24. Electric Force & Field; Gauss' Law

Charge $q$ is distributed uniformly throughout the volume of an insulating sphere of radius $R=4.00$ cm. At a distance of $r=8.00$ cm from the center of the sphere, the electric field due to the charge distribution has magnitude $E=940$ N/C. What is the volume charge density for the sphere?

The nuclei of large atoms, such as uranium, with $92$ protons, can be modeled as spherically symmetric spheres of charge. The radius of the uranium nucleus is approximately $7.4\times {10}^{-15}$ m. What is the electric field this nucleus produces just outside its surface?

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?

(II) Suppose the thick spherical shell of Problem 29 is a conductor. It carries a total net charge Q and at its center there is a point charge +q. What total charge is found on the inner surface of the shell?

An infinitely long cylindrical conductor has radius $r$ and uniform surface charge density $\sigma$. In terms of $\sigma$, what is the magnitude of the electric field produced by the charged cylinder at a distance $r>R$ from its axis? Then, express the result in terms of $\lambda$ and show that the electric field outside the cylinder is the same as if all the charge were on the axis.

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.

A very large, horizontal, nonconducting sheet of charge has uniform charge per unit area $\(\sigma\)=5.00\(\times\)10^{-6}$ C/m². A small sphere of mass $m=8.00\(\times\)10^{-6}$ kg and charge $q$ is placed $3.00$ cm above the sheet of charge and then released from rest. What is $q$ if the sphere is released $1.50$ cm above the sheet?

You measure an electric field of $1.25\times {10}^6$ N/C at a distance of $0.150$ m from a point charge. There is no other source of electric field in the region other than this point charge.

(a) What is the electric flux through the surface of a sphere that has this charge at its center and that has radius $0.150$ m?

(b) What is the magnitude of this charge?

A spherical, thin conducting shell of radius 8cm has a charge of –6C. If a 4C charge were placed at the center of the shell, what is the electric field 4 cm from the center? At 12 cm?

A hollow, conducting sphere with an outer radius of $0.250$ m and an inner radius of $0.200$ m has a uniform surface charge density of $+6.37\times {10}^{-6}$ C/m². A charge of $-0.500$ $\mu$C is now introduced at the center of the cavity inside the sphere. What is the new charge density on the outside of the sphere?

Charge $q$ is distributed uniformly throughout the volume of an insulating sphere of radius $R = 4.00$ cm. At a distance of $r = 8.00$ cm from the center of the sphere, the electric field due to the charge distribution has magnitude $E = 940$ N/C. What is the electric field at a distance of $2.00$ cm from the sphere's center?

An infinite slab of charge of thickness 2𝒵₀ lies in the xy-plane between 𝒵 = -𝒵₀ and 𝒵 = +𝒵₀ . The volume charge density p (C/m³) is a constant. Find an expression for the electric field strength above the slab (𝒵 ≥ 𝒵₀).

Some planetary scientists have suggested that the planet Mars has an electric field somewhat similar to that of the earth, producing a net electric flux of $-3.63\(\times\)10^{16}$ Nm²/C at the planet's surface. Calculate the electric field at the planet's surface (refer to the astronomical data inside the back cover).