An electron is moving at a velocity of 100 m/s along the X-axis in the presence of an electric field of 30 V/m directed along the Y-axis and a magnetic field of 300 G directed along the Z-axis. The magnitude and direction of force acting on the electron is
(Given: the charge on the electron is “e”)
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( 27\,e\,\mathrm{N} \) along the negative Y-axis
The force on a charge \( q \) moving in electric and magnetic fields is the Lorentz force. For the electron the charge is \( q = -e \).
\( \displaystyle \vec{F} = q\left(\vec{E} + \vec{v}\times\vec{B}\right) = -e\left(\vec{E} + \vec{v}\times\vec{B}\right) \)
\( B = 300\,\mathrm{G} = 300\times10^{-4}\,\mathrm{T} = 0.03\,\mathrm{T} \)
\( \vec{v} = 100\,\hat{\imath}\,\mathrm{m\,s^{-1}} \)
\( \vec{E} = 30\,\hat{\jmath}\,\mathrm{V\,m^{-1}} \)
\( \vec{B} = 0.03\,\hat{k}\,\mathrm{T} \)
\( \displaystyle \vec{v}\times\vec{B} = (100\,\hat{\imath})\times(0.03\,\hat{k}) = 3\,(\hat{\imath}\times\hat{k}) = -3\,\hat{\jmath}\,\mathrm{V\,m^{-1}} \)
\( \displaystyle \vec{E} + \vec{v}\times\vec{B} = (30 - 3)\,\hat{\jmath} = 27\,\hat{\jmath}\,\mathrm{V\,m^{-1}} \)
\( \displaystyle \vec{F} = -e \times 27\,\hat{\jmath} = -27\,e\,\hat{\jmath}\,\mathrm{N} \)
The force is \( 27e\,\mathrm{N} \) along the negative Y-axis, because the electron is negative and both forces act opposite to the way they would on a positive charge.