Magnetics 1 Question 4
4. A particle of mass $m$ and charge $q$ is in an electric and magnetic field is given by
$$ \mathbf{E}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}, \mathbf{B}=4 \hat{\mathbf{j}}+6 \hat{\mathbf{k}} $$
The charged particle is shifted from the origin to the point $P(x=1 ; y=1)$ along a straight path. The magnitude of the total work done is
(Main 2019, 11 Jan II)
(a) $(0.35) q$
(b) $(0.15) q$
(c) $(2.5) q$
(d) $5 q$
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Answer:
Correct Answer: 4. (d)
Solution:
- Here, $E=2 \hat{i}+3 \hat{j}, B=4 \hat{j}+6 \hat{k}$,
$q=$ charge on a particle.
Initial position, $r _1=(0,0)$
Final position, $r _2=(1,1)$
Net force experienced by charge particle inside electromagnetic field is
$$ \begin{aligned} & F _{net}=q E+q(v \times B) \\ & =q(2 \hat{i}+3 \hat{j}) \quad[\text { Here, } v \times B=0] \\ & =(2 q \hat{i}+3 q \hat{j}) \\ & \therefore \quad d W=F _{\text {net }} \cdot \mathbf{d r} \\ & \Rightarrow \quad \int d W=\int _{r _1}^{r _2}(2 q \hat{i}+3 q \hat{j}) \cdot(d x \hat{i}+d y \hat{j}) \\ & {[\text { Here, } d r=d x \hat{i}+d y \hat{j}]} \\ & \Rightarrow \quad W=2 q \int _0^{1} d x+3 q \int _0^{1} d y \\ & \text { or } \quad W=2 q+3 q \text { or } W=5 q \end{aligned} $$
