Ampere's Law - Level 5 Physics - IIS Cremona€™s Law Level 5 Physics January 2013 Material...

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Ampere’s Law Ampere’s Law Level 5 Physics January 2013 Material adapted from MIT 8.02 course notes

Transcript of Ampere's Law - Level 5 Physics - IIS Cremona€™s Law Level 5 Physics January 2013 Material...

Page 1: Ampere's Law - Level 5 Physics - IIS Cremona€™s Law Level 5 Physics January 2013 Material adapted from MIT 8.02 course notes Ampere’s Law Theory Review Biot-Savart Law Biot-Savart

Ampere’s Law

Ampere’s LawLevel 5 Physics

January 2013

Material adapted from MIT 8.02 course notes

Page 2: Ampere's Law - Level 5 Physics - IIS Cremona€™s Law Level 5 Physics January 2013 Material adapted from MIT 8.02 course notes Ampere’s Law Theory Review Biot-Savart Law Biot-Savart

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

Page 3: Ampere's Law - Level 5 Physics - IIS Cremona€™s Law Level 5 Physics January 2013 Material adapted from MIT 8.02 course notes Ampere’s Law Theory Review Biot-Savart Law Biot-Savart

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.

Page 4: Ampere's Law - Level 5 Physics - IIS Cremona€™s Law Level 5 Physics January 2013 Material adapted from MIT 8.02 course notes Ampere’s Law Theory Review Biot-Savart Law Biot-Savart

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).

Page 5: Ampere's Law - Level 5 Physics - IIS Cremona€™s Law Level 5 Physics January 2013 Material adapted from MIT 8.02 course notes Ampere’s Law Theory Review Biot-Savart Law Biot-Savart

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).

Page 6: Ampere's Law - Level 5 Physics - IIS Cremona€™s Law Level 5 Physics January 2013 Material adapted from MIT 8.02 course notes Ampere’s Law Theory Review Biot-Savart Law Biot-Savart

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

Page 7: Ampere's Law - Level 5 Physics - IIS Cremona€™s Law Level 5 Physics January 2013 Material adapted from MIT 8.02 course notes Ampere’s Law Theory Review Biot-Savart Law Biot-Savart

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.

Page 8: Ampere's Law - Level 5 Physics - IIS Cremona€™s Law Level 5 Physics January 2013 Material adapted from MIT 8.02 course notes Ampere’s Law Theory Review Biot-Savart Law Biot-Savart

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.

Page 9: Ampere's Law - Level 5 Physics - IIS Cremona€™s Law Level 5 Physics January 2013 Material adapted from MIT 8.02 course notes Ampere’s Law Theory Review Biot-Savart Law Biot-Savart

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.

Page 10: Ampere's Law - Level 5 Physics - IIS Cremona€™s Law Level 5 Physics January 2013 Material adapted from MIT 8.02 course notes Ampere’s Law Theory Review Biot-Savart Law Biot-Savart

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.

Page 11: Ampere's Law - Level 5 Physics - IIS Cremona€™s Law Level 5 Physics January 2013 Material adapted from MIT 8.02 course notes Ampere’s Law Theory Review Biot-Savart Law Biot-Savart

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.)

Page 12: Ampere's Law - Level 5 Physics - IIS Cremona€™s Law Level 5 Physics January 2013 Material adapted from MIT 8.02 course notes Ampere’s Law Theory Review Biot-Savart Law Biot-Savart

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.)

Page 13: Ampere's Law - Level 5 Physics - IIS Cremona€™s Law Level 5 Physics January 2013 Material adapted from MIT 8.02 course notes Ampere’s Law Theory Review Biot-Savart Law Biot-Savart

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.)

Page 14: Ampere's Law - Level 5 Physics - IIS Cremona€™s Law Level 5 Physics January 2013 Material adapted from MIT 8.02 course notes Ampere’s Law Theory Review Biot-Savart Law Biot-Savart

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.)

Page 15: Ampere's Law - Level 5 Physics - IIS Cremona€™s Law Level 5 Physics January 2013 Material adapted from MIT 8.02 course notes Ampere’s Law Theory Review Biot-Savart Law Biot-Savart

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

Page 16: Ampere's Law - Level 5 Physics - IIS Cremona€™s Law Level 5 Physics January 2013 Material adapted from MIT 8.02 course notes Ampere’s Law Theory Review Biot-Savart Law Biot-Savart

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.)

Page 17: Ampere's Law - Level 5 Physics - IIS Cremona€™s Law Level 5 Physics January 2013 Material adapted from MIT 8.02 course notes Ampere’s Law Theory Review Biot-Savart Law Biot-Savart

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.)

Page 18: Ampere's Law - Level 5 Physics - IIS Cremona€™s Law Level 5 Physics January 2013 Material adapted from MIT 8.02 course notes Ampere’s Law Theory Review Biot-Savart Law Biot-Savart

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.)