Problem 15
Let \(V=10(\rho+1) z^{2} \cos \phi \mathrm{V}\) 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 \(\phi=\) \(0.2 \pi\) and \(z=1.5 .(c)\) Find \(\left|\rho_{S}\right|\) at that point.
Problem 16
A coaxial transmission line has inner and outer conductor radii \(a\) and \(b\).
Between conductors \((a<\rho
Problem 17
Given the potential field \(V=100 x z /\left(x^{2}+4\right) \mathrm{V}\) 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
Problem 18
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.
Problem 19
Let \(V=20 x^{2} y z-10 z^{2} \mathrm{~V}\) 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 \(0 \leq V \leq 60 \mathrm{~V}\) 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.
Problem 21
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 \(P\).
Problem 24
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.
Problem 25
Electron and hole concentrations increase with temperature. For pure silicon, suitable expressions are \(\rho_{h}=-\rho_{e}=6200 T^{1.5} e^{-7000 / T} \mathrm{C} / \mathrm{m}^{3}\). 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) \(0^{\circ} \mathrm{C} ;(b) 40^{\circ} \mathrm{C} ;(c) 80^{\circ} \mathrm{C}\).
Problem 27
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 \(\epsilon_{r}\).
Problem 29
A coaxial conductor has radii \(a=0.8 \mathrm{~mm}\) and \(b=3 \mathrm{~mm}\) and a polystyrene dielectric for which \(\epsilon_{r}=2.56 .\) 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 \(\rho ;(b) V_{a b}\) and \(\chi_{e} \cdot(c)\) If there are \(4 \times 10^{19}\) molecules per cubic meter in the dielectric, find \(\mathbf{p}(\rho)\)
