Physics 122 Problem Set 5 No need to submit this problem set. This is only for you to practice. 1.

The fraction of light incident on a circular aperture (normal incidence) that is transmitted is given by 2𝑘𝑎

𝑇= 0

2 1 − 𝑑𝑥 𝑥 2𝑘𝑎

𝐽2 𝑥

where a is the radius of the aperture, and k is the wave number 2/. Show that 𝑇=1−

1 2𝑘𝑎

2𝑘𝑎 0

𝐽0 𝑥 𝑑𝑥

2. The differential cross-section in a nuclear scattering experiment is given by d/d = f()2. An approximate treatment leads to: −𝑖𝑘 2𝜋 𝑅 𝑓 𝜃 = exp 𝑖𝑘𝜌 sin 𝜃 sin 𝜑 𝜌𝑑𝜌 𝑑𝜑 2𝜋 0 0 where  is the angle through which the particle is scattered, and R is the nuclear radius. Show that 𝑑𝜍 1 𝐽1 (𝑘𝑅 sin 𝜃) = 𝜋𝑅2 𝑑 𝜋 sin 𝜃

2

3. Using the Laplace transform method, find the current in an RL series circuit for an input voltage v(t) = t if 0 < t < 4 and v(t) = 0 if t > 4 assuming that i(0) = 0. 4. Solve the given initial value problem by Laplace transforms. a. y” + 4y = u(t - 3), y(0) = 1 and y’(0) = 0 b. y” + 9y = 6 sin t if 0 t  and 0 if t > , y(0) = 0 and y’(0) = 0