Question

A wire of length l carries a current i long the x-axis. A magnetic field exists, which is given as $\overrightarrow{B}=B_0(\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k})$ T. Find the magnitude of the magnetic force acting on the wire.

Answer 1

Given:
A wire of length l cm
Electric current through the wire = i
Magnetic field, $\overrightarrow{B}=B_0(\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k})$ T.
As per the question, the current is passing along the X-axis.
Magnetic force,
$\overrightarrow{F}=i\overrightarrow{l}\times \overrightarrow{\mathrm{B}}$
Putting the repective values in the above equation, we get:
= i{(l $\overrightarrow{i}$) × (B₀ $\overrightarrow{i}$ + B₀ $\overrightarrow{j}$ + B₀ $\overrightarrow{k}$)}
= i{lB₀ $\overrightarrow{k}$ − lB₀ $\overrightarrow{j}$}
= ilB₀{−$\overrightarrow{j}$ + $\overrightarrow{k}$}
The magnitude of the magnetic force,
$\left|\overrightarrow{\mathit{F}}\right|$ = $\sqrt{2i^2{l}^2{B}_0^2}$
∴ $\left|\overrightarrow{\mathit{F}}\right|$ = $\sqrt{2}$ilB₀