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You are working for the summer at a research laboratory. Your research director has devised a scheme for holding small charged particles at fixed positions. The scheme is shown in Figure P23.36. An insulating cylinder of radius a and length L >> a is positively charged and carries a uniform volume charge density ρ. A very thin tunnel is drilled through a diameter of the cylinder and two small spheres with charge q are placed in the tunnel. These spheres are represented by the blue dots in the figure. They find equilibrium positions at a distance of r on opposite sides of the axis of the cylinder. Your research director has had great success with this scheme. (a) Determine the specific value of rat which equilibrium exists. (b) Your research director asks you see if he can extend the system as follows. Determine if it is possible to add transparent plastic tubes as extensions of the tunnel and have the small spheres be in equilibrium at a position for which r > a.
Figure P23.36
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Chapter 23 Solutions
Physics for Scientists and Engineers
- In Figure P24.49, a charged particle of mass m = 4.00 g and charge q = 0.250 C is suspended in static equilibrium at the end of an insulating thread that hangs from a very long, charged, thin rod. The thread is 12.0 cm long and makes an angle of 35.0 with the vertical. Determine the linear charge density of the rod. FIGURE P24.49arrow_forwardThe infinite sheets in Figure P25.47 are both positively charged. The sheet on the left has a uniform surface charge density of 48.0 C/m2, and the one on the right has a uniform surface charge density of 24.0 C/m2. a. What are the magnitude and direction of the net electric field at points A, B, and C? b. What is the force exerted on an electron placed at points A, B, and C? FIGURE P25.47arrow_forwardA thin, semicircular wire of radius R is uniformly charged with total positive charge Q (Fig. P24.63). Determine the electric field at the midpoint O of the diameter.arrow_forward
- A very long, thin wire fixed along the x axis has a linear charge density of 3.2 C/m. a. Determine the electric field at point P a distance of 0.50 m from the wire. b. If there is a test charge q0 = 12.0 C at point P, what is the magnitude of the net force on this charge? In which direction will the test charge accelerate?arrow_forward(a) Find the total electric field at x = 1.00 cm in Figure 18.52(b) given that q =5.00 nC. (b) Find the total electric field at x = 11.00 cm in Figure 18.52(b). (c) If the charges are allowed to move and eventually be brought to rest by friction, what will the final charge configuration be? (That is, will there be a single charge, double charge; etc., and what will its value(s) he?)arrow_forwarda. Figure 24.22A shows a rod of length L and radius R with excess positive charge Q. The excess charge is uniformly distributed over the entire outside surface of the rod. Write an expression for the surface charge density . Write an expression in terms of for the amount of charge dq contained in a small segment of the rod of length dx. b. Figure 24.22B shows a very narrow rod of length L with excess positive charge Q. The rod is so narrow compared to its length that its radius is negligible and the rod is essentially one-dimensional. The excess charge is uniformly distributed over the length of the rod. Write an expression for the linear charge density . Write an expression in terms of for the amount of charge dq contained in a small segment of the rod of length dx. Compare your answers with those for part (a). Explain the similarities and differences.arrow_forward
- A When we find the electric field due to a continuous charge distribution, we imagine slicing that source up into small pieces, finding the electric field produced by the pieces, and then integrating to find the electric field. Lets see what happens if we break a finite rod up into a small number of finite particles. Figure P24.77 shows a rod of length 2 carrying a uniform charge Q modeled as two particles of charge Q/2. The particles are at the ends of the rod. Find an expression for the electric field at point A located a distance above the midpoint of the rod using each of two methods: a. modeling the rod with just two particles and b. using the exact expression E=kQy12+y2 c. Compare your results to the exact expression for the rod by finding the ratio of the approximate expression to the exact expression. FIGURE P24.77 Problems 77 and 78.arrow_forwardFigure P23.49 shows two identical small, charged spheres. One of mass 4.0 g is hanging by an insulating thread of length 20.0 cm. The other is held in place and has charge q1 = 3.6 C. The thread makes an angle of 18 with the vertical, resulting in the spheres being aligned horizontally, a distance r apart. Determine the charge q2 on the hanging sphere. Figure P23.49arrow_forwardThree charged spheres are at rest in a plane as shown in Figure P23.70. Spheres A and B are fixed, but sphere C is attached to the ceiling by a lightweight thread. The tension in the string is 0.240 N. Spheres A and B have charges qA = 28.0 nC and qB = 28.0 nC. What charge is carried by sphere C?arrow_forward
- A circular ring of charge with radius b has total charge q uniformly distributed around it. What is the magnitude of the electric field at the center of the ring? (a) 0 (b) keq/b2 (c) keq2/b2 (d) keq2/b (e) none of those answersarrow_forwardA coaxial cable is formed by a long, straight wire and a hollow conducting cylinder with axes that coincide. The wire has charge per unit length = 20, and the hollow cylinder has net charge per unit length = 30. Use Gausss law to answer these questions: What are the charges per unit length on a. the inner surface and b. the outer surface of the hollow cylinder? c. What is the electric field a radial distance d from the axis of the coaxial cable?arrow_forwardA solid, insulating sphere of radius a has a uniform charge density throughout its volume and a total charge Q. Concentric with this sphere is an uncharged, conducting, hollow sphere whose inner and outer radii are b and c as shown in Figure E23.1. We wish to understand completely the charges and electric fields at all locations. (a) Find the charge contained within a sphere of radius r < a. (b) From this value, find the magnitude of the electric field for r < a. (c) What charge is contained within a sphere of radius r when a < r < b? (d) From this value, find the magnitude of the electric field for r when a < r < b. (e) Now consider r when b < r< c. What is the magnitude of the electric field for this range of values of r? (f) From this value, what must be the charge on the inner surface of the hollow sphere? (g) From part (f), what must be the charge on the outer surface of the hollow sphere? (h) Consider the three spherical surfaces of radii a, b, and c.…arrow_forward
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