Prerequisites

More Total Charge

Recall that the volume element in spherical coordinates is given by $dV=r^2\sin\theta\,dr\,d\theta\,d\phi$. If the charge density on the spherical shell is $\rho®=10kr^2$ (in spherical coordinates), then the total charge on the shell is given by \begin{eqnarray} q_{\hbox{tot}} &=& \int_B\rho\,dV \nonumber\\ &=& \int_0^{2\pi} \int_0^\pi \int_a^b 10k\,r^2 r^2\sin\theta\,dr\,d\theta\,d\phi \nonumber\\ &=& 8k\pi\, (b^5-a^5) \end{eqnarray}

The volume in cylindrical coordinates is given by $dV=r\,dr\,d\phi\,dz$. If there are no limits on $z$, the cylinder is infinite, and the total charge will also be infinite. So assume instead that the cylinder has height $Z$. If the charge density on the cylindrical shell is $\rho®=10kr^2$ (in cylindrical coordinates), then the total charge on the shell is given by \begin{eqnarray} q_{\hbox{tot}} &=& \int_B\rho\,dV \nonumber\\ &=& \int_0^L \int_0^{2\pi} \int_a^b 10k\,r^2 r\,dr\,d\phi\,dz \nonumber\\ &=& 5k\pi\, (b^4-a^4) L \end{eqnarray}

What are the dimensions of $k$? You should verify in each case that $q_{\hbox{tot}}$ has the correct dimensions.


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