Once Rutherford and Geiger determined the charge-to mass ratio of the \(\alpha\) particle (Problem 71 ), they performed another experiment to determine its charge. An \(\alpha\) source was placed in an evacuated chamber with a fluorescent screen. Through a glass window in the chamber, they could see a flash on the screen every time an \(\alpha\) particle hit it. They used a magnetic field to deflect \(\beta\) particles away from the screen so they were sure that every flash represented an alpha particle. (a) Why is the deflection of a \(\beta\) particle in a magnetic field much larger than the deflection of an \(\alpha\) particle moving at the same speed? (b) By counting the flashes, they could determine the number of \(\alpha\) s per second striking the screen (R). Then they replaced the screen with a metal plate connected to an electroscope and measured the charge \(Q\) accumulated in a time \(\Delta t .\) What is the \(\alpha\) -particle charge in terms of \(R, Q,\) and \(\Delta t ?\)

This difference in deflection can be attributed to the difference in both their charges and masses. Alpha particles have a charge of +2e while beta particles have a charge of -e (where e = 1.6 x 10^{-19} C is the elementary charge) and beta particles have a much lower mass than alpha particles. When particles move in a magnetic field, the force they experience is given by the Lorentz force: F = q(v x B) where F is the force, q is the charge of the particle, v is the velocity of the particle, and B is the magnetic field. For a charged particle moving in a magnetic field perpendicular to its velocity, the centripetal force is given by: F = (mv^2)/r where m is the mass of the particle and r is the radius of the circular path. Now let's equate the Lorentz force and the centripetal force for both alpha and beta particles: mv^2/r = q(v x B) => r = (mv)/(qB) By comparing the radii of both alpha and beta particles in the magnetic field, we can understand how much deflection each particle experiences: r_alpha/r_beta = (m_alpha)(q_beta) / (m_beta)(q_alpha) Given that the mass of the alpha particle is nearly 8000 times the mass of the beta particle and q_alpha = 2q_beta, we have: r_alpha/r_beta = 8000/2 r_alpha = 4000 * r_beta This result shows that the alpha particles will have 4000 times larger radius, hence the beta particles are much more deflected in the magnetic field than alpha particles at the same speed.

Let's denote the charge of the alpha particle as αQ. The total charge (Q) accumulated by all the alpha particles will be equal to the charge carried by each alpha particle times the total number of alpha particles (R * Δt) striking the screen. Q = (αQ) * (R * Δt) To find the charge of an individual alpha particle (αQ), we can rearrange the equation: αQ = Q / (R * Δt) The charge of an alpha particle can be determined using values of R, Q, and Δt derived from the experiment.