The three parallel planes of charge shown in FIGURE P24.44 have surface charge densities ─ ½ η , η , and ─ ½ η. Find the electric fields $\overrightarrow{E_A}$ to $\overrightarrow{E_D}$ in regions A to D. The upward direction is the + y-direction.

A rod of length $L$ lies along the $y$-axis with its center at the origin. The rod has a nonuniform linear charge density $λ=a|y|$, where a is a constant with the units C/m². Determine the constant a in terms of $L$ and the rod's total charge $Q$.

A thin rod of length L and total charge Q has the nonuniform linear charge distribution $\lambda \left(x\right)={\lambda }_{0}x/L$, where x is measured from the rod's left end. What is ${\lambda }_0$ in terms of Q and L?

FIGURE EX24.2 shows a cross section of two concentric spheres. The inner sphere has a negative charge. The outer sphere has a positive charge larger in magnitude than the charge on the inner sphere. Draw this figure on your paper, then draw electric field vectors showing the shape of the electric field.

The electric field is constant over each face of the tetrahedron shown in FIGURE EX24.4. Does the box contain positive charge, negative charge, or no charge? Explain.

Charges $q_1=-4Q$ and $q_2=+2Q$ are located at $\text{𝓍}=-a$ and $\text{𝓍}=+a$, respectively. What is the net electric flux through a sphere of radius $2a$ centered (a) at the origin and (b) at $\text{𝓍}=2a$?

A Geiger counter is used to detect charged particles emitted by radioactive nuclei. It consists of a thin, positively charged central wire of radius Rₐ surrounded by a concentric conducting cylinder of radius Rᵦ with an equal negative charge (Fig. 23–57). The charge per unit length on the inner wire is λ (units C/m). The interior space between wire and cylinder is filled with low-pressure inert gas. Charged particles ionize some of these gas atoms; the resulting free electrons are attracted toward the positive central wire. If the radial electric field is strong enough, the freed electrons gain enough energy to ionize other atoms, causing an “avalanche” of electrons to strike the central wire, generating an electric “signal.” Find the expression for the electric field between the wire and the cylinder, and (b) show that the potential difference between Rₐ and Rᵦ is Vₐ - Vᵦ = ( λ / 2π∊₀ ) ln( Rᵦ/Rₐ) .

An infinite slab of charge is centered in the xy-plane. It has charge density $\rho ={\rho }_0{e}^{-\mid z\mid /z_e}$, where ρ₀ and z₀ are constants. This is a charge density that decreases exponentially as you move away from z = 0 in either the positive or negative direction. Find the electric field strength at distance z from the center of the slab.

A sphere of radius R has total charge Q. The volume charge density (C/m³) within the sphere is p(r) = C/r², where C is a constant to be determined. Use Gauss’s law to find an expression for the electric field strength E inside the sphere, r ≤ R, in terms of Q and R.