Chapter 8.2, Problem 8.3P

Problem 9.9 Write down the (real) electric and magnetic fields for a monochro- matic plane wave of amplitude Eo, frequency w, and phase angle zero that is (a) traveling in the negative x direction and polarized in the z direction; (b) traveling in the direction from the origin to the point (1, 1, 1), with polarization parallel to the xz plane. In each case, sketch the wave, and give the explicit Cartesian components of k and în.

1.4 Show that the distance of closest approach d, in Rutherford scattering leading to an angle of deflection 0, is given by d=(1+ cosec 0,2). where p is detined in Fig. 1.6. (Use the conservation of energy and angular momentum.]

A charged particle of mass m and positive charge q moves in uniform electric 3. and magnetic fields, E and B, both pointing in the z direction. Ignoring gravity, use Newton's second law to write down the equation of motion for the velocity of the particle into its three components (a set of equations containing velocities and their derivatives). Solve the equations for v with initial velocity (vo, 0, 0).

7.10 A particle of charge e moves in a central potential V(r) superimposed onto a uniform magnetic field B whose vector potential is A = B × r/2. (a) Show that if B is a weak field, so that effects proportional to B² can be neglected, then where H(r, p): p² 2m + V(r) − µ · B, e 2mc (b) Write down Hamilton's equations in this approximation. μ = -rxp.

Question 2 a. Electrostatic fields have many applications, but most are low “low-power" applications; that is relatively low forces are involved. Explain why this is so. b. Given that D= (10ră ) ar (c/m²) in cylindrical coordinates, evaluate both sides of the divergence theorem for the volume enclosed by r= Im, r= 2m and Z= 0 and Z= 10

(Problem 4.10) A sphere of radius R carries a static radial polarization density P(r) = kr, r < R where k is a constant and r is the radial vector from the center of the sphere. (a) What are the dimensional units of the constant k? (b) Calculate the surface areal bound charge density o(R, 0, ø) and the volume bound charge density p(r). 2 (c) Find the electric field inside and outside the sphere.

A particle of charge −q and mass m is placed at the center of a uniformly charged ring of total charge Q and radius R. The particle is displaced a small distance along the axis perpendicular to the plane of the ring and released. Assuming that the particle is constrained to move along the axis, show that the particle oscillates in simple harmonic motion with a frequency (in picture attached).

Problem 7.63 Prove Alfven's theorem: In a perfectly conducting fluid (say, a gas of free electrons), the magnetic flux through any closed loop moving with the fluid is constant in time. (The magnetic field lines are, as it were, "frozen" into the fluid.) (a) Use Ohm's law, in the form of Eq. 7.2, together with Faraday's law, to prove that if o ∞o and J is finite, then 7.3 Maxwell's Equations a slightly different approach to the same problem, see W. K. Terry, Am. J. Phys. 50, 742 (1982). R ƏB at S do= =V x (v x B). FIGURE 7.58 (b) Let S be the surface bounded by the loop (P) at time t, and S' a surface bounded by the loop in its new position (P) at time t + dt (see Fig. 7.58). The change in flux is S' dt B(t + dt) da + do = dt S B(t + dt) da - Use VB0 to show that S √₂³ (where R is the "ribbon" joining P and P'), and hence that ав l vdt at dt B(t + dt) da = S B(t). da. da - R S B(t + dt) da B(t + dt) da (for infinitesimal dt). Use the method of Sect. 7.1.3 to rewrite the second inte-…