JEE Main & Advanced Physics Electrostatics & Capacitance Electrostatic Potential Energy

(1) Work done in bringing the given charge from infinity to a point in the electric field is known as potential energy of the charge. Potential can also be written as potential energy per unit charge. i.e. \[V=\frac{W}{Q}=\frac{U}{Q}\].

(2) Potential energy of a system of two charge 

Potential energy of \[{{Q}_{1}}=\] Potential energy of \[{{Q}_{2}}=\] potential energy of system \[U=k\frac{{{Q}_{1}}{{Q}_{2}}}{r}\]

In C.G.S.  \[U=\frac{{{Q}_{1}}{{Q}_{2}}}{r}\]

(3) Potential energy of a system of n charge

It is given by \[U=\frac{k}{2}\sum\limits_{\begin{smallmatrix}  i,\,j \\  i\,\ne j \end{smallmatrix}}^{n}{\frac{{{Q}_{i}}{{Q}_{j}}}{{{r}_{ij}}}}\]         \[\left( k=\frac{1}{4\pi {{\varepsilon }_{0}}} \right)\]

The factor of \[\frac{1}{2}\] is applied only with the summation sign because on expanding the summation each pair is counted twice.

For a system of 3 charges \[U=k\,\left( \frac{{{Q}_{1}}{{Q}_{2}}}{{{r}_{12}}}+\frac{{{Q}_{2}}{{Q}_{3}}}{{{r}_{23}}}+\frac{{{Q}_{1}}{{Q}_{3}}}{{{r}_{13}}} \right)\]

(4) Work energy relation : If a charge moves from one position to another position in an electric field so it's potential energy change and work done by external force for this change is \[W={{U}_{f}}-{{U}_{i}}\]

(5) Electron volt (eV) : It is the smallest practical unit of energy used in atomic and nuclear physics. As electron volt is defined as "the energy acquired by a particle having one quantum of charge (1e), when accelerated by 1volt" i.e. \[1eV=1.6\times {{10}^{-19}}C\times \frac{1J}{C}\]\[=1.6\times {{10}^{-19}}J\]\[=1.6\times {{10}^{-12}}erg\]

(6) Electric potential energy of a uniformly charged sphere : Consider a uniformly charged sphere of radius R having a total charge Q. The electric potential energy of this sphere is equal to the work done in bringing the charges from infinity to assemble the sphere.   \[U=\frac{3{{Q}^{2}}}{20\pi {{\varepsilon }_{0}}R}\]

(7) Electric potential energy of a uniformly charged thin spherical shell : It is given by the following formula  \[U=\frac{{{Q}^{2}}}{8\pi {{\varepsilon }_{0}}R}\]

(8) Energy density : The energy stored per unit volume around a point in an electric field is given by

\[{{U}_{e}}=\frac{U}{\text{Volume}}=\frac{1}{2}{{\varepsilon }_{0}}{{E}^{2}}\]. If in place of vacuum some medium is present then \[{{U}_{e}}=\frac{1}{2}{{\varepsilon }_{0}}{{\varepsilon }_{r}}{{E}^{2}}\]