Chapter 3

Suppose that the Faraday concentric sphere experiment is performed in free space using a central charge at the origin, $Q_1$, and with hemispheres of radius a. A second charge $Q_2$ (this time a point charge) is located at distance $R$ from $Q_1$, where $R>>a.(a)$ What is the force on the point charge before the hemispheres are assembled around $Q_1?$ (b) What is the force on the point charge after the hemispheres are assembled but before they are discharged? ( $c$ ) What is the force on the point charge after the hemispheres are assembled and after they are discharged? ( $d$ ) Qualitatively, describe what happens as $Q_2$ is moved toward the sphere assembly to the extent that the condition $R>>a$ is no longer valid.

An electric field in free space is $\mathbf{E}=\left(5z^2/{ϵ}_0\right){\hat{\mathbf{a}}}_z\mathrm{V}/\mathrm{m}$. Find the total charge contained within a cube, centered at the origin, of $4-\mathrm{m}$ side length, in which all sides are parallel to coordinate axes (and therefore each side intersects an axis at $\pm 2$ ).

The cylindrical surface $\rho =8\mathrm{cm}$ contains the surface charge density, ${\rho }_S=$ $5e^{-20|z|}\mathrm{nC}/{\mathrm{m}}^2.(a)$ What is the total amount of charge present? $(b)$ How much electric flux leaves the surface \(\rho=8 \mathrm{~cm}, 1 \mathrm{~cm}

In free space, a volume charge of constant density ${\rho }_v={\rho }_0$ exists within the region \(-\infty

Volume charge density is located in free space as ${\rho }_v=2e^{-1000r}\mathrm{nC}/{\mathrm{m}}^3$ for \(0

Use Gauss's law in integral form to show that an inverse distance field in spherical coordinates, $\mathbf{D}=A{a}_r/r$, where $A$ is a constant, requires every spherical shell of $1\mathrm{m}$ thickness to contain $4\pi A$ coulombs of charge. Does this indicate a continuous charge distribution? If so, find the charge density variation with $r$.

A uniform volume charge density of $80\mu \mathrm{C}/{\mathrm{m}}^3$ is present throughout the region \(8 \mathrm{~mm}10 \mathrm{~mm}\), find $D_r$ at $r=20\mathrm{mm}$.

An infinitely long cylindrical dielectric of radius $b$ contains charge within its volume of density ${\rho }_v=a{\rho }^2$, where $a$ is a constant. Find the electric field strength, $\mathbf{E}$, both inside and outside the cylinder.

The sun radiates a total power of about $3.86\times {10}^{26}$ watts $(\mathrm{W})$. If we imagine the sun's surface to be marked off in latitude and longitude and assume uniform radiation, $(a)$ what power is radiated by the region lying between latitude ${50}^{\circ }\mathrm{N}$ and ${60}^{\circ }\mathrm{N}$ and longitude ${12}^{\circ }\mathrm{W}$ and ${27}^{\circ }\mathrm{W}?(b)$ What is the power density on a spherical surface $93,000,000$ miles from the sun in $\mathrm{W}/{\mathrm{m}}^2?$

Spherical surfaces at $r=2,4$, and $6\mathrm{m}$ carry uniform surface charge densities of $20\mathrm{nC}/{\mathrm{m}}^2,-4\mathrm{n}\mathrm{C}/{\mathrm{m}}^2$, and ${\rho }_{\mathrm{so}}$, respectively. $(a)$ Find $\mathbf{D}$ at $r=1$, 3 , and $5\mathrm{m}$. (b) Determine ${\rho }_{S0}$ such that $\mathbf{D}=0$ at $r=7\mathrm{m}$.