24. Electric Force & Field; Gauss' Law

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?

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 $-3.63\times10^{16}$ Nm²/C at the planet's surface. Calculate the charge density on Mars, assuming all the charge is uniformly distributed over the planet's surface.

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 $-3.63\times10^{16}$ 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).

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 total electric charge on the planet.

A very long uniform line of charge has charge per unit length $4.80$ $\mu$C/m and lies along the $x$-axis. A second long uniform line of charge has charge per unit length $-2.40$ $\mu$C/m and is parallel to the $x$-axis at $y=0.400$ m. What is the net electric field (magnitude and direction) at the following points on the $y$-axis: (a) $y=0.200$ m and (b) $y=0.600$ m?

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 $+6.37\times10^{-6}$ C/m². A charge of $−0.500$ $\mu$C is now introduced at the center of the cavity inside the sphere. What is the electric flux through a spherical surface just inside the inner surface of the sphere?

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 $+6.37\times10^{-6}$ C/m². A charge of $−0.500$ $\mu$C is now introduced at the center of the cavity inside the sphere. Calculate the strength of the electric field just outside the sphere?

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?

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?

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?

The earth has a vertical electric field at the surface, pointing down, that averages 100 N/C. This field is maintained by various atmospheric processes, including lightning. What is the excess charge on the surface of the earth?

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).

An early model of the atom, proposed by Rutherford after his discovery of the atomic nucleus, had a positive point charge + Ze (the nucleus) at the center of a sphere of radius R with uniformly distributed negative charge -Ze. Z is the atomic number, the number of protons in the nucleus and the number of electrons in the negative sphere. A uranium atom has Z = 92 and 𝑅 = 0.10nm . What is the electric field strength at r = ½𝑅?

A spherical ball of charge has radius R and total charge Q. The electric field strength inside the ball (r ≤ R ) is $E\left(r\right)=r^4{E}_{max}/R^4$. Find an expression for the volume charge density ρ(r) inside the ball as a function of r.

FIGURE EX24.18 shows three charges. Draw these charges on your paper four times. Then draw two-dimensional cross sections of three-dimensional closed surfaces through which the electric flux is (a) $2q/{ϵ}_0$, (b) $q/{ϵ}_0$, (c) 0, and (d) $5q/{ϵ}_0$.