24. Electric Force & Field; Gauss' Law

What is the magnitude of the net force on charge A in FIGURE EX22.17?

Two 1.0 g spheres are charged equally and placed 2.0 cm apart. When released, they begin to accelerate at 150 m/s². What is the magnitude of the charge on each sphere?

A 3.00-cm-long spring has a small plastic bead glued to each end. Charging each bead to −25 nC expands the spring by 0.50 cm. What is the value of the spring constant?

A small glass bead charged to +6.0 nC is in the plane that bisects a thin, uniformly charged, 10-cm-long glass rod and is 4.0 cm from the rod's center. The bead is repelled from the rod with a force of 840 μN. What is the total charge on the rod?

INT In a classical model of the hydrogen atom, the electron orbits the proton in a circular orbit of radius 0.053 nm. What is the orbital frequency in rev/s? The proton is so much more massive than the electron that you can assume the proton is at rest.

Suppose that electrical attraction, rather than gravity, was responsible for holding the Moon in orbit around the Earth. If equal and opposite charges Q were placed on the Earth and the Moon, what should be the value of Q to maintain the present orbit? Use data given on the inside front cover of this book. Treat the Earth and Moon as point particles.

Four equal positive point charges, each of charge 5.8 μC, are at the corners of a square of side 9.2 cm. What charge should be placed at the center of the square so that all charges are at equilibrium? Is this a stable or an unstable equilibrium (Section 12–4) in the plane?

Two small, identical conducting spheres A and B are a distance R apart; each carries the same charge Q. (a) What is the force sphere B exerts on sphere A? (b) An identical sphere with zero charge, sphere C, makes contact with sphere B and is then moved very far away. What is the net force now acting on sphere A? (c) Sphere C is brought back and now makes contact with sphere A and is then moved far away. What is the force now on sphere A?

A one-dimensional row of positive ions, each with charge +Q and separated from its neighbors by a distance d, occupies the right-hand half of the x axis. That is, there is a +Q charge at x = 0, x = +d, x = +2d, x = +3d, and so on out to ∞. (a) If an electron is placed at the position x = - d, determine F, the magnitude of the electric force that this row of charges exerts on the electron. (b) If the electron is instead placed at x = ―3d, what is the value of F ? [Hint: The infinite sum, 

${\sum }_{n=1}^{\infty }\frac{1}{n^2}=\frac{{\pi }^2}{6}$ where n is a positive integer.]

Near the surface of the Earth, there is a downward electric field of 150 N/C and a downward gravitational field of 9.8 N/kg. A charged 1.0-kg mass is observed to fall with acceleration 8.0 m/s². Determine the magnitude and sign of its charge.

As a rough rule, anything traveling faster than about 0.1c is called relativistic—that is, special relativity is a significant effect. Determine the speed of an electron in a hydrogen atom (radius 0.53 x 10^-10 m) and state whether or not it is relativistic. (Treat the electron as though it were in a circular orbit around the proton.)

What is the magnitude of the electric force of attraction between an iron nucleus (q = +26e) and its innermost electron if the distance between them is 1.5 x 10^-12 m?

Compare the electric force holding the electron in orbit (r = 0.53 x 10^-10) around the proton nucleus of the hydrogen atom, with the gravitational force between the same electron and proton. Give the ratio of these two forces.

(III) Two charges, -2Q and -3Q, are a distance ℓ apart. These two charges are free to move but do not because there is a third (fixed) charge nearby. What must be the magnitude of the third charge and its placement in order for the first two to be in equilibrium?

Two point charges, Q₁ = ― 6.7 μC and Q₂ = 2.6 μC, are located between two oppositely charged parallel plates, as shown in Fig. 21–74. The two charges are separated by a distance of 𝓍 = 0.47 m. Assume that the electric field produced by the charged plates is uniform and equal to E = 53,000 N/C . Calculate the net electrostatic force on Q₁ and give its direction.