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Electrostatic Potential Energy

Electrostatic Potential Energy :- Just like gravitational potential energy of a mass in a gravitational field, we can derive an expression for electrostatic potential energy of a charge in an electric field.

Let us assume that a charge +Q is placed at the origin which produces electric filed $\displaystyle \overrightarrow{E}$ . Let a small positive test charge +q (assumed so small that it does not disturb the configuration of charge +Q at the origin) is moved from point A to point B against the electric field $\displaystyle \overrightarrow{E}$ by applying an external force $\displaystyle {{\overrightarrow{F}}_{ext}}$ .

We assume that the applied external force $\displaystyle {{\overrightarrow{F}}_{ext}}$ is just sufficient to counter the repulsive electrostatic force $\displaystyle {{\overrightarrow{F}}_{E}}$ on the test charge +q so that net force on test charge +q is zero and it moves from A to B without any acceleration. In this situation, work done by external force is negative of work done by the electrostatic force and gets fully stored in the charge +q in the form of its potential energy.

Work done by external force in moving the test charge +q from point A to C through small distance dr is :

$\displaystyle dW=\left( {{\overrightarrow{F}}_{_{ext}}} \right).\left( \overrightarrow{dr} \right)=\left( -{{\overrightarrow{F}}_{_{E}}} \right).\left( \overrightarrow{dr} \right)$

$\displaystyle dW=-{{\overrightarrow{F}}_{_{E}}}.\text{ }\overrightarrow{dr}$

Total work done to move the test charge +q from A to B :

$\displaystyle {{W}_{AB}}=\int\limits_{A}^{B}{dW=}-\int\limits_{A}^{B}{{{\overrightarrow{F}}_{_{E}}}.\text{ }\overrightarrow{dr}}$ …..(1)

This work done against electrostatic force is the electrostatic potential energy difference of charge +q , i.e.,

$\displaystyle {{W}_{AB}}=\Delta U={{U}_{B}}-{{U}_{A}}$ …..(2)

Hence,

Electrostatic potential energy difference between two points A and B as the minimum work done by an external force in moving a test charge q from A to B without acceleration.

Note that work done by an electric field in moving a given charge from one point to another, depends only on the initial and final positions of points. It does not depend on the path chosen in going from one point to the other.

Now suppose that initially the charge +q is at infinite distance away from source charge +Q so that there is no interaction between +q and +Q. We can say that the charge +q possess zero potential energy at this point. So we choose zero potential energy at infinity. Therefore, when point A is at infinity, then from equation (2) we get,

$\displaystyle {{W}_{\infty B}}={{U}_{B}}-{{U}_{\infty }}={{U}_{B}}-0={{U}_{B}}$ …..(3)

Hence,

Electrostatic Potential Energy of a charge q at a point (B) in the electrostatic field due to any charge configuration may be defined as the work done by an external force in bringing the charge q from infinity to that point without acceleration.

Electrostatic Potential Energy

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