Chapter 21

In a region where there is a uniform electric field that is upward and has magnitude 3.60 \(\times 10^4 \space N/\)C, a small object is projected upward with an initial speed of 1.92 m\(/\)s. The object travels upward a distance of 6.98 cm in 0.200 s. What is the object's charge-to-mass ratio \(q/m\)? Assume \(g = 9.80 \space m/s^2\), and ignore air resistance.

A negative point charge \(q_1 = -4.00\) nC is on the \(x\)-axis at \(x =\) 0.60 m. A second point charge \(q_2\) is on the \(x\)-axis at \(x = -\)1.20 m. What must the sign and magnitude of \(q_2\) be for the net electric field at the origin to be (a) 50.0 N\(/\)C in the \(+x\)-direction and (b) 50.0 N\(/\)C in the \(-\)x-direction?

A small sphere with mass \(m\) carries a positive charge \(q\) and is attached to one end of a silk fiber of length \(L\). The other end of the fiber is attached to a large vertical insulating sheet that has a positive surface charge density \(\sigma\). Show that when the sphere is in equilibrium, the fiber makes an angle equal to arctan (\(q\sigma/2mg\epsilon_0\)) with the vertical sheet.

Negative charge \(-Q\) is distributed uniformly around a quarter-circle of radius a that lies in the first quadrant, with the center of curvature at the origin. Find the \(x\)- and \(y\)-components of the net electric field at the origin.

Two very large parallel sheets are 5.00 cm apart. Sheet \(A\) carries a uniform surface charge density of \(-8.80 \space \mu\)C\(/m^2\), and sheet \(B\), which is to the right of \(A\), carries a uniform charge density of \(-11.6 \space \mu\)C\(/m^2\). Assume that the sheets are large enough to be treated as infinite. Find the magnitude and direction of the net electric field these sheets produce at a point (a) 4.00 cm to the right of sheet \(A\); (b) 4.00 cm to the left of sheet \(A\); (c) 4.00 cm to the right of sheet \(B\).

Two very large horizontal sheets are 4.25 cm apart and carry equal but opposite uniform surface charge densities of magnitude \(\sigma\). You want to use these sheets to hold stationary in the region between them an oil droplet of mass 486 \(\mu\)g that carries an excess of five electrons. Assuming that the drop is in vacuum, (a) which way should the electric field between the plates point, and (b) what should \(\sigma\) be?

Inkjet printers can be described as either continuous or drop-on-demand. In a continuous inkjet printer, letters are built up by squirting drops of ink at the paper from a rapidly moving nozzle. You are part of an engineering group working on the design of such a printer. Each ink drop will have a mass of 1.4 \(\times \space 10^{-8}\) g. The drops will leave the nozzle and travel toward the paper at 50 m\(/\)s, passing through a charging unit that gives each drop a positive charge \(q\) by removing some electrons from it. The drops will then pass between parallel deflecting plates, 2.0 cm long, where there is a uniform vertical electric field with magnitude 8.0 \(\times \space 10^4 \space N/C\). Your team is working on the design of the charging unit that places the charge on the drops. (a) If a drop is to be deflected 0.30 mm by the time it reaches the end of the deflection plates, what magnitude of charge must be given to the drop? How many electrons must be removed from the drop to give it this charge? (b) If the unit that produces the stream of drops is redesigned so that it produces drops with a speed of 25 \(m/s\), what \(q\) value is needed to achieve the same 0.30-mm deflection?

Two small spheres, each carrying a net positive charge, are separated by \(0.400 m\). You have been asked to perform measurements that will allow you to determine the charge on each sphere. You set up a coordinate system with one sphere (\(charge \space q_1\)) at the origin and the other sphere (\(charge \space q_2\)) at \(x = +\)0.400 m. Available to you are a third sphere with net charge \(q_3 = 4.00 \times 10^{-6}\) C and an apparatus that can accurately measure the location of this sphere and the net force on it. First you place the third sphere on the \(x\)-axis at \(x =\) 0.200 m; you measure the net force on it to be 4.50 N in the \(+ x\)-direction. Then you move the third sphere to \(x = +\)0.600 m and measure the net force on it now to be 3.50 N in the \(+ x\)-direction. (a) Calculate \(q_1\) and \(q_2\). (b) What is the net force (magnitude and direction) on \(q_3\) if it is placed on the \(x\)-axis at \(x = -\)0.200 m? (c) At what value of \(x\) (other than \(x = \pm \infty\)) could \(q_3\) be placed so that the net force on it is zero?

Two thin rods of length \(L\) lie along the \(x\)-axis, one between \(x = \frac{1}{2} a\) and \(x = \frac{1}{2} a + L\) and the other between \(x = -\frac{1}{2} a\) and \(x = -\frac{1}{2} a - L\). Each rod has positive charge \(Q\) distributed uniformly along its length. (a) Calculate the electric field produced by the second rod at points along the positive x-axis. (b) Show that the magnitude of the force that one rod exerts on the other is $$F = {Q^2 \over 4\pi\epsilon_0 L^2} ln [ {(a + L)^2 \over a(a + 2L)} ]$$ (c) Show that if \(a\) \(\gg\) \(L\), the magnitude of this force reduces to \(F = Q^2/4\pi\epsilon_0 a^2\). (\(Hint\): Use the expansion ln \((1 + z) = z - \frac{1}{2} z^2 + \frac{1}{3} z^3 - \cdot\cdot\cdot\), valid for \(\mid z \mid\ll1\). Carry \(all\) expansions to at least order \(L^2/a^2.\)) Interpret this result.

What is the best explanation for the observation that the electric charge on the stem became positive as the charged bee approached (before it landed)? (a) Because air is a good conductor, the positive charge on the bee's surface flowed through the air from bee to plant. (b) Because the earth is a reservoir of large amounts of charge, positive ions were drawn up the stem from the ground toward the charged bee. (c) The plant became electrically polarized as the charged bee approached. (d) Bees that had visited the plant earlier deposited a positive charge on the stem.