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Ampere’s Law

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Page 1: Ampere’s Law. Outline Introduce Ampere’s Law as an analogy to Gauss’ Law. Define Ampere’s Law. Show how to use Ampere’s Law for cases with symmetry

Ampere’s Law

Page 2: Ampere’s Law. Outline Introduce Ampere’s Law as an analogy to Gauss’ Law. Define Ampere’s Law. Show how to use Ampere’s Law for cases with symmetry

Outline

• Introduce Ampere’s Law as an analogy to Gauss’ Law.

• Define Ampere’s Law.

• Show how to use Ampere’s Law for cases with symmetry.

Page 3: Ampere’s Law. Outline Introduce Ampere’s Law as an analogy to Gauss’ Law. Define Ampere’s Law. Show how to use Ampere’s Law for cases with symmetry

Student Objectives

• Recognise Ampere’s Law to be analogous to Gauss’ Law.

• Recognise similar concepts: (1) draw an imaginary shape enclosing the current carrying conductor, (2) current enclosed.

Page 4: Ampere’s Law. Outline Introduce Ampere’s Law as an analogy to Gauss’ Law. Define Ampere’s Law. Show how to use Ampere’s Law for cases with symmetry

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.

Page 5: Ampere’s Law. Outline Introduce Ampere’s Law as an analogy to Gauss’ Law. Define Ampere’s Law. Show how to use Ampere’s Law for cases with symmetry

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.

Page 6: Ampere’s Law. Outline Introduce Ampere’s Law as an analogy to Gauss’ Law. Define Ampere’s Law. Show how to use Ampere’s Law for cases with symmetry

Ampere’s Law

• ie,

• ie ienc

N

n

nisdB1

0.

N

nni

1

Page 7: Ampere’s Law. Outline Introduce Ampere’s Law as an analogy to Gauss’ Law. Define Ampere’s Law. Show how to use Ampere’s Law for cases with symmetry

Ampere’s Law

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

1i

3i 2i

Page 8: Ampere’s Law. Outline Introduce Ampere’s Law as an analogy to Gauss’ Law. Define Ampere’s Law. Show how to use Ampere’s Law for cases with symmetry

Ampere’s Law

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

1i

3i 2i

Page 9: Ampere’s Law. Outline Introduce Ampere’s Law as an analogy to Gauss’ Law. Define Ampere’s Law. Show how to use Ampere’s Law for cases with symmetry

Ampere’s Law

• An arbitrary direction for the integration is chosen.

3i

1i

2i

Direction ofintegration

Page 10: Ampere’s Law. Outline Introduce Ampere’s Law as an analogy to Gauss’ Law. Define Ampere’s Law. Show how to use Ampere’s Law for cases with symmetry

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

Page 11: Ampere’s Law. Outline Introduce Ampere’s Law as an analogy to Gauss’ Law. Define Ampere’s Law. Show how to use Ampere’s Law for cases with symmetry

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

Page 12: Ampere’s Law. Outline Introduce Ampere’s Law as an analogy to Gauss’ Law. Define Ampere’s Law. Show how to use Ampere’s Law for cases with symmetry

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).

Page 13: Ampere’s Law. Outline Introduce Ampere’s Law as an analogy to Gauss’ Law. Define Ampere’s Law. Show how to use Ampere’s Law for cases with symmetry

Ampere’s Law

• Consider the integral,

3i

1i

2i

Direction ofintegration

sd

B

N

n

nisdB1

0. dsB cos

Page 14: Ampere’s Law. Outline Introduce Ampere’s Law as an analogy to Gauss’ Law. Define Ampere’s Law. Show how to use Ampere’s Law for cases with symmetry

Ampere’s Law

• Applying the screw rule,

3i

1i

2i

Direction ofintegration

sd

B

210cos iidsB

Page 15: Ampere’s Law. Outline Introduce Ampere’s Law as an analogy to Gauss’ Law. Define Ampere’s Law. Show how to use Ampere’s Law for cases with symmetry

Ampere’s Law

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

r

I

Page 16: Ampere’s Law. Outline Introduce Ampere’s Law as an analogy to Gauss’ Law. Define Ampere’s Law. Show how to use Ampere’s Law for cases with symmetry

Ampere’s Law

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

Wire surface

Amperian Loop

Direction ofIntegration

B

sd

0

Page 17: Ampere’s Law. Outline Introduce Ampere’s Law as an analogy to Gauss’ Law. Define Ampere’s Law. Show how to use Ampere’s Law for cases with symmetry

Ampere’s Law

• Recall,

• Therefore,

• The equation derived earlier.

N

n

nisdB1

0.

rBdsBdsB 2cos IrB 02

r

IB

2

0

Page 18: Ampere’s Law. Outline Introduce Ampere’s Law as an analogy to Gauss’ Law. Define Ampere’s Law. Show how to use Ampere’s Law for cases with symmetry

Ampere’s Law

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

Page 19: Ampere’s Law. Outline Introduce Ampere’s Law as an analogy to Gauss’ Law. Define Ampere’s Law. Show how to use Ampere’s Law for cases with symmetry

Ampere’s Law

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

Wire surface

Amperian Loopr

R

B

sd

Page 20: Ampere’s Law. Outline Introduce Ampere’s Law as an analogy to Gauss’ Law. Define Ampere’s Law. Show how to use Ampere’s Law for cases with symmetry

Ampere’s Law

• Ampere’s Law,

N

n

nisdB1

0.

rBdsBdsB 2cos

Page 21: Ampere’s Law. Outline Introduce Ampere’s Law as an analogy to Gauss’ Law. Define Ampere’s Law. Show how to use Ampere’s Law for cases with symmetry

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

Page 22: Ampere’s Law. Outline Introduce Ampere’s Law as an analogy to Gauss’ Law. Define Ampere’s Law. Show how to use Ampere’s Law for cases with symmetry

Ampere’s Law

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

2

2

02R

rirB

rR

iB

20

2


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