A flat sheet of paper of area $0.250$ m² is oriented so that the normal to the sheet is at an angle of $60$° to a uniform electric field of magnitude $14$ N/C.

(a) Find the magnitude of the electric flux through the sheet.

(b) Does the answer to part (a) depend on the shape of the sheet? Why or why not?

A flat sheet of paper of area $0.250$ m² is oriented so that the normal to the sheet is at an angle of $60$° to a uniform electric field of magnitude $14$ N/C. For what angle $\varphi$ between the normal to the sheet and the electric field is the magnitude of the flux through the sheet (i) largest and (ii) smallest? Explain your answers.

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 charged paint is spread in a very thin uniform layer over the surface of a plastic sphere of diameter $12.0$ cm, giving it a charge of $-49.0$ $\mu$C. Find the electric field just inside the paint layer.

A charged paint is spread in a very thin uniform layer over the surface of a plastic sphere of diameter $12.0$ cm, giving it a charge of $−49.0$ $μ$C. Find the electric field just outside the paint layer;

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?

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 $7.4\times10^{-15}$ m. The electrons can be modeled as forming a uniform shell of negative charge. What net electric field do they produce at the location of the nucleus?

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.

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 $-3.63\times10^{16}$ 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.

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 $+6.37\times {10}^{-6}$ 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?

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 $+6.37\times10^{-6}$ 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?

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?

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?

A conductor with an inner cavity, like that shown in Fig. $22.23$c, carries a total charge of $+5.00$ nC. The charge within the cavity, insulated from the conductor, is $-6.00$ nC. How much charge is on (a) the inner surface of the conductor and (b) the outer surface of the conductor?

An infinitely long cylindrical conductor has radius $r$ and uniform surface charge density $\sigma$. In terms of $\sigma$ and $R$, what is the charge per unit length $\lambda$ for the cylinder?

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.

A very long conducting tube (hollow cylinder) has inner radius $A$ and outer radius $b$. It carries charge per unit length $+α$, where $\alpha$ is a positive constant with units of C/m. A line of charge lies along the axis of the tube. The line of charge has charge per unit length. Calculate the electric field in terms of $\alpha$ and the distance $r$ from the axis of the tube for (i) ; (ii) ; (iii) $r>b$. Show your results in a graph of $E$ as a function of $R$.

A very long conducting tube (hollow cylinder) has inner radius $A$ and outer radius $b$. It carries charge per unit length $+\alpha$, where $\alpha$ is a positive constant with units of C/m. A line of charge lies along the axis of the tube. The line of charge has charge per unit length. What is the charge per unit length on (i) the inner surface of the tube and (ii) the outer surface of the tube?