Electric potential of a point charge is the building block for electric potential by more complex charge distributions as we will see in the next section.
Since all points a distance \(r\) away from the point charge are at the same potential, equipotential surfaces here are spherical shells with center at the charge. This equation is the building block for electric potential by more complex charge distributions as we will see in the next section.
Subsection31.3.1(Calculus) Derivation of the Formula for Electric Potential for Point Charge
To derive formula in (31.9) from the definition of electric potential given in Section 31.1, we need to perform an integral from the reference point, which is at \(r=\infty\) to the field point P, which is at \(r=r\text{.}\) The integral is required to obtain work done because the electric force varies along the path. Setting the test charge \(Q\) to unity makes the integral equal to the electric potential at P.
Since electric field is radial, the dot product with \(d\vec r\) gives
\begin{equation*}
\vec E \cdot d\vec r = E\: dr = \frac{1}{4\pi\epsilon_0}\,\frac{q}{r^2}dr,
\end{equation*}
where I have replaced the magitude of electric field \(E\) by its formula obtained in previous chapter. With reference at \(r=\infty\text{,}\) upon doing the integration we get the desired result.
(b) What is the potential energy of an electron when placed \(5.29\times 10^{−11}\text{ m}\) from a proton? State both in \(\text{eV}\) and \(\text{J}\text{.}\)
(a) Use \(\phi\) for a point charge, (b) \(Q\times \phi\) gives potential energu of \(Q\text{,}\) (c) No additional work required, just the sign needs to be right.
(c) This requires no additional work since absolute value of the charge of an electron is same as that of proton. The only difference from (a) is the sign.
