Ampere’s Law
Ampere’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 Law
~
The magnetic field contribution, d B,
from a current source, Id~s, at a field
point P a distance r in the direction of
r̂ is given by
~ =
dB
µ0 Id~s × r̂
4π r 2
where µ0 is the permeability of free
space constant with value
µ0 = 4π × 10−7 T · 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 wire
with current I is
B=
µ0 I
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.
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
H law discovered by Ampere in 1826 results from
~ · d~s along a closed path such as this (called an
calculating B
Amperian loop).
Ampere’s Law
Theory
Ampere
Ampere’s Law - Special Case
H
~ · d~s around a circular loop with
Calculating the line integral of B
radius r yields
I
I
µ0 I
~ · d~s =
B
∗ ds
2πr
I
µ0 I
=
ds
2πr
µ0 I
∗ 2πr
=
2πr
= µ0 I
Ampere’s Law
Theory
Ampere
Ampere’s Law - General Case
The more general form of Ampere’s Law involves integrating over
an arbitrary closed loop such as the one below.
Ampere’s Law
Theory
Ampere
Ampere’s Law - General Case
The more general form of Ampere’s Law involves integrating over
an arbitrary closed loop such as the one below.
It
H can be shown with a proof involving cylindrical coordinates that
~ · d~s = µ0 I for this general case as well.
B
Ampere’s Law
Theory
Ampere
Ampere’s Law
Ampere’s Law
A closed loop around a current carrying wire relates the
surrounding magnetic field to the current by
H
~ · d~s = µ0 Ienclosed
B
where Ienclosed is the current encircled by the loop.
Ampere’s Law
Theory
Ampere
Ampere’s Law
Ampere’s Law
A closed loop around a current carrying wire relates the
surrounding magnetic field to the current by
H
~ · d~s = µ0 Ienclosed
B
where Ienclosed is the current encircled by the loop.
Ampere’s Law can be used to easily calculate the magnetic field in
certain situations when an appropriate Amperian loop is chosen.
Typically a good choice allows the magnetic field magnitude to be
treated as a constant or for the integral to be evaluated in
sections. Biot-Savart’s Law can be used in other situations to find
the magnetic field.
Ampere’s Law
Questions
Concept Questions
Toroid Central Region
What is the best description of the magnetic field in the central
circular region of a toroid?
1
Negative
2
Zero
3
Positive
Ampere’s Law
Questions
Concept Questions
Toroid Central Region
What is the best description of the magnetic field in the central
circular region of a toroid?
1
Negative
2
Zero
3
Positive
Correct Answer: 2
(There is symmetry so Ampere’s Law can be used. Since
Ienclosed = 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 region
surrounding a toroid?
1
Negative
2
Zero
3
Positive
Ampere’s Law
Questions
Concept Questions
Toroid Outer Region
What is the best description of the magnetic field in the region
surrounding 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 of
the cable has a 1.0A total current going into the page. The outer
region has a uniform current density of 1.0A/m2 pointing out of
the page. At what distance in meters from the center of the cable
is the magnetic field equal to zero?
3 1/3
4π )
1
r = (1 +
2
r=
3
r=
4
The magnetic field is never
zero
1
2π
q
+1
1
π
+1
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 equation
π(r 2 − 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 shown
below. The current flows counterclockwise. What is the direction
of the force on this segment?
1
Left
2
Right
3
Into page
4
Out of page
Ampere’s Law
Questions
Concept Questions
One Current Loop
Consider a small segment of a circular current loop as shown
below. The current flows counterclockwise. What is the direction
of 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 segment
that points out of the page.)