Chapter 21

Compute $\mathrm{\Delta }\psi$ for a capacitor. You have a parallelplate capacitor filled with an oil of dielectric constant $D=2$, with a distance of $0.20\mathrm{cm}$ between two $20\mathrm{cm}\times$ $20\mathrm{cm}$ plates. The upper plate has a total charge of ${10}^{-10}\mathrm{C}$ relative to the lower plate. Neglecting edge effects, what is the potential difference between the upper and lower plates?

A dipole moment. What is the dipole moment of two charges - e and +e that are $1\mathrm{A}$ apart? (The unit of dipole moment is 1 debye $=1\mathrm{D}=3.33\times {10}^{-30}\mathrm{Cm}$.)

The charge-dipole interaction of ${\mathrm{Na}}^4$ with water. What is the maximum attraction in units of $RT$ : (a) between a bare ${\mathrm{Na}}^{+}$ion and a water molecule in air? (b) between a hydrated ${\mathrm{Na}}^{+}$ion in water and a water molecule? Take a sphere with radius $0.95$ for the bare ${\mathrm{Na}}^{+}$ion and a sphere with radius $2.3$ A for the hydrated ${\mathrm{Na}}^{+}$ion. Model a water molecule as a sphere with radius $1.4\text{AA}$ and a point dipole of moment $1.85\mathrm{D}$ at its center.

Water is a dipole. A water molecule in vacuum has a dipole moment of $0.62\times {10}^{-29}\mathrm{C}\mathrm{m}$. (a) Assuming that this dipole moment is due to charges $+e$ and $-e$ at distance $\ell$, calculate $\ell$ in $A$. (b) Consider the water molecule as a sphere with radius 1.5A. If the dipole moment in (a) is due to charges $+q$ and $-q$ at the north and south poles of the sphere, how large is $q$ ?

The potential around a charged sphere. You have an evenly charged spherical surface with radius $a$ and a total charge $q$ in a medium with dielectric constant $D$. Derive the potential inside and outside the surface by using Coulomb's law and integrating over the charged surface.

The potential around a cylindrical shell. You have a hollow inner cylinder with radius $a_1$ surrounded by a concentric outer cylinder with radius $a_2>a_1$. Charges $-\lambda$ and $+\lambda$ per unit length are distributed uniformly over the inner and outer cylinders, respectively. The shells are in a medium with dielectric constant $D$. (a) What is the potential $\psi (r)$ as a function of the axial distance $r$ ? (b) What is the capacitance per unit length of the arrangement of cylinders?