Ampere’s Law

Ampere’s LawLevel 5 Physics

January 2013

Material adapted from MIT 8.02 course notes

Ampere’s Law

Theory

Review

Biot-Savart Law

Biot-Savart Law

The magnetic field contribution, d~B,from a current source, Id~s, at a fieldpoint P a distance r in the direction ofr̂ is given by

d~B =µ04π

Id~s× r̂

r2

where µ0 is the permeability of freespace constant with value

µ0 = 4π × 10−7T ·m/A

Ampere’s Law

Theory

Review

Magnetic Field for Current-Carrying Wire

Infinite Length Wire

The magnitude of the magnetic field a distance a away from a wirewith current I is

B =µ0I

2πa

The magnetic field line direction is given by a right hand rule.

Ampere’s Law

Theory

Ampere

Circular Closed Loop

Consider a circular closed loop around an infinitely long, straight,current carrying wire as shown below.

An important law discovered by Ampere in 1826 results fromcalculating

∮~B · d~s along a closed path such as this (called an

Amperian loop).

Ampere’s Law

Theory

Ampere

Circular Closed Loop

Consider a circular closed loop around an infinitely long, straight,current carrying wire as shown below.

An important law discovered by Ampere in 1826 results fromcalculating

∮~B · d~s along a closed path such as this (called an

Amperian loop).

Ampere’s Law

Theory

Ampere

Ampere’s Law - Special Case

Calculating the line integral of∮~B · d~s around a circular loop with

radius r yields ∮~B · d~s =

∮µ0I

2πr∗ ds

=µ0I

2πr

∮ds

=µ0I

2πr∗ 2πr

= µ0I

Ampere’s Law

Theory

Ampere

Ampere’s Law - General Case

The more general form of Ampere’s Law involves integrating overan arbitrary closed loop such as the one below.

It can be shown with a proof involving cylindrical coordinates that∮~B · d~s = µ0I for this general case as well.

Ampere’s Law

Theory

Ampere

Ampere’s Law - General Case

The more general form of Ampere’s Law involves integrating overan arbitrary closed loop such as the one below.

It can be shown with a proof involving cylindrical coordinates that∮~B · d~s = µ0I for this general case as well.

Ampere’s Law

Theory

Ampere

Ampere’s Law

Ampere’s Law

A closed loop around a current carrying wire relates thesurrounding magnetic field to the current by∮

~B · d~s = µ0Ienclosed

where Ienclosed is the current encircled by the loop.

Ampere’s Law can be used to easily calculate the magnetic field incertain situations when an appropriate Amperian loop is chosen.Typically a good choice allows the magnetic field magnitude to betreated as a constant or for the integral to be evaluated insections. Biot-Savart’s Law can be used in other situations to findthe magnetic field.

Ampere’s Law

Theory

Ampere

Ampere’s Law

Ampere’s Law

A closed loop around a current carrying wire relates thesurrounding magnetic field to the current by∮

~B · d~s = µ0Ienclosed

where Ienclosed is the current encircled by the loop.

Ampere’s Law can be used to easily calculate the magnetic field incertain situations when an appropriate Amperian loop is chosen.Typically a good choice allows the magnetic field magnitude to betreated as a constant or for the integral to be evaluated insections. Biot-Savart’s Law can be used in other situations to findthe magnetic field.

Ampere’s Law

Questions

Concept Questions

Toroid Central Region

What is the best description of the magnetic field in the centralcircular region of a toroid?

1 Negative

2 Zero

3 Positive

Correct Answer: 2(There is symmetry so Ampere’s Law can be used. SinceIenclosed = 0, if an Amperian loop is drawn in the central region,B = 0.)

Ampere’s Law

Questions

Concept Questions

Toroid Central Region

What is the best description of the magnetic field in the centralcircular region of a toroid?

1 Negative

2 Zero

3 Positive

Correct Answer: 2(There is symmetry so Ampere’s Law can be used. SinceIenclosed = 0, if an Amperian loop is drawn in the central region,B = 0.)

Ampere’s Law

Questions

Concept Questions

Toroid Outer Region

What is the best description of the magnetic field in the regionsurrounding a toroid?

1 Negative

2 Zero

3 Positive

Correct Answer: 2(Ienclosed = 0 since Ienclosed is the net current encircled by the loop.)

Ampere’s Law

Questions

Concept Questions

Toroid Outer Region

What is the best description of the magnetic field in the regionsurrounding a toroid?

1 Negative

2 Zero

3 Positive

Correct Answer: 2(Ienclosed = 0 since Ienclosed is the net current encircled by the loop.)

Ampere’s Law

Questions

Concept Questions

Coaxial Cable

A cross-section of a coaxial cable is shown below. The center ofthe cable has a 1.0A total current going into the page. The outerregion has a uniform current density of 1.0A/m2 pointing out ofthe page. At what distance in meters from the center of the cableis the magnetic field equal to zero?

1 r = (1 + 34π )1/3

2 r = 12π + 1

3 r =√

1π + 1

4 The magnetic field is neverzero

Ampere’s Law

Questions

Concept Questions

Coaxial Cable, cont’d.

Correct Answer: 3(The magnetic field is zero if the enclosed current is zero,according to Ampere’s Law. Solving the equationπ(r2 − 1)(1) − 1 = 0 for r yields the answer.)

Ampere’s Law

Questions

Concept Questions

One Current Loop

Consider a small segment of a circular current loop as shownbelow. The current flows counterclockwise. What is the directionof the force on this segment?

1 Left

2 Right

3 Into page

4 Out of page

Correct Answer: 2(The rest of the loop generates a magnetic field at the segmentthat points out of the page.)

Ampere’s Law

Questions

Concept Questions

One Current Loop

Consider a small segment of a circular current loop as shownbelow. The current flows counterclockwise. What is the directionof the force on this segment?

1 Left

2 Right

3 Into page

4 Out of page

Correct Answer: 2(The rest of the loop generates a magnetic field at the segmentthat points out of the page.)