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A cube of side a has point charges q placed at each of its eight corners. The cube is rotated with constant angular velocity omega about one of its body diagonals as the axis. Which of the following statements about the magnetic field at the centre of the cube 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*mu₀*q*omega / (3*sqrt(3)*pi*a) (D) If the sign of any four of the charges is reversed, the magnetic field at the centre becomes zero
- Net magnetic field at the centre of the cube is zero
- Net magnetic field at the centre is sqrt(2)*mu₀*q*omega / (pi*a)
- Net magnetic field at the centre is 8*mu₀*q*omega / (3*sqrt(3)*pi*a)
- If the sign of any four charges are reversed, the magnetic field at the centre becomes zero
Correct answer: Net magnetic field at the centre is 8*mu₀*q*omega / (3*sqrt(3)*pi*a)
Solution
Each corner charge rotates about the body diagonal. The two charges ON the diagonal (at distance 0 from axis) contribute nothing. The other six charges rotate at equal radii. The distance of each corner from the body diagonal of a cube of side a is a*sqrt(2/3). Each is equivalent to current I = q*omega/(2*pi), producing B = mu₀*I/(2*r) along the axis. Six such contributions add up to give the net B.
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