Question

Electromagnetic Induction is proving hard for me to fully grasp. I've got lots of conceptual questions, mainly dealing with the shapes of conductors, the direction of currents and magnetic fields and how to piece them altogether. If anyone knows of some good resources for me to check out (websites, videos, etc...) please post them.

Thanks!

Answer 1

This question is making my brain hurt:

Two circular concentric wires in the same plane have different currents. The smaller wire, with radius 18.00 cm has a clockwise current 11.500 A. What current in the larger wire, radius 32.00 cm is required so the total magnetic field at the center is zero?

Should I treat it like two parallel wires???

Answer 2

No - two concentric circular wires is a bit of a different situation from two parallel straight wires. However, this question is a lot easier than it looks! I'm going to give a detailed explanation so you can understand the thought process.

The nice thing about circular wires is that there's a simple formula to calculate the strength of the magnetic field in the CENTRE of the circle:

\[B=\frac{ \mu_{0}I }{ 2r }\]
Where B is the magnitude of the magnetic field, mu(0) is the permeability of free space (a constant, just like R is the gas constant), I is the current, and r is the radius of the wire loop.

The question is asking you to find the current in the outer wire that will cause the magnetic field generated by the inner wire to be cancelled out. We don't really need to worry about the right hand rule here; all that's important is that the currents are moving in opposite directions. That means for the inner (i) and outer (o) wires:

\[B_{i}=\frac{ \mu_{0}I_{i} }{ 2r_{i} }\]\[-B_{o}=\frac{ \mu_{0}(-I_{o}) }{ 2r_{o} }\]
Notice that for the outer wire, the I is negative (relative to I of the inner wire), since the current is travelling in the opposite direction. This therefore makes B negative (relative to B of the inner wire). Since we're only worried about the magnitude (we already know the current direction will be counterclockwise), we can just divide out the negatives in that second equation to make things easier and then set the two equations equal to each other since B(i) must equal B(o) so that the magnetic fields cancel out:

\[\frac{ \mu_{0}I_{i} }{ 2r_{i} }=\frac{ \mu_{0}I_{o} }{ 2r_{o} }\]
We can take out the mu(0) since it's a constant, and the 2 in the denominator, leaving:

\[\frac{ I_{i} }{ r_{i} }=\frac{I_{o} }{ r_{o} }\]
Now by plugging in the numbers given in the question you should be able to get your answer!

If you have any questions please let me know!