Question

An How does the magnetic field of a moving point charge fall off with distance at a given angle: like $\frac{\mathbf{1}}{\mathbf{r}}$ , $\frac{\mathbf{1}}{{\mathbf{r}}^{\mathbf{2}}}$ or $\frac{\mathbf{1}}{{\mathbf{r}}^{\mathbf{3}}}$ ?

Short answer

The magnetic field of a moving point charge fall off with distance at a given angle is$\frac{1}{{\mathrm{r}}^2}$ .

Step-by-step solution

Significance of magnetic field due to a point charge

Whenever a point charge moves at a specific speed, then there would be a magnetic field set up around the point charge. The relation between the magnetic field and the distance from the point charge is an inverse one.

Determination the magnetic field of a moving point charge

The expression of magnetic field due to a point charge is expressed as,

$\begin{array}{rcl}\overrightarrow{\mathrm{B}}& =& \left(\frac{{\mathrm{\mu }}_0}{4\mathrm{\pi }}\right)\left(\frac{\mathrm{q}\times \hat{\mathrm{r}}}{{\mathrm{r}}^2}\right)\\ & \propto & \frac{1}{{\mathrm{r}}^2}\end{array}$

Here, $\overrightarrow{\mathrm{B}}$is the magnetic field of a moving point charge, $\mathrm{r}$represents the distance from the origin and ${\mathrm{\mu }}_0$ represents the vacuum permeability.

Thus, the magnetic field of a moving point charge fall off with distance at a given angle is $\frac{1}{{\mathrm{r}}^2}$.