Total Charge

Section 9.3 Total Charge

If we know the charge density \(\rho\) in some volume of space, we can find the total charge by chopping up the volume into lots of small (actually infinitesimal) volumes \(d\tau\text{,}\) finding the charge on each \(\rho(\vec{r})\, d\tau\text{,}\) and adding up (integrating) the charges from each of the small volumes.

\begin{equation} Q_{\textrm{tot}} = \int_{\textrm{vol.}} \rho(\vec{r})\, d\tau .\tag{9.3.1} \end{equation}

Similarly, if a surface charge density \(\sigma\) in distributed across some surface, we can find the total charge by chopping up the surface into lots of small (actually infinitesimal) volumes \(dA\text{,}\) finding the charge on each \(\sigma(\vec{r})\, dA\text{,}\) and adding up (integrating) the charges from each of the small volumes.

\begin{equation} Q_{\textrm{tot}} = \int_{\textrm{surf.}} \sigma(\vec{r})\, dA .\tag{9.3.2} \end{equation}

Alternatively, if a linear charge density \(\lambda\) in distributed along some curve, we can find the total charge by chopping up the curve into lots of small (actually infinitesimal) volumes \(d\vert\vec{r}\vert\text{,}\) finding the charge on each \(\lambda(\vec{r})\, d\vert\vec{r}\vert\text{,}\) and adding up (integrating) the charges from each of the small volumes.

\begin{equation} Q_{\textrm{tot}} = \int_{\textrm{surf.}} \lambda(\vec{r})\, d\vert\vec{r}\vert .\tag{9.3.3} \end{equation}

Activity 9.3.1. Total charge on a spherical shell.

Suppose that a spherical shell has inner radius \(a\) and outer radius \(b\) with a spherically symmetric internal charge density \(\rho(r)=10k\, r^2\text{.}\) What is the total charge?

Answer.

Recall that the volume element in spherical coordinates is given by \(d\tau=r^2\sin\theta\,dr\,d\theta\,d\phi\text{.}\) If the charge density on the spherical shell is \(\rho(r)=10k\, r^2\) (in spherical coordinates), then the total charge on the shell is given by

\begin{align*} Q_{\textrm{tot}} \amp= \int_{\textrm{shell}}\rho(\vec{r})\,d\tau\\ \amp= \int_0^{2\pi} \int_0^\pi \int_a^b 10k\,r^2 (r^2\sin\theta\,dr\,d\theta\,d\phi)\\ \amp= 8k\pi\, (b^5-a^5) . \end{align*}

What are the dimensions of \(k\text{?}\) You should verify that \(Q_{\textrm{tot}}\) has the correct dimensions.

Activity 9.3.2. Total charge on a cylindrical shell.

Suppose that a cylindrical shell has inner radius \(a\) and outer radius \(b\) with a cylindrically symmetric internal charge density \(\rho(r)=10k\, s^2\text{.}\) What is the total charge?

Answer.

The volume in cylindrical coordinates is given by \(d\tau=s\,ds\,d\phi\,dz\text{.}\) If there are no limits on \(z\text{,}\) the cylinder is infinite, and the total charge will also be infinite. So assume instead that the cylinder has height \(L\text{.}\) If the charge density on the cylindrical shell is \(\rho(s)=10k\,s^2\) (in cylindrical coordinates), then the total charge on the shell is given by

\begin{align*} Q_{\textrm{tot}} \amp= \int_{\textrm{shell}}\rho(\vec{r})\,dV\\ \amp= \int_0^L \int_0^{2\pi} \int_a^b 10k\, s^2 (s\,ds\,d\phi\,dz)\\ \amp= 5k\pi\, (b^4-a^4) L . \end{align*}

You can find the total charge per unit length by dividing this answer by \(L\text{.}\)

What are the dimensions of \(k\text{?}\) You should verify that \(Q_{\textrm{tot}}\) has the correct dimensions.

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