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Question

# A charge q of 10.0 C is placed somewhere in space. What is the work required to bring a charge of 1.0 mC from a point X, 10.0 m from q, to a point Y, 2.0 m from q? Does the answer depend on which path the charge follows?

Solution

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The work done in moving the charge from $\textbf{X}$ to $\textbf{Y}$ can be calculated as follows

$W=q\Delta V=q(V_{Y}-V_{X})=q\left[ \frac{kQ}{r_{y}}-\frac{KQ}{r_{x}} \right]$

where $(q)$ is the charge we would like to move, and $(Q)$ is the $(10$ C)

\begin{align*} W&=KqQ\left[ \frac{1}{r_{y}}-\frac{1}{r_{x}} \right] \\ &=(8.99 \times 10^{9} \mathrm{~ N\cdot m^{2}/C^{2}})\times ( 1\times 10^{-3} \mathrm{~ C}) \times ( 10 \mathrm{~ C}) \left[ \frac{1}{2 \mathrm{~ m}}-\frac{1}{10 \mathrm{~ m}} \right] \end{align*}

$W=3.6 \times 10^{7} \mathrm{~ J}$

The work done doesn't depend on the path followed by the charge.

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