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Exploration 28.1: A Long Wire with Uniform Current
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The gray circle in the center represents a cross section of a wire carrying current coming out of the computer screen. The current is uniformly distributed throughout the wire (position is given in centimeters and magnetic field strength is given in millitesla). The black circle is an Amperian loop with a radius you can change with the slider. Restart.
Begin with the Amperian loop with a radius larger than the radius of the wire.
- What is the radius of the Amperian loop?
- What is the magnetic field at this radius?
You will use Ampere's law to find the total current in the wire:
∫ B · dl = μ0I, μ0 = 4π x 10-7 T·m/A,
where the integration is over a closed loop (closed path), dl is an element of the path in the direction of the path, and I is the total current enclosed in the path. Pick a point on the Amperian loop and draw both the direction of the magnetic field at that point and the direction of dl (tangent to the path).
- The magnetic field and dl should be parallel to each other. What is B · dl?
Pick another point on the Amperian loop.
- What is the magnitude of the magnetic field at that point? At any point on the loop?
- This means you can write ∫ B · dl = B ∫ dl. Why?
∫dl is simply the length of the Amperian loop (in this case the circumference). Therefore, B = μ0I/2πr outside the wire.
- Calculate the current carried by the wire from your measurement of the magnetic field.
- Change the radius of the loop (but leave it still bigger than the wire) and predict the magnetic field on that loop. Measure the value to verify your answer.
Make the Amperian loop smaller so it fits inside the wire. This time, the current inside the loop is not equal to the total current, but instead it is equal to the total current times the fraction of the area inside the loop, Ir2/a2, where a is the radius of the wire.
- Use this fraction and Ampere's law again to predict the magnetic field inside the loop.
- Measure the field to verify your answer.
- Show that the general expression for the magnetic field inside the wire is μ0I r/2πa2.
Exploration authored by Anne J. Cox.