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Section 12.3 Highly Symmetric Surfaces

One of the most fundamental examples in electromagnetism is the electric field of a point charge.

The electric field of a point charge \(q\) at the origin is given by

\begin{equation} \EE = \frac{q}{4\pi\epsilon_0} \frac{\rhat}{r^2} = \frac{q}{4\pi\epsilon_0} \frac{x\,\xhat+y\,\yhat+z\,\zhat}{(x^2+y^2+z^2)^{3/2}}\tag{12.3.1} \end{equation}

where \(\rhat\) is now the unit vector in the radial direction in spherical coordinates. The first expression clearly indicates both the spherical symmetry of \(\EE\) and its \(\frac{1}{r^2}\) fall-off behavior, whereas the second expression does neither. Given the electric field, Gauss' Law allows one to determine the total charge inside any closed surface, namely

\begin{equation} \frac{q}{\epsilon_0} = \Sint \EE\cdot d\AA\tag{12.3.2} \end{equation}

which is of course just the Divergence Theorem.

It is easy to determine \(d\AA\) on the sphere by inspection; we nevertheless go through the details of the differential approach for this case. We use “physicists' conventions” for spherical coordinates, so that \(\theta\) is the angle from the North Pole, and \(\phi\) the angle in the \(xy\)-plane. We use the obvious families of curves, namely the lines of latitude and longitude. Starting either from the general formula for \(d\rr\) in spherical coordinates, namely

\begin{equation} d\rr = dr\,\rhat + r\,d\theta\,\that + r\sin\theta\,d\phi\,\phat ,\tag{12.3.3} \end{equation}

or directly using the geometry behind that formula, one quickly arrives at

\begin{align*} d\rr_1 \amp= r\,d\theta\,\that ,\\ d\rr_2 \amp= r\sin\theta\,d\phi\,\phat ,\\ d\AA \amp= d\rr_1 \times d\rr_2 = r^2\sin\theta\,d\theta\,d\phi\,\rhat , \end{align*}

so that

\begin{equation} \Sint \EE\cdot d\AA = \int_0^{2\pi} \int_0^\pi \frac{q}{4\pi\epsilon_0} \frac{\rhat}{r^2} \cdot r^2 \sin\theta\,d\theta\,d\phi \,\rhat = \frac{q}{\epsilon_0}\tag{12.3.4} \end{equation}

as expected.