4.Moving Charges and Magnetism
hard

A charged particle of mass $\mathrm{m}$ and charge $q$ moving under the influence of uniform electric field $E\hat{i }$ and a uniform magnetic field $B\hat{k}$ follows a trajectory from point $\mathrm{P}$ to $\mathrm{Q}$ as shown in figure. The velocities at $P$ and $Q$ are respectively, $v\hat i$ and $-2 v \hat j$. Then which of the following statements $(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D})$ are the correct $?$ (Trajectory shown is schematic and not to scale)
$(A)$ $\mathrm{E}=\frac{3}{4}\left(\frac{\mathrm{mv}^{2}}{\mathrm{qa}}\right)$
$(B)$ Rate of work done by the electric field at $\mathrm{P}$ is $\frac{3}{4}\left(\frac{\mathrm{mv}^{3}}{\mathrm{a}}\right)$
$(C)$ Rate of work done by both the fields at $\mathrm{Q}$ is zero
$(D)$ The difference between the magnitude of angular momentum of the particle at $\mathrm{P}$ and $Q$ is $2 mav$.

A$(A), (B), (C), (D)$
B$(A), (B), (C)$
C$(B), (C), (D)$
D$(A), (C), (D)$
(JEE MAIN-2020)

Solution

Option $(\mathrm{A})$
$\mathrm{W}=\mathrm{k}_{\mathrm{f}}-\mathrm{k}_{\mathrm{i}}$
$q E(2 a-0)=\frac{1}{2} m(2 V)^{2}-\frac{1}{2} m V^{2}$
$\mathrm{q} \mathrm{E} 2 \mathrm{a}=\frac{3}{2} \mathrm{mV}^{2}$
$\mathrm{E}=\frac{3}{4} \frac{\mathrm{mv}^{2}}{\mathrm{qa}}$
Option $(\mathrm{B})$
Rate of work done $\mathrm{P}=\overrightarrow{\mathrm{F}} \cdot \overrightarrow{\mathrm{V}}=\mathrm{FV} \cos \theta=\mathrm{FV}$
Power $=q \mathrm{EV}$
Power $=\mathrm{q}\left(\frac{3}{4} \frac{\mathrm{mV}^{2}}{\mathrm{qa}}\right) \mathrm{V}$
Power $=9 \frac{3}{4} \frac{\mathrm{mV}^{3}}{\mathrm{qa}}$
Power $=\frac{3}{4} \frac{\mathrm{mV}^{3}}{\mathrm{a}}$
Option $(\mathrm{C})$
Angle between electric force and velocity is $90^{\circ},$ hence rate of work done will be zero at $\mathrm{Q} .$
Option (D)
Initial angular momentum $\mathrm{L}_{\mathrm{i}}=\mathrm{mVa}$
Final angular momentum $\mathrm{L}_{\mathrm{f}}=\mathrm{m}(2 \mathrm{V})$ (2a)
Change in angular momentum $\mathrm{L}_{\mathrm{f}}-\mathrm{L}_{\mathrm{i}}=3 \mathrm{mVa}$
Standard 12
Physics

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