Question

Biot-Savart law indicates that the moving electrons (velocity$\overrightarrow{\mathrm{v}}$) produce a magnetic field $\overrightarrow{\mathrm{B}}$ such that:

• A. $\overrightarrow{\mathrm{B}}\perp \overrightarrow{\mathrm{v}}$
• B. $\overrightarrow{\mathrm{B}}\parallel \overrightarrow{\mathrm{v}}$
• C. It obeys the inverse cube law
• D. It is along the line joining the electron and the point of observation

Answer 1

The correct option is A

$\overrightarrow{\mathrm{B}}\perp \overrightarrow{\mathrm{v}}$

The explanation for the correct answer:

Option(A): $\overrightarrow{\mathrm{B}}\perp \overrightarrow{\mathrm{v}}$

According to Biot-Savart law, the magnitude $\overrightarrow{\mathrm{B}}$ is given as:

$\mathrm{B}=\frac{{\mathrm{\mu }}_0}{4\mathrm{\pi }}\frac{\left|\mathrm{q}\right|\mathrm{vsin\varphi }}{{\mathrm{r}}^2}$
The magnetic field as a vector is given as:

$\overrightarrow{\mathrm{B}}=\frac{{\mathrm{\mu }}_0}{4\mathrm{\pi }}\frac{\left|\mathrm{q}\right|(\overrightarrow{\mathrm{v}}\times \overrightarrow{\mathrm{r}})}{{\mathrm{r}}^3}\hat{\mathrm{\eta }}$

Clearly, B is perpendicular to v.

The explanation for the incorrect answer:

Option(B): $\overrightarrow{\mathrm{B}}\parallel \overrightarrow{\mathrm{v}}$

The magnetic field produced by charges moving with velocity $\overline{\mathrm{v}}$, at a distance $\mathrm{r}$is $\overrightarrow{\mathrm{B}}$$\left(\frac{{\mu }_0}{4\pi }\right)·\mathrm{q}\frac{\overline{v}\times \hat{r}}{r^2}$

Option(C): It obeys the inverse cube law

The direction of $\overrightarrow{\mathrm{B}}$is along $(\overrightarrow{\mathrm{v}}\times \overrightarrow{\mathrm{r}})$, i.e., perpendicular to the plane containing $\overrightarrow{\mathrm{v}}$ and $\overrightarrow{\mathrm{r}}.\overrightarrow{\mathrm{B}}$ at a point obeys inverse square law and not inverse cube law.

Option(D): It is along the line joining the electron and the point of observation

According to Biot-Savart law, we get $\mathrm{B}$ is perpendicular to $\mathrm{v}$

Therefore, option A is the correct choice.