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An infinite sheet carrying a uniform surface charge density $\sigma$ lies on the $x y$ -plane. The work done to carry a charge $q$ from the point $\mathbf{A}=a(\mathbf{i}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})$ to the point $\mathbf{B}=a(\mathbf{i}-2 \hat{\mathbf{j}}+6 \hat{\mathbf{k}})$
(where $a$ is a constant with the dimension of length and $\varepsilon_{0}$ is the permittivity of free space) is
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(where $a$ is a constant with the dimension of length and $\varepsilon_{0}$ is the permittivity of free space) is
Solution:
2852 Upvotes
Verified Answer
The correct answer is:
$\frac{3 \sigma a q}{2 \varepsilon_{0}}$
The given
$$
\begin{aligned}
\mathbf{A} &=a(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}) \\
\mathbf{B} &=a(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}) \\
\mathbf{A B} &=\mathbf{O B}-\mathbf{O A} \\
&=a(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+6 \hat{\mathbf{k}})-a(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}) \\
\mathbf{A B} &=a(-4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})
\end{aligned}
$$
Work done $=q\left(\frac{\sigma}{2 \varepsilon_{0}}\right) \hat{\mathbf{k}} \cdot \mathbf{A B} \quad$ (along to $Z$ -axisi
$$
\begin{array}{r}
=q\left(\frac{\sigma}{2 \varepsilon_{0}}\right) \hat{\mathbf{k}} \cdot a(-4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})=\frac{3 q \sigma a}{2 \varepsilon_{0}} \\
(\because \mathbf{i} \cdot \mathbf{i}=\mathbf{j} \cdot \mathbf{j}=\mathbf{i} \cdot \mathbf{k}=1 \text { and } \mathbf{i} \cdot \mathbf{j}=\mathbf{j} \cdot \hat{\mathbf{k}}=\hat{\mathbf{k}} \cdot \hat{\mathbf{i}}=0)
\end{array}
$$
$$
\begin{aligned}
\mathbf{A} &=a(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}) \\
\mathbf{B} &=a(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}) \\
\mathbf{A B} &=\mathbf{O B}-\mathbf{O A} \\
&=a(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+6 \hat{\mathbf{k}})-a(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}) \\
\mathbf{A B} &=a(-4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})
\end{aligned}
$$
Work done $=q\left(\frac{\sigma}{2 \varepsilon_{0}}\right) \hat{\mathbf{k}} \cdot \mathbf{A B} \quad$ (along to $Z$ -axisi
$$
\begin{array}{r}
=q\left(\frac{\sigma}{2 \varepsilon_{0}}\right) \hat{\mathbf{k}} \cdot a(-4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})=\frac{3 q \sigma a}{2 \varepsilon_{0}} \\
(\because \mathbf{i} \cdot \mathbf{i}=\mathbf{j} \cdot \mathbf{j}=\mathbf{i} \cdot \mathbf{k}=1 \text { and } \mathbf{i} \cdot \mathbf{j}=\mathbf{j} \cdot \hat{\mathbf{k}}=\hat{\mathbf{k}} \cdot \hat{\mathbf{i}}=0)
\end{array}
$$
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