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Which of the options represents Biot – Savart’s Law for a point charge particle Q moving with velocity $\overrightarrow V$ ?

$\begin {array} {1 1} (a)\;\large\frac{\mu_o}{4 \pi} Q\large\frac{\overrightarrow V \times \hat r}{r^3} & \quad (b)\;\large\frac{\mu}{4 \pi} Q\large\frac{\overrightarrow V \times \hat r}{r^2} \\ (c)\;\large\frac{\mu_o}{4 \pi} Q\large\frac{\overrightarrow V \times \{\hat r \times ( \overrightarrow V \times \hat r) \}}{r^2} & \quad (d)\;\large\frac{\mu_o}{4 \pi} Q\large\frac{\overrightarrow V \times \overrightarrow r}{r^3} \end {array}$

In the case of a point charged particle $Q$ moving at a constant velocity $V$, Maxwell's equations give the following expression for the electric field and magnetic field:
$E = \large\frac{Q}{4\pi\varepsilon_0}$$\large\frac{1-\frac{v^2}{c^2}}{(1-v^2 \frac{\sin^2\theta}{c^2})^{3/2}}$$\large\frac{\hat r}{r^2}$, where $\hat r$ is the vector pointing from the current position of the particle to the point at which the field is being measured, and $\theta$ is the angle between $V$ and $r$.
$B = \overrightarrow V \times \large\frac{1}{c^2}$$E \Rightarrow E = \large\frac{Q}{4\pi\varepsilon_0}$$\large\frac{\hat r}{r^2}$$\rightarrow B = \large\frac{\mu_o}{4 \pi}$$Q\large\frac{\overrightarrow V \times \hat r}{r^2}$
edited Mar 14, 2014