JEE Advance - Physics (2012 - Paper 1 Offline - No. 12)
Consider the motion of a positive point charge in a region, there are simultaneous uniform electric and magnetic fields $$\overrightarrow E = {E_0}\widehat j$$ and $$\overrightarrow B = {B_0}\widehat j$$. At time t = 0, this charge has velocity $$\overrightarrow v $$ in the xy-plane, making an angle $$\theta$$ with the x-axis. Which of the following option(s) is(are) correct for time t > 0 ?
If $$\theta$$ = 0$$^\circ$$, the charge moves in a circular path in the xy-plane.
If $$\theta$$ = 0$$^\circ$$, the charge undergoes helical motion with constant pitch along the y-axis.
If $$\theta$$ = 10$$^\circ$$, the charge undergoes helical motion with its pitch increasing with time, along the y-axis.
If $$\theta$$ = 90$$^\circ$$, the charge undergoes linear but accelerated motion along the y-axis.
Explanation
For $$\theta$$ = 90$$^\circ$$ : $$\overrightarrow v \,||\,\overrightarrow E $$ and $$\overrightarrow v \,||\,\overrightarrow B $$
. Therefore, magnetic field $$\overrightarrow B $$ exerts no force on the point charge :$${\overrightarrow F _B} = q(\overrightarrow v \times \overrightarrow B ) = 0$$
$${\overrightarrow F _E} = q\overrightarrow E $$
Therefore, the motion is along y-axis and the particle accelerates.
For $$\theta$$ = 0$$^\circ$$ and $$\theta$$ = 10$$^\circ$$ : The motion is helical with increasing pitch, along with increasing time (along y-axis), as $$p = (v\cos \theta )t$$, where v increases due to $${\overrightarrow F _E}$$. (For $$\theta$$ = 0$$^\circ$$, the change cannot move in a circular path.)
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