Classical Electrodynamics II: Solutions

Problem 1
Solution 1
At Fermilab anti-protons p' are produced by colliding a beam of protons p with protons at rest in the reaction p+p → p+p+p+ p'. In such a “fixed target experiment” the center-of-mass system is defined as the reference frame where the total momentum pCM of the system is zero.

(a) Show that for a particle of mass m incident with kinetic energy TL (in the lab frame) on a target of mass M, the total center-of-mass energy ECM is given by E2CM = (M+m)2+2MTL.
(b) What minimum kinetic beam energy TB is required to produce anti-protons with mass mp = 938 MeV in the process described above.
(c) In the actual experiment at Fermilab, the protons are collided with a block of heavy metal. This means the protons in the target are bound inside heavy nuclei with the result that the protons have a kinetic energy TT . What incident beam energy EB would be required to produce anti-protons off of those protons with TT , if their velocity was directed toward the beam. Express your answer in terms of ET , pT and mp.
(d) Suppose the kinetic energy TT of such protons inside heavy nuclei is 40 MeV. What is the velocity β of these protons.

Problem 2
Solution 2
Suppose coordinate system K′ is moving with respect to coordinate system K with velocity K in arbitrary direction. Determine the Lorentz transformation which connects the coordinates (ct′, x′, y′, z′) in K′ to the coordinates (ct, x, y, z) in K.

Problem 3
Solution 3
A cone-shaped region of angle α is obtained from a circle with radius a. Consider the electrostatic potential in this two-dimensional region where the potential φ is zero along the edges (the two radial parts and the arc of the circle). Find the Green’s function for this boundary value problem. Note that the two-dimensional Green’s function is defined by

2G(x, x′) = δ2(xx′)

with

2 = ∂2/∂x2 + ∂2/∂y2∂2 = (1/r)(∂/∂r)[r(∂/∂r)] + (1/r2)(∂2/∂θ2)

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