24. Electric Force & Field; Gauss' Law

If the force between two charges is F when the distance is d, what will the force between the two charges be if they were moved to a distance of 2d?

In which direction will the −1 C charge move? If it has a mass of 10 g, what will its initial acceleration be?

What is the direction of the net force on the charge at the center of the square in the following figure?

Just How Strong Is the Electric Force? Suppose you had two small boxes, each containing 1.0 g of protons. (a) If one were placed on the moon by an astronaut and the other were left on the earth, and if they were connected by a very light (and very long!) string, what would be the tension in the string? Express your answer in newtons and in pounds. Do you need to take into account the gravitational forces of the earth and moon on the protons? Why? (b) What gravitational force would each box of protons exert on the other box?

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 , 

n=∞

∑ 1/n² = π² / 6 where n is a positive integer.]

n= 1

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.)

(I) 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?

(II) 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 = . 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.

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(II) 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.

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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 10⁻⁶ 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.

(II) At each corner of a square of side ℓ there are point charges of magnitude Q, 2Q, 3Q, and 4Q (Fig. 21–57). Determine the magnitude and direction of the force on the charge 2Q.

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