Figure $23-61$ shows a Geiger counter, a device used to detect ionizing radiation, which causes ionization of atoms. A thin, positively charged central wire is surrounded by a concentric, circular, con- ducting cylindrical shell with an equal negative charge, creating a strong radial electric field. 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 that the radius of the central wire is $25 \mu \mathrm{m},$ the inner radius of the shell 1.4 $\mathrm{cm},$ and the length of the shell 16 $\mathrm{cm} .$ If the electric field at the
shell's inner wall is $2.9 \times 10^{4} \mathrm{N} / \mathrm{C}$ , what is the total positive charge on the central wire?