• # question_answer Direction: Light having photons energy $hv$ is incident on a metallic plate having work function $\phi$ to eject the electrons. The most energetic electrons are then allowed to enter in a region of uniform magnetic field B as shown in the figure.   The electrons are projected in XZ-plane making an angle of$\theta$ with X-axis and magnetic field is $\mathbf{B}={{B}_{0}}\,i$ along X-axis. Maximum pitch of the helix described by electron is found to be p. Take mass of electron as m and charge as q. Based on above information, answer the following questions Considering the instant of crossing origin at $t=0,$ the Z-coordinate of location of electron as a function of time is A)  $-\frac{-\sqrt{2m\,(hv-\phi )}}{q{{B}_{0}}}\sin \,\theta \,\left[ 1-\cos \,\left( \frac{q{{B}_{0}}t}{m} \right) \right]$ B)  $\frac{\sqrt{2m\,(hv-\phi )}}{q{{B}_{0}}}\sin \,\theta \,\times \sin \,\left[ \frac{q{{B}_{0}}t}{m} \right]$ C) $\frac{-\sqrt{2m\,(hv-\phi )}}{q{{B}_{0}}}\sin \,\theta \,\times \sin \,\left[ \frac{q{{B}_{0}}t}{m} \right]$ D)  $\frac{\sqrt{2m\,(hv-\phi )}}{q{{B}_{0}}}\times \sin \,\left( \frac{q{{B}_{0}}t}{m} \right)$

$X-$coordinate, $x=v\,\,\cos \,\,\theta \times t$ $Y-$coordinate, $y=-[R-R\,\cos \,\omega t]$ Z-coordinate, $z=R\,\sin \,\omega t$ $z=\frac{mv\,\sin \,\theta }{q{{B}_{0}}}\times \sin \,\left[ \frac{q{{B}_{0}}}{m}t \right]$ $=\sqrt{\frac{2m(hv-\phi )\times \,sin\,\theta }{q{{B}_{0}}}}\times \sin \,\left[ \frac{q{{B}_{0}}t}{m} \right]$ As v increases, slope of $x$ versus $t$ graph (a straight line increases).