A charge of \( 5\,\mu\mathrm{C} \) moves with a speed of \( 2 \times 10^{6}\,\mathrm{m/s} \) along the positive y-axis. A magnetic field \( \vec{B} \) of strength \( (2\hat{\imath} + \hat{k})\,\mathrm{T} \) exists in space. The magnetic force acting on the charge is :
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( (-20\hat{k} + 10\hat{\imath})\,\mathrm{N} \)
The magnetic (Lorentz) force on a moving charge is \( \vec{F} = q\,(\vec{v} \times \vec{B}) \).
\( q = 5\,\mu\mathrm{C} = 5\times10^{-6}\,\mathrm{C} \)
\( \vec{v} = 2\times10^{6}\,\hat{\jmath}\,\mathrm{m\,s^{-1}} \)
\( \vec{B} = (2\hat{\imath} + \hat{k})\,\mathrm{T} \)
Using the cyclic cross products \( \hat{\jmath}\times\hat{\imath} = -\hat{k} \) and \( \hat{\jmath}\times\hat{k} = \hat{\imath} \):
\( \hat{\jmath} \times (2\hat{\imath} + \hat{k}) = -2\hat{k} + \hat{\imath} \)
\( q\,v = 5\times10^{-6} \times 2\times10^{6} = 10 \)
\( \vec{F} = 10\,(-2\hat{k} + \hat{\imath}) = (-20\hat{k} + 10\hat{\imath})\,\mathrm{N} \)