Magnetics 1 Question 19
19. Consider the motion of a positive point charge in a region where there are simultaneous uniform electric and magnetic fields $\mathbf{E}=E _0 \hat{\mathbf{j}}$ and $\mathbf{B}=B _0 \hat{\mathbf{j}}$. At time $t=0$, this charge has velocity $\mathbf{v}$ in the $x-y$ plane, making an angle $\theta$ with the $x$-axis. Which of the following option(s) is(are) correct for time $t>0$ ?
(2012)
(a) If $\theta=0^{\circ}$, the charge moves in a circular path in the $x-z$ plane.
(b) If $\theta=0^{\circ}$, the charge undergoes helical motion with constant pitch along the $y$-axis
(c) If $\theta=10^{\circ}$, the charge undergoes helical motion with its pitch increasing with time, along the $y$-axis.
(d) If $\theta=90^{\circ}$, the charge undergoes linear but accelerated motion along the $y$-axis.
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Answer:
Correct Answer: 19. (c, d)
Solution:
- and Magnetic field will rotate the particle in a circular path (in $x-z$ plane or perpendicular to $B$ ). Electric field will exert a constant force on the particle in positive $y$-direction. Therefore, resultant path is neither purely circular nor helical or the options (a) and (b) both are wrong.
$v _{\perp}$ and $\mathbf{B}$ will rotate the particle in a circular path in $x-z$ plane (or perpendicular to $\mathbf{B}$ ). Further $v _{|}$and $\mathbf{E}$ will move the particle (with increasing speed) along positive $y$-axis (or along the axis of above circular path). Therefore, the resultant path is helical with increasing pitch, along the $y$-axis (or along $\mathbf{B}$ and $\mathbf{E}$ ). Therefore option (c) is correct.
(d)
Magnetic force is zero, as $\theta$ between $\mathbf{B}$ and $\mathbf{v}$ is zero. But electric force will act in $y$-direction. Therefore, motion is $1-D$ and uniformly accelerated (towards positive $y$-direction).
