Assume that a ball of charged particles has a uniformly distributed negative charge density except for a narrow radial tunnel through its center, from the surface on one side to the surface on the opposite side. Also assume that we can position a proton anywhere along the tunnel or outside the ball. Let FR be the magnitude of the electrostatic force on the proton when it is located at the ball's surface, at radius R. As a multiple of R, how far from the surface is there a point where the force magnitude is 0.31FR if we move the proton (a) away from the ball and (b) into the tunnel?

Chapter 23, problem 075 the figure shows a geiger counter, a device used to detect ionizing radiation (radiation that causes ionization of atoms). the counter consists of a thin, positively charged central wire surrounded by a concentric, circular, conducting cylindrical shell with an equal negative charge. thus, a strong radial electric field is set up inside the shell. the shell contains a low-pressure inert gas. a particle of radiation entering the device through the shell wall ionizes a few of the gas atoms. the resulting free electrons (e) are drawn to the positive wire. however, the electric field is so intense that, between collisions with gas atoms, the free electrons gain energy sufficient to ionize these atoms also. more free electrons are thereby created, and the process is repeated until the electrons reach the wire. the resulting "avalanche" of electrons is collected by the wire, generating a signal that is used to record the passage of the original particle of radiation. suppose the radius of the central wire is 24 âµm, the inner radius of the shell 2.3 cm, and the length of the shell 14 cm. if the electric field at the shell's inner wall is 2.8 ă— 104 n/c, what is the total positive charge on the central wire?