PHYS 332 Homework #1 Maxwell's Equations, Poynting Vector, Stress

Home , Displacement current, Maxwell stress tensor, Poynting vector

PHYS 332 Homework #1

Maxwell’s Equations, Poynting Vector, Stress Energy Tensor, Electromagnetic Momentum, and Angular Momentum

1 : Wangsness 21-1. A parallel plate capacitor consists of two circular plates of area S (an effectively infinite area) with a vacuum between them. It is connected to a battery of constant emf . The plates are then slowly oscillated so that the separation d between ~ them is described by d = d0 + d1 sin ωt. Find the magnetic field H between the plates produced by the displacement current. Similarly, find H~ if the capacitor is first disconnected from the battery and then the plates are oscillated in the same manner.

2 : Wangsness 21-7. Find the form of Maxwell’s equations in terms of E~ and B~ for a linear isotropic but non-homogeneous medium. Do not assume that Ohm’s Law is valid.

3 : Wangsness 21-9. Figure 21-3 shows a charging parallel plate capacitor. The plates are circular with radius a (effectively infinite). Find the Poynting vector on the bounding surface of region 1 of the figure. Find the total rate at which energy is entering region 1 and then show that it equals the rate at which the energy of the capacitor is increasing.

4a: Extra-2. You have a parallel plate capacitor (assumed inﬁnite) with charge densities σ and −σ and plate separation d. The capacitor moves with velocity v in a direction perpendicular to the area of the plates (NOT perpendicular to the area vector for the plates). Determine S~.

b: This time the capacitor moves with velocity v parallel to the area of the plates. De- termine S~. Then determine the power (P ) passing through a cross-sectional area (of dU ˙ width l) between the plates and perpendicular to v. Lastly, determine dt = U where U is the total electromagnetic energy (not the energy density) which passes through that cross section in a time t and show that U˙ = P/2. The other half of the power is transmitted to the plates themselves. You may assume that v c.

5 : Wangsness 21-10. Suppose that the very long coaxial line of Figure 20-14 is used as a transmission line between a battery and resistor. The battery of emf has its two terminals connected to the two conductors at one end of the line. At the other end of the line, the two conductors are connected through a resistor of resistance R. Find S in the region between the conductors. Show that the total power passing across a cross section of the line equals 2/R and interpret this result. Will the direction of the energy ﬂow change if the connections to the battery are interchanged?

6a: Griﬃths 8-9. A very long solenoid of radius a and n turns per unit length carries a current Is. Coaxial with the solenoid, at radius b a is a circular ring of wire with resistance R. When the current in the solenoid is gradually decreased, a current Ir dIs is induced in the ring. Determine Ir in terms of dt . You should assume that any displacement currents are negligible.

2 b: The power Ir R delivered to the ring must have come from the solenoid. Confirm this by calculating the Poynting vector at ρ = a. Integrate this Poynting vector over the surface of the solenoid and verify that you recover the correct total power. (Note that the electric field is due to the changing flux in the solenoid while the magnetic field is due to the current in the ring. Since b a, you can assume that you just need the magnetic field on the axis of a ring of current.)

7: Griffiths 8-4b. Consider two equal but opposite point charges separated by a distance 2a. Construct the infinite plane equidistant from the two charges and the infinite hemisphere that surrounds one of the charges. Integrate the Maxwell stress tensor over this infinite surface and effectively verify Coulomb’s Law.

8 : Griﬃths 8-5. Consider an inﬁnite parallel-plate capacitor, with the lower plate (at z = −d/2) carrying a charge density −σ, and the upper plate (at z = +d/2) carrying a charge density +σ.

a: Determine all nine elements of the stress tensor in the region between the plates. Display your answer as a 3 × 3 matrix:

  Txx Txy Txz  Tyx Tyy Tyz  Tzx Tzy Tzz

b: Determine the force per unit area on the top plate.

c: What is the momentum per unit area, per unit time, crossing the xy plane (or any other plane parallel to that one, between the plates)?

d: At the plates, this momentum is absorbed and the plates recoil (unless there is some nonelectrical force holding them in position). Find the recoil force per unit area on the top plate, and compare your answer to (b). (Note: This is not an additional force, but rather an alternative way of calculating the same force–in (b) we obtained it from the force law, and now we are obtaining it from conservation of momentum.)

9a: Extra-1. You have an inﬁnite current I in the z-direction. And similarly you have an inﬁnite line charge λ coincident with that current. Determine S~. (Note that I is not necessarily related to λ. You could easily have a net positive charge on a wire but still have electrons creating the current. So assume that λ and I are independent parameters.)

b: Determine the power passing through a donut-shaped cross section of inner radius R1 and outer radius R2 surrounding the current. What happens to this power if R1 → 0? Does this make sense? Explain. c: Determine the electromagnetic momentum density due to this conﬁguration and make a drawing illustrating the momentum density vectors. d: Determine the total electromagnetic angular momentum. You should be able to do this without actually doing any integration.