Ampere’s law

Ampere’s Law

Ampere’s Law

• Gauss’ law allowed us to find the net electric field due to any charge distribution (with little effort) by applying symmetry.

• Similarly the net magnetic field can be found with little effort if there is symmetry using Ampere’s law.

Ampere’s Law

• Ampere’s law,

• Where the integral is a line integral.• B.ds is integrated around a closed loop

called an Amperian loop.• The current ienc is net current enclosed by

the loop.

encisdB 0.

Ampere’s Law

• ie,

• ie ienc

N

n

nisdB1

0.

N

nni

1

Ampere’s Law

• The figure shows the cross section of 3 arbitrary long straight wires with current as shown.

1i

3i 2i

Ampere’s Law

• Two of the currents are enclosed by an Amperian loop.

1i

3i 2i

Ampere’s Law

• An arbitrary direction for the integration is chosen.

3i

1i

2i

Direction ofintegration

Ampere’s Law

• The loop is broken into elements of length ds (choose in the direction of the integration).

• Direction of B doesn’t need to be known before the integration!

3i

1i

2i

Direction ofintegration

sd

B

Ampere’s Law

• B can be in an arbitrary direction at some angle to ds as shown (from the right hand grip rule we know B must in the plane of page).

• We choose B to be in the

direction as ds.

3i

1i

2i

Direction ofintegration

sd

B

Ampere’s Law

• The right hand screw (grip) rule is used to assign a direction to the enclosed currents.

• A current passing through the loop in the same direction as the thumb are positive ( in the opposite direction -ve).

Ampere’s Law

• Consider the integral,

3i

1i

2i

Direction ofintegration

sd

B

N

n

nisdB1

0. dsB cos

Ampere’s Law

• Applying the screw rule,

3i

1i

2i

Direction ofintegration

sd

B

210cos iidsB

Ampere’s Law

• Example. Find the magnetic field outside a long straight wire with current.

r

I

Ampere’s Law

• We draw an Amperian loop and the direction of integration.

Wire surface

Amperian Loop

Direction ofIntegration

B

sd

0

Ampere’s Law

• Recall,

• Therefore,

• The equation derived earlier.

N

n

nisdB1

0.

rBdsBdsB 2cos IrB 02

r

IB

20

Ampere’s Law

• The positive sign for the current collaborates that the direction of B was correct.

Ampere’s Law

• Example. Magnetic Field inside a Long Straight wire with current.

Wire surface

Amperian Loopr

R

B

sd

Ampere’s Law

• Ampere’s Law,

N

n

nisdB1

0.

rBdsBdsB 2cos

Ampere’s Law

• Ampere’s Law,

• The charge enclosed is proportional to the area encircled by the loop,

N

n

nisdB1

0.

rBdsBdsB 2cos

iR

rienc 2

2

Ampere’s Law

• The current enclosed is positive from the right hand rule.

2

2

02R

rirB

rR

iB

20

2

Applications of Ampere’s Law

Applications of Ampere’s law

• Long-straight wire• Insider a long straight wire• Toroidal coil• Solenoid

Toroidal Coil

Toroidal Coil

r

I0

Ampere Loop, circle radius r

No current flowing through loop thus B = 0 inside the Toroid

Toroid has N loops of wire, carrying a current I0

Toroidal Coil

rI0

Ampere Loop, circle radius r

For each wire going in there is another wire comeing out Thus no nett current flowing through loop thus B = 0 outside the Toroid

Toroidal Coil

rI0

Ampere Loop, circle radius r

For each loop of the coil an extra I0 of current passes through the Ampere Loop

Zoom

B2Circle

r dsB I000NI

Toroid has N loops of wire

r

NIB

2

00

Solenoid

Infinitely Long Solenoid

Zoom looks very similar to the toroid with a very large radius

Wire carrying a current of I0 wrapped around with n coils per unit length

Toroidal Coil: Revisited

r

I0

Central radius R circumference is 2pR

Toroid has N loops of wire, carrying a current I0

Number of coils per unit length n is

R

Nn

2

0000

2nI

R

NIB

From earlier:

Independent of R

Infinitely Long Solenoid

Field at centre is same as torus of infinite radius

Wire carrying a current of I0 wrapped around with n coils per unit length

00nIB