Note 18 Ampere’s Law
Ampere’s Law
A current will produce a magnetic field. The description of the amount of magnetic field produced
from a current is Ampere’s law. Ampere’s law works much like Gauss’ law except for the geometry
involved. Gauss’ law says what source charges do which is producing electric fields through a
closed surface. Ampere’s law says what source currents do which is producing magnetic fields
along a closed line.
currents through
loop surface
amperean loop
Here is Ampere’s law. The total magnetic field along a close loop, called the ampere loop, is
determined from the currents that pass through the surface formed by the close loop.
Σclosed loopBat the loop ΔL cos θbetween B and ΔL = µoI enclosed
I2
I1
I3
amperean loop
∆L1
B1
∆L2 ∆L3
B2
B3
The constant, µo, is called the permeability of free space.
µo = 4π ×10−7 T ⋅ m
A
Just like Gauss’ law, the application of Ampere’s law is difficult for non-uniform and asymmetric
loops. However, where the geometry is simple, Ampere’s law can produce results quickly.
Here are the geometries that we will be dealing with.
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Magnetic Field from a Moving Charge
If a magnetic field can be generated from a current, it can be generated from a moving charge.
This is described in the Biot-Savart law.
Magnetic Fields Around an Infinite Wire
Let me apply Ampere’s law to an infinitely long wire carrying a constant current. The ampere loop,
just like the gaussian surface, should fit the symmetry of the source. Again, for a wire, the
symmetry is the circular cylinder. As a loop, it is just a circle centered at the wire.
current (up)
∆L
amperean loop
B
At the ampere loop, the magnetic field is constant and circular and so is the loop so that the
cosine returns a value of 1. The sum over the loop is just the circumference times the magnetic
field.
Bat the loop (2πr) = µoI
⇒ Bat the loop =
µo I
2π r
Magnetic Fields at the Center of a Current Loop
Here is what this looks like. I curved the current into a loop.
current
B
The magnetic field at the center of the current loop is the following.
Bat the loop center =
µo I
2 R
Along the center axis away from the center of the loop, the magnetic field is the following. R is the
radius of the loop and z is the distance on the axis from the center of the loop.
Balong loop center axis =
µo
R2
2 (z 2 + R 2 )3/2
Outside the center of the loop, the magnetic field looks like this. The shape is called a magnetic
dipole. The vector through the center points at the magnetic north pole.
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Away from the central axis of the current loop, the magnetic field has the shape of a dipole.
current loop
magnetic fields
Magnetic Fields at the Center of a Solenoid
A solenoid is just a number of current loops in the shape of a coil. The number of coils per unit
length is the coil density, n.
current
B
Bat the solenoid center = µonI
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