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A conducting wire shaped as the parabola y = x² moves with constant velocity V = V0 i in a non-uniform magnetic field B = B0*(1 + (y/L)^b) k, where V0, B0, L and b are positive constants. Let |Dphi| be the magnitude of the potential difference between the ends of the wire. Which statement(s) is/are correct?

1. |Dphi| is unchanged if the parabola is replaced by a straight wire y = x of length sqrt(2)*L.
2. |Dphi| is proportional to the length of the wire projected onto the y-axis.
3. |Dphi| = (1/2)*B0*V0*L when b = 0.
4. |Dphi| = (4/3)*B0*V0*L when b = 2.

Correct answer: |Dphi| is proportional to the length of the wire projected onto the y-axis.

Solution

Since V x B points along the y-direction (V0 i x B0 k = -V0*B0 j), only dy parts of the wire matter. Thus Dphi = V0 * integral of B dy over the y-projection of the wire, so it depends only on the y-extent, not the path shape. Carrying out the integral for general b and the special cases shows the y-projection statement is the robust correct conclusion.

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