1
Ampere’s Circuital Law
The magnetic field in space around an electric current is proportional to the electric current which serves as its source, just as the electric field in space is proportional to the charge which serves as its source.
I
B
2
I
B
r
Ampere’s “Law”
Just for kicks, let’s evaluate the line integral along the direction of B over a closed circular path around a current-carrying wire.
ds
The above calculation is only for the special case of a long straight wire, but you can show that the result is valid in general.
3
Ampere’s “Law”
I is the total current that passes through a surface bounded by the closed (and not necessarily circular) path of integration.
Ampere’s “Law” is useful for calculating the magnetic field due to current configurations that have high symmetry.
I
B
r
ds
The current I passing through a loop is positive if the direction of integration is the same as the direction of B from the right hand rule.
I
B
r
ds
positive I
negative I
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Your text writes
General form of Ampere’s “Law”:
I1
ds
If your path includes more than one source of current, add all the currents (with correct sign).
I2
because the current that you use is the current “enclosed” by the closed path over which you integrate.
The reason for the 2nd term on the right will become apparent later. Ignore it for now.
5
Example: a cylindrical wire of radius R carries a current I that is uniformly distributed over the wire’s cross section. Calculate the magnetic field inside and outside the wire.
I
R
Cross-section of the wire:
R
r
◉
direction of I
B
6
R
r
◉
direction of I
B
Over the closed circular path r:
Solve for B:
B is linear in r.
7
R
r
◉
direction of I
B
Outside the wire:
(as expected).
B
r
R
Plot:
A lot easier than using the Biot-Savart “Law”!
8
Calculating Electric and Magnetic Fields
Electric Field
in general: Coulomb’s “Law”
for high symmetry configurations: Gauss’ “Law”
Magnetic Field
in general: Biot-Savart “Law”
for high symmetry configurations: Ampere’s “Law”
This analogy is rather flawed because Ampere’s “Law” is not really the “Gauss’ “Law” of magnetism.”
9
Ampere’s “Law.”
You must be able to use Ampere’s “Law” to calculate the magnetic field for high-symmetry current configurations.
Solenoids.
You must be able to use Ampere’s Law to calculate the magnetic field of solenoids and toroids. You must be able to use the magnetic field equations derived with Ampere’s “Law” to make numerical magnetic field calculations for solenoids and toroids.
10
Magnetic Field of a Solenoid
A solenoid is made of many loops of wire, packed closely together. Here’s the magnetic field from a single loop of wire:
Some images in this section are from hyperphysics.
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Stack many loops to make a solenoid:
This ought to remind you of the magnetic field of a bar magnet.
Magnetic Field at the centre of a Straight Solenoid:
I
I
x
x
x
x
x
x
x
P
Q
R
S
∫
B . dl = μ0 I0
(where I0 is the net current threading through the solenoid)
∫
B . dl =
∫
B . dl +
PQ
∫
B . dl +
QR
∫
B . dl +
RS
∫
B . dl
SP
B
B . dl cos 0° +
∫
∫
B . dl cos 90° +
∫
0 . dl cos 0° +
∫
B . dl cos 90°
=
= B
∫
dl = B.a
and μ0 I0 = μ0 n a I
(where n is no. of turns per unit length, a is the length of the path and I is the current passing through the lead of the solenoid)
a
a
B = μ0 n I
Magnetic Field due to Toroidal Solenoid (Toroid):
I
dl
B
P
O
Q
B = 0
B = 0
∫
B . dl = μ0 I0
B . dl cos 0°
∫
= B
∫
dl = B (2π r)
r
And μ0 I0 = μ0 n (2π r) I
B = μ0 n I
∫
B . dl =
NOTE:
The magnetic field exists only in the tubular area bound by the coil and it does not exist in the area inside and outside the toroid.
i.e. B is zero at O and Q and non-zero at P.
B ≠ 0