24. Electric Force & Field; Gauss' Law

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.

Three positive particles of equal charge, +17.0 μC, are located at the corners of an equilateral triangle of side 15.0 cm (Fig. 21–55). Calculate the magnitude and direction of the net force on each particle due to the other two.

A $-4.00$-nC point charge is at the origin, and a second $-5.00$-nC point charge is on the $x$-axis at $x = 0.800$ m. Find the net electric force that the two charges would exert on an electron placed at each point in part (a). Note: Part (a) asked to find the electric field (magnitude and direction) at each of the following points on the $x$-axis: (i) $x = 0.200$ m; (ii) $x = 1.20$ m; (iii) $x = -0.200$ m.

Two protons are released from rest when they are $0.750$ nm apart. What is the maximum acceleration they will achieve and when does this acceleration occur?

A positive charge $q$ is fixed at the point $x=0$, $y=0$, and a negative charge $-2q$ is fixed at the point $x=a$, $y=0$. Show the positions of the charges in a diagram.

A positive point charge Q₁ = 2.5 x 10^-5 C is fixed at the origin of coordinates, and a negative point charge Q₂ = -5.0 x 10^-8 C is fixed to the 𝓍 axis at 𝓍 = +3.2 m. Find the location of the place(s) along the 𝓍 axis where the electric field due to these two charges is zero.

Two positive point charges are a fixed distance apart. The sum of their charges is Qₜ. What charge must each have in order to

(a) maximize the electric force between them, and

(b) minimize it?