Biot-Savart Law, Ampère’s Law

• Fields and forces for current in straight wires

• Ampère’s Law

Example 1: Long parallel Example 1: Long parallel wireswires

Find force on current I1 in segment ab due to current I2 (magnitude & direction)

L

I1

d

(Take I1 = 20A, I2=30A, d=0.01m, L=1.5m)

a b

I2

Solution

QuizQuiz

I1

I2

The two long wires shown cross at right angles. The horizontal wire 1 is fixed.

How will the red wire move when the currents are turned on?

A) LeftB) RightC) Rotate clockwiseD) Rotate counter-clockwiseE) It won’t move

For any closed path:

Ampère’s Law (an integral theorem)

The current flows through the surface bordered by the path.

direction of integration determinespositive currentdirection

Positive current is out of the page through the shaded surface (right hand rule)

encircledneto

path

IsdB ,

Calculating B from Ampere’s Law

For a few simple shapes, we can calculate the field B easily from Ampere’s Law. The steps:

i) use symmetry to choose an appropriate pathii) determine the direction of B from RHR.iii) calculate (in terms of |B|)

iv) calculate Iencircled

v) apply Ampere’s Law to get B

dsB

Ampere’s Law holds only for steady currents

C 1

C 2

C 3

I

B

Question: Long Straight WireQuestion: Long Straight Wire

On which of the curves

C1, C2, and C3 is

i) smallest

ii) largest

iii) zero

path

sdB

Example 2: Long Straight WireExample 2: Long Straight Wire

r

IB

2

o

Show that the field produced by a current I in a long straight wire at distance r from the wire is:

Steps:

a) Symmetry: choose a circle for the path,

depends only on .B r

b) Calculate along this circle of radius B ds r::::::::::::::::::::::::::::

net, encircledc) Calculate I through the circle

BIdsB

for solve d) o ,

B

ds

path of integration

field line

I

Solution

Example 3:Example 3:

Wire, radius R, uniform current density

x x xx x x x xx x x x x

xx x x x x

xx x x x x

x x x

R

r

Total

2

IJ

R

Find for i)

ii)

B r R

r R

J

Solution

Example 4

What is the magnetic field inside and outside:

A hollow copper pipe carrying current I?

I

Example 5

What is the magnetic field inside and outside:

A coaxial cable carrying equal currents (in opposite directions) on the inner and outer conductors?

II

Solution

Infinitely Long “Solenoid” (long straight coil/helix)

Current I flows in each turn of the coil. The number of turns ofwire per meter of solenoid is called the turn density n.

I

B inside the solenoid isuniform and has magnitude

and is uniform.

What is the direction of B?

0B nI

Field patterns for circular coilsField patterns for circular coils

Single Coil:

IB

Several Coils:

xx x x x