Chapter Questions
According to this theorem, the Poisson equation $\boldsymbol{\nabla}^{2} V=-\rho_{t} / \epsilon_{0}$ can have only one solution if the potential $V$ is defined at the boundaries of the field.Show that two solutions can differ at most by a constant if the normal component of $\boldsymbol{\nabla} V$ is defined at the boundaries.
A point charge $Q$ lies at a distance $D$ above a grounded conducting plate.(a) Calculate the surface charge density induced on the plate as a function of the radius $r$ from the foot of the perpendicular drawn from the charge.(b) Show that the total induced charge is $-Q$.
A line charge of $\lambda$ coulombs/meter is parallel to a flat conducting plate, at a distance $a$, as in Fig. 11-9. Find $\boldsymbol{E}$ at a point $(x, y)$. The surface charge density on the plate is given by $\epsilon_{0} E$ at $(x, 0)$.
A hollow conducting sphere of radius $a$ contains a point charge $Q$ at the radius $b$ as in Fig. 11-10.(a) Show that the field inside the sphere is the same as if there was no sphere and, instead, a charge $Q^{\prime}=-(a / b) Q$ at $D=(a / b) a$. You can prove this by showing that the $V$ of $Q$ plus $Q^{\prime}$ is uniform over the surface of the sphere.(b) Calculate the force of attraction.(c) Calculate the surface charge density on the inside surface of the conducting sphere.
Show that the potential at the surface of the dielectric, opposite $Q$ in Fig. 11-3, is the same, whether one calculates the field in air, as in Fig. 11-3(a), or in the dielectric, as in Fig. 11-3(c). Set $\epsilon_{r}=3$.
Show that there exist solutions of Laplace's equation that are of the form $X(x)+Y(y)+Z(z)$
(a) Show that $\boldsymbol{V}^{2}(1 / r)=0$.(b) Use this fact to show that$$\frac{\partial}{\partial x} \frac{1}{r}, \quad \frac{\partial^{2}}{\partial x^{2}} \frac{1}{r}, \quad \frac{\partial^{2}}{\partial x \partial y} \frac{1}{r}$$are also solutions of Laplace's equation.
Plot $\exp (-n \pi x / b)$ as a function of $x / b$ from $x / b=0$ to 1 and for $n=1$, 2. and 3 .
Plot $V / V_{0}$ as a function of $y / b$ for the field of Fig. $11-8$ for $x=0$, $x=0.5 b$, and $x=b$, up to $n=100$. You can truncate the series when $\exp (-n \pi x / b)$ is less than $0.01$.