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05:34 Brazilian Center for Research in Physics

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Problem 57

(a) An insulating sphere with radius a has a uniform charge density $\rho$. The sphere is not centered at the origin but at $\vec{r} = \vec{b}$. Show that the electric field inside the sphere is given by $\overrightarrow{$} = \rho(\vec{r} - \vec{b} )/3\epsilon_0$. (b) An insulating sphere of radius R has a spherical hole of radius a located within its volume and centered a distance b from the center of the sphere, where a$< b < R$(a cross section of the sphere is shown in$\textbf{Fig. P22.57}$). The solid part of the sphere has a uniform volume charge density$\rho$. Find the magnitude and direction of the electric field$\overrightarrow{E}$inside the hole, and show that$\overrightarrow{E}$is uniform over the entire hole. [$Hint:$Use the principle of superposition and the result of part (a).] Answer a)$E_{x}(0)=0$b)$\overrightarrow{\boldsymbol{E}}=\left(\rho_{0} d / 3 \boldsymbol{\epsilon}_{0}\right)(x /|x|) \hat{i}$c)$\overline{\boldsymbol{E}}=\left(\rho_{0} x^{3} / 3 \epsilon_{0} d^{2}\right) \hat{\boldsymbol{i}}\$

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