Question
A point charge of $10^{- 8}C$ is placed at origin. The work done in moving a point charge $2\mu C$ from point $A(4,4,2)m$ to $B(2,2,1)m$ is $\_\_\_\_$ J.
( $\frac{1}{4\pi\epsilon_{0}} = 9 \times 10^{9}$ in SI units)
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Solution
Work done by external agent :
$$W_{ext} = \Delta U; $$$\Delta U \rightarrow$ Change in potential energy in taking the charge from initial to final configuration
$$\Rightarrow W_{\text{ext~}} = \frac{1}{4\pi\epsilon_{0}}\frac{q_{1}q_{2}}{r_{f}} - \frac{1}{4\pi\epsilon_{0}}\frac{q_{1}q_{2}}{r_{i}}$$
Now, $r_{f} = \sqrt{(2 - 0)^{2} + (2 - 0)^{2} + (1 - 0)^{2}} = 3\text{ }m$
$$\begin{matrix} r_{i} & \ = \sqrt{(4 - 0)^{2} + (4 - 0)^{2} + (2 - 0)^{2}} = 6\text{ }m \\ \therefore\ {\text{ }W}_{ext} & \ = \left( 9 \times 10^{9} \right) \times \left( 10^{- 8} \times 2 \times 10^{- 6} \right)\left\lbrack \frac{1}{3} - \frac{1}{6} \right\rbrack \\ & \ = 3 \times 10^{- 5} \\ & \ = 30 \times 10^{- 6}\text{ }J \end{matrix}$$
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