Amperes Law

Calculating The Magnetic Field

The main question in this topic is how can we calculate the magnetic field that a given distribution of currents produces in the surrounding space?

An array of compass needles near a central wire carrying a strong current.

. i

Calculating The Magnetic Field

The quantitative relationship between current i and the magnetic field B as

Which is known as Amperes law

Unit in Tesla = Wb/m2

o = permeability = 4x10-7 H/m = 1.26x10-6 H/m

Amperes law is a basic equation of electromagnetism that be extended by Maxwell

Calculating The Magnetic Field

From the experiment of the compass surrounding the wire flowed by current, finally, this experiment result leads to the right-hand rule for finding the direction of B near a wire carrying a current i.

Grasp the wire with the right hand, the thumb pointing in the direction of the current, the fingers will curl around the wire in the direction of B.

Sample Problem 1

Derive an expression for B at a distance r from the center of a long cylindrical wire of radius R, where r

Line of B

.

B

Line of B

Two Parallel Conductor

What is Ba around wire b?

What happen with wire b?

Can you measure Force per unit length?

d

a b

ia ib

L

Sample Problem 2

A long horizontal rigidity supported wire carries a current ia of 100A. Directly above it and parallel to it is a fine wire that carries a current ib of 20 A and weighs 0.073 N/m. How far above the lower wire should this second wire be strung if we hope to support it by magnetic repulsion? (5.5 x 10-3 m)

Sample Problem 3

Two parallel wires a distance d apart carry equal currents I in opposite directions. Find the magnetic field for points between the wires and at a distance x from one wire.

(B = Ba + Bb = ..)

Solenoid

A solenoid is a long wire wound in a close-packed helix and carrying a current i.

We assume that the helix is very long compared with its diameter

Solenoid

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

a b

c d h

Solenoid

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

a b

c d h

= B.h + 0 + 0 + 0 = 0i i = i0(nh)

B = 0i0n

Solenoid

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

B = 0i0n Flux magnetic ???

Sample Problem 4

A solenoid is 1.0 m long and 3.0 cm in inner diameter. It has five layers of winding of 850 turns each and carries a current of 5A.

What is B at its center? (2.7x10-5 Wb/m2)

What is the magnetic flux for a cross section of the solenoid of its center? (1.9x10-5 Wb)

Toroid

The Biot-Savart Law

We can use Amperes law to calculate magnetic fields only if the symmetry of the current distribution is high enough to permit the easy evaluation of the line integral

The Biot-Savart Law

According to the Biot-Savart law, dB is:

i

i

dB r

P i dl

A long straight wire, what is B at point P?

r

R P

i

x

dx

Sample Problem 6

What is B at point P? (0i/8R)

P

i

R