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