The magnitude of the magnetic field produced by a long straight wire?

Question

The magnitude of the magnetic field produced by a long straight wire is proportional to the current passing through the wire and inversely proportional to the distance from the wire. Consider the five pairs of long parallel wires shown below. The arrows indicate the direction of the current in each wire, with small arrows representing a current of 3 A and large arrows representing a current of 9 A. For each pair, determine in which one of the following regions the net magnetic field is zero someplace: A) on the left of both wires B) between the two wires C) on the right of both wires D) none of the above (i.e. nowhere) Do not consider distances very far from the wires since in this case the magnetic field always goes to zero.

I place the first case - 3A,3A anti pararell to (D)

I place the second case - 9A,3A anti parallel to (C)

I place the third case - 3A,9A anti parallel to (A)

I place the fouth case - 9A,3A parallel to (D)

I place the fifth case - 3A,3A parallel to (B)

But one or more case is wrong. I can't figure out which one. Could someone tell me which one is wrong.

Thank you

I place the first case - 3A,3A anti pararell to (D)
I place the second case - 9A,3A anti parallel to (C)
I place the third case - 3A,9A anti parallel to (A)
I place the fouth case - 9A,3A parallel to (D)
I place the fifth case - 3A,3A parallel to (B)
But one or more case is wrong. I can't figure out which one. Could someone tell me which one is wrong.
Thank you

? · Accepted Answer

Place the left wire at x=0.  Let B1=k/x be the field as a function of x from a 3 amp current in the wire when the current vector is pointing down.  Place the right wire at x=a. The total field B will contain components from both wires.

First case:
B = k/x +k/(a-x) = [k(a-x) + kx]/[x(a-x)] = ak/[x(a-x)] which is never 0 --> D

Second case:
B = 3k/x + k/(a-x) = [3k(a-x) + kx]/[x(a-x)] = [3ka-2kx]/[x(a-x)] which is zero at 3a/2 --> C

Third case:
B = -k/x - 3k/(a-x) = [-k(a-x) - 3kx]/[x(a-x)] = [-ka-2kx]/[x(a-x)] which is zero at -a/2 --> A

Fourth case:
B = -3k/x + k/(a-x) = [-3k(a-x) + kx]/[x(a-x)] = [-3ka+4kx]/[x(a-x)] which is zero at 3a/4 --> B

Fifth case:
B = k/x - k/(a-x) = [k(a-x) - kx]/[x(a-x)] = [ka-2kx]/[x(a-x)] which is zero at a/2 --> B

I get the same answers as you except for the fourth case.

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The magnitude of the magnetic field produced by a long straight wire?

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