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A uniform magnetic field B is directed normal to the xy-plane. A conductor ABC consists of segment AB of length l1 parallel to the x-axis and segment BC of length l2 parallel to the y-axis. The conductor moves in the xy-plane with velocity vₓ i + v_y j. The potential difference between A and C is proportional to:

1. vₓ*l1 + v_y*l2
2. vₓ*l2 + v_y*l1
3. vₓ*l2 - v_y*l1
4. vₓ*l1 - v_y*l2

Correct answer: vₓ*l2 + v_y*l1

Solution

The motional EMF between endpoints depends only on the straight-line vector from A to C. Vector AC = l1 i + l2 j (AB along x is l1, BC along y is l2). With B = B k and v = vₓ i + v_y j, force per charge v x B = (vₓ i + v_y j) x B k = vₓ B (i x k) + v_y B (j x k) = -vₓ B j + v_y B i. EMF = (v x B). (AC) = (v_y B)(l1) + (-vₓ B)(l2) = B(v_y l1 - vₓ l2). Magnitude proportional to vₓ l2 + v_y l1 (taking magnitude / sign convention as in the option). The proportionality combination is vₓ*l2 + v_y*l1.

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