Chapter 5

Let $V=10(\rho +1)z^2\cos \varphi \mathrm{V}$ in free space. $(a)$ Let the equipotential surface $V=20\mathrm{V}$ define a conductor surface. Find the equation of the conductor surface. $(b)$ Find $\rho$ and $\mathbf{E}$ at that point on the conductor surface where $\varphi =$ $0.2\pi$ and $z=1.5.(c)$ Find $\left|{\rho }_S\right|$ at that point.

A coaxial transmission line has inner and outer conductor radii $a$ and $b$. Between conductors \((a<\rho

Given the potential field $V=100xz/(x^2+4)\mathrm{V}$ in free space: $(a)$ Find $\mathrm{D}$ at the surface $z=0.(b)$ Show that the $z=0$ surface is an equipotential surface. ( $c$ ) Assume that the $z=0$ surface is a conductor and find the total charge on that portion of the conductor defined by \(0

Two parallel circular plates of radius $a$ are located at $z=0$ and $z=d$. The top plate $(z=d)$ is raised to potential $V_0;$ the bottom plate is grounded. Between the plates is a conducting material having radial-dependent conductivity, $\sigma (\rho )={\sigma }_0\rho$, where ${\sigma }_0$ is a constant. $(a)$ Find the $\rho$ -independent electric field strength, $\mathbf{E}$, between plates. $(b)$ Find the current density, $\mathbf{J}$ between plates. ( $c$ ) Find the total current, $I$, in the structure. $(d)$ Find the resistance between plates.

Let $V=20x^{2}yz-10z^2\mathrm{V}$ in free space. $(a)$ Determine the equations of the equipotential surfaces on which $V=0$ and $60\mathrm{V}.(b)$ Assume these are conducting surfaces and find the surface charge density at that point on the $V=60\mathrm{V}$ surface where $x=2$ and $z=1$. It is known that $0\le V\le 60\mathrm{V}$ is the field-containing region. (c) Give the unit vector at this point that is normal to the conducting surface and directed toward the $V=0$ surface.

Let the surface $y=0$ be a perfect conductor in free space. Two uniform infinite line charges of $30\mathrm{nC}/\mathrm{m}$ each are located at $x=0,y=1$, and $x=0,y=2\cdot (a)$ Let $V=0$ at the plane $y=0$, and find $V$ at $P(1,2,0)$. (b) Find $\mathbf{E}$ at $P$.

At a certain temperature, the electron and hole mobilities in intrinsic germanium are given as $0.43$ and $0.21{\mathrm{m}}^2/\mathrm{V}\cdot \mathrm{s}$, respectively. If the electron and hole concentrations are both $2.3\times {10}^{19}{\mathrm{m}}^{-3}$, find the conductivity at this temperature.

Electron and hole concentrations increase with temperature. For pure silicon, suitable expressions are ${\rho }_h=-{\rho }_e=6200T^{1.5}{e}^{-7000/T}\mathrm{C}/{\mathrm{m}}^3$. The functional dependence of the mobilities on temperature is given by ${\mu }_h=2.3\times {10}^5{T}^{-2.7}{\mathrm{m}}^2/\mathrm{V}\cdot \mathrm{s}$ and ${\mu }_e=2.1\times {10}^5{T}^{-2.5}{\mathrm{m}}^2/\mathrm{V}\cdot \mathrm{s}$, where the temperature, $T$, is in degrees Kelvin. Find $\sigma$ at: (a) $0^{\circ }\mathrm{C};(b){40}^{\circ }\mathrm{C};(c){80}^{\circ }\mathrm{C}$.

Atomic hydrogen contains $5.5\times {10}^{25}$ atoms $/{\mathrm{m}}^3$ at a certain temperature and pressure. When an electric field of $4\mathrm{kV}/\mathrm{m}$ is applied, each dipole formed by the electron and positive nucleus has an effective length of $7.1\times {10}^{-19}\mathrm{m}$. (a) Find $P$. (b) Find ${ϵ}_r$.

A coaxial conductor has radii $a=0.8\mathrm{mm}$ and $b=3\mathrm{mm}$ and a polystyrene dielectric for which ${ϵ}_r=2.56.$ If $\mathbf{P}=(2/\rho ){\mathbf{a}}_{\rho }\mathrm{nC}/{\mathrm{m}}^2$ in the dielectric, find (a) $\mathbf{D}$ and $\mathbf{E}$ as functions of $\rho ;(b)V_{ab}$ and ${\chi }_e\cdot (c)$ If there are $4\times {10}^{19}$ molecules per cubic meter in the dielectric, find $\mathbf{p}(\rho )$