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A cube of side a has identical point charges +q placed at all eight corners. The cube rotates about a symmetry axis with constant angular velocity omega. Which statement(s) about the net magnetic field at the centre of the cube is/are correct? (A) Net magnetic field at the centre is zero. (B) Net magnetic field at the centre is sqrt(2)*mu0*q*omega / (pi*a). (C) Net magnetic field at the centre is [8 / (3*sqrt(3))] * mu0*q*omega / (pi*a). (D) If the sign of any four charges is reversed (alternating), the magnetic field at the centre becomes zero.
- Net magnetic field at the centre is zero.
- Net magnetic field at the centre is sqrt(2)*mu0*q*omega / (pi*a).
- Net magnetic field at the centre is [8 / (3*sqrt(3))] * mu0*q*omega / (pi*a).
- If the sign of any four charges is reversed, the magnetic field at the centre becomes zero.
Correct answer: Net magnetic field at the centre is [8 / (3*sqrt(3))] * mu0*q*omega / (pi*a).
Solution
Each corner charge q rotates with omega; effective current I = q*omega/(2*pi). Taking the axis along a body diagonal through centre: 2 charges lie on the axis (no contribution), 6 charges orbit at perpendicular distance r_perp = a*sqrt(2)/sqrt(3)... The exact calculation for all 8 charges equally off-axis gives the net field along the rotation axis. Each charge at corner (±a/2, ±a/2, ±a/2) at distance d=a*sqrt(3)/2 from centre, each orbiting at r_perp from axis. The standard result for rotation about a face-centred axis (each corner at r_perp = a/sqrt(2), distance to centre = a*sqrt(3)/2) gives the formula in option C.
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