Bl NI    



NI B nI l    

Springr2004 OSU Sources of Magnetic Field 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.

• utside 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 NI B   2 

2 B dl B r NI     



Springr2004 OSU Sources of Magnetic Field 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

Springr2004 OSU Sources of Magnetic Field 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

Springr2004 OSU Sources of Magnetic Field 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

Springr2004 OSU Sources of Magnetic Field 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

Springr2004 OSU Sources of Magnetic Field Chapter 32 19

B Field Inside a Long Wire

Inside the wire: (r < R)

R

r

1

• Outside the wire:( r>R )

B =



0 I

2  r

2

π 2 μ R r I B 

Springr2004 OSU Sources of Magnetic Field 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

• Bw

Bw Bw l d B 2      



 

• nwi

I 

therefore,

I μ l d B  



 



2

0ni

μ B 

x x x x x x x x x x x x x y

constant constant

Calculate using Ampere's law for a