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A small ball of charge q and mass m rests on a smooth horizontal plane at a distance b from the origin. A time-varying magnetic field B = B₀*(10 + t) directed along the negative z-axis exists inside the circular region x² + y² = b². In which direction does the ball begin to move, and what is its acceleration?

1. Toward positive x-axis with acceleration b*q*B₀ / (2m)
2. Toward negative x-axis with acceleration b*q*B₀ / (2m)
3. Toward positive y-axis with acceleration b*q*B₀ / (2m)
4. Toward negative y-axis with acceleration b*q*B₀ / (2m)

Correct answer: Toward negative y-axis with acceleration b*q*B₀ / (2m)

Solution

dB/dt = B₀ in the -z direction. By Faraday's law: E*(2*pi*b) = pi*b²*B₀. So E = B₀*b/2. The field B_z is becoming more negative (increasing magnitude in -z), so by Lenz's law the induced E field opposes this change and circulates clockwise when viewed from +z. At the ball's position on the +x-axis (r=b), the clockwise direction is -y. Force on ball = qE in -y direction. Acceleration a = qE/m = q*B₀*b/(2m) toward -y axis.

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