Question #42955

A short magnetic dipole of magnetic moment $\vec{m} = m_0 \hat{k}$ (where $m_0 > 0$ is in $\text{A}\cdot\text{m}^2$) is placed at the origin of a Cartesian coordinate system. In the same region, an electrostatic field is established in free space such that its electric potential is given by $V(x,y,z) = V_0 (x^2 + y^2 - 2z^2)$, where $V_0 > 0$ is a constant in $\text{V}\cdot\text{m}^{-2}$ and $x, y, z$ are in meters.

Consider a point $P$ in space, with a position vector making an angle $\theta$ ($0 < \theta < \pi/2$) with the positive $z$-axis. It is observed that at point $P$, the magnetic field vector $\vec{B}$ is strictly normal to the electrostatic equipotential surface passing through $P$.

What is the value of $\cos\theta$?