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Six identical electric dipoles, each with dipole moment P directed along the +z axis, are placed at the six corners of a regular hexagon of side a (with a much larger than the dipole size, so the dipole approximation holds). The centre of the hexagon is at origin O. Point A is at (0, 0, a) and point B is at (0, 0, -a). Which of the following statements is/are correct?

1. The electric field at origin O due to all six dipoles has magnitude 6kP / (sqrt(2) * a³) and points along -z.
2. The electric potential at origin O due to all six dipoles is zero.
3. The electric potential at point A due to all six dipoles is +3kP / (sqrt(2) * a²).
4. The electric potential at point B due to all six dipoles is +3kP / (sqrt(2) * a²).

Correct answer: The electric potential at origin O due to all six dipoles is zero.

Solution

The electric potential at O is zero for all six dipoles because the angle between each dipole axis and the line to O is 90 degrees. For point A, each dipole is at distance a*sqrt(2) with cos(theta) = 1/sqrt(2), giving a total potential of +3kP/(sqrt(2)*a²). For point B, cos(theta) = -1/sqrt(2), so the total potential is -3kP/(sqrt(2)*a²), making option D wrong.

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