24. Electric Force & Field; Gauss' Law

The earth has a net electric charge that causes a field at points near its surface equal to $150$ $N/C$ and directed in toward the center of the earth. What would be the force of repulsion between two people each with the charge calculated in part (a) and separated by a distance of $100$ m? Is use of the earth's electric field a feasible means of flight? Why or why not? Note: Part (a) asked for what magnitude and sign of charge would a $60$-kg human have to acquire to overcome his or her weight by the force exerted by the earth's electric field.

A charge of $-6.50$ nC is spread uniformly over the surface of one face of a nonconducting disk of radius $1.25$ cm. Why is the field in part (a) stronger than the field in part (b)? Why is the field in part (c) the strongest of the three fields? Note: Part (a) asked to find the magnitude and direction of the electric field this disk produces at a point $P$ on the axis of the disk a distance of $2.00$ cm from its center. Part (b) asked to find the magnitude and direction of the electric field at point $P$, supposing that the charge were all pushed away from the center and distributed uniformly on the outer rim of the disk. Part (c) asked to find the magnitude and direction of the electric field at point $P$ if the charge is all brought to the center of the disk.

The electric field at a point in space is $E=(400\hat{i}+100\hat{j})$ N/C. What is the electric force on a proton at this point? Give your answer in component form.

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. What magnitude of electric field does it produce at the distance of the electrons, which is about $1.0\times {10}^{-10}$ m?

A Van de Graaff generator is a device for generating a large electric potential by building up charge on a hollow metal sphere. A typical classroom-demonstration model has a diameter of 30 cm. What is the electric field strength just outside the surface of the sphere when it is charged to 500,000 V?

What are the strength and direction of the electric field 1.0 mm from a proton?

A hollow copper sphere has inner radius 1.0 cm and outer radius 2.5 cm. A 5.0 A current flows radially outward from the inner surface to the outer surface. What is the electric field strength at r=2.0 cm?

Two point charges, Q₁ = - 32 μC and Q₂ = +45μC, are separated by a distance of 12 cm. The electric field at the point P (see Fig. 21–61) is zero. How far from Q₁ is P?

You are given two unknown point charges, Q₁ and Q₂. At a point on the line joining them, one-third of the way from Q₁ to Q₂, the electric field is zero (Fig. 21–64). What is the ratio Q₁/Q₂?

(II) A positive charge q is placed at the center of a circular ring of radius R. The ring carries a uniformly distributed negative charge of total magnitude -Q. (a) If the charge q is displaced from the center a small distance x as shown in Fig. 21–71, show that it will undergo simple harmonic motion when released. (b) If its mass is m, what is its period?

A point charge of mass 0.145 kg, and net charge +3.40 μC, hangs at rest at the end of an insulating cord above a large sheet of charge. The horizontal sheet of fixed uniform charge creates a uniform vertical electric field in the vicinity of the point charge. The tension in the cord is measured to be 5.18 N. (a) Calculate the magnitude and direction of the electric field due to the sheet of charge (Fig. 21–80). (b) What is the surface charge density σ(C/m²) on the sheet?

Three very large square charged planes are arranged as shown (on edge) in Fig. 21–78. From left to right, the planes have charge densities per unit area of -0.50μC/m², +0.25 μC/m² and -0.35 μC/m². Find the total electric field (direction and magnitude) at the points A, B, C, and D. Assume the plates are much larger than the distance AD.

(II) Determine the direction and magnitude of the electric field at the point P shown in Fig. 21–66. The two point charges are separated by a distance of 2a. Point P is on the perpendicular bisector of the line joining the charges, a distance x from the midpoint between them. Express your answers in terms of Q, x, a, and k.

(III) A thin rod of length ℓ carries a total charge Q distributed uniformly along its length. See Fig. 21–69. Determine the electric field along the axis of the rod starting at one end—that is, find E(𝓍) for 𝓍 ≥ 0 in Fig. 21–69.

(II) A very long uniformly charged wire (linear charge density λ = 2.5 C/m) lies along the x-axis in Fig. 21–59. A small charged sphere (Q = -2.0 C) is at the point x = 0 cm, y = -5.0 cm. What is the electric field at the point x = 7.0 cm, y = 7.0 cm? ${\overrightarrow{E}}_{\text{wire}}$ and ${\overrightarrow{E}}_q$ represent fields due to the long wire and the charge Q, respectively.