1 of 2 PHYS3320 Fall 2009 Phys 3320, HW #6, Due at start of class, Wed Oct. 14th Q1. CHARGING CAPACITOR Consider a parallel plate capacitor with large circular plates of radius a and plate separation distance d. The capacitor is being charged at a constant rate by a constant current I, so that Q = I t. A. Calculate the electric and magnetic fields inside the capacitor. For simplicity assume that the magnetic field is produced only by the displacement current. You can ignore edge effects and fringe fields in these calculations. B. Calculate uEM and S inside the gap space of the capacitor. Does the direction of S make sense? Does your answer satisfy energy conservation in the fields, i.e., t(umech+ uEM) =   S ? Hint: What is umech in the gap space? C. Now find the total energy in the gap space between the plates as a function of time. Also calculate the total power flux into or out of the gap space by integrating S over the surface enclosing the gap space. Compare the total power flux to the rate of change of the total energy. Q2. FORCE BETWEEN TWO POINT CHARGES USING MAXWELL STRESS TENSOR A. Consider two equal point charges q separated by a distance 2a. Instead of using Coulomb’s law to find the force on one of the charges, use the Maxwell stress tensor  T integrated on the surface around a volume containing the point charge. You might guess that a small spherical surface around one of the charges would be best, but you’d be wrong! Since any volume containing the charge will work, use the volume defined by a circular flat surface of radius R on the plane in which each point is equidistant from the two charges, plus a hemispherical surface of radius R around one of the charges (centered on the point halfway between the charges) in order to form a closed surface (this is just like the surface used in Example 8.2 in Griffiths, p. 353). B. Now perform the calculation again, but let R first. Then your closed surface is the entire “equidistant” plane between the charges and a hemispherical surface “at infinity.” Give an argument for why it might be safe to assume that the surface integral on the hemisphere vanishes in this limit, so that we don’t need to actually calculate it. Assuming that the hemispherical part of the flux vanishes, now calculate the flux integral on just the infinite plane. Does your answer agree with that of part A?2 of 2 PHYS3320 Fall 2009 Q3. STRESS IN A CHARGED PARALLEL PLATE CAPACITOR Consider a large (“infinite”) parallel plate capacitor, with the lower plate (at z = -d/2) carrying the charge density -, and the upper plate (at z = +d/2) carrying the charge density +. A. Calculate all nine elements of the Maxwell stress tensor  T in the region between the plates, and write out the results in a 3x3 matrix. Also determine the Poynting vector S in the region between the plates. B. Find the force per unit area on the top plate using your  T , using the relation: F = T  da 0Sμ0ddtSd V In this formula, F is the total force on the charges in V. To use it in this case, just use a volume which is a very thin slab of + of area A and neglible thickness (so that we don’t have to calculate the thin edges of the slab. Does the direction of F/A make sense, using what you know about the Coulomb force? Are there any shear forces? C. Using the information you already have calculate, but using the interpretation in terms of densities and fluxes, what is the momentum per unit time per unit area crossing the xy plane (z = 0) plane? What is the direction of this momentum? Is your answer the same for a plane parallel to the xy plane, but located just below z = d/2? D. When the charges on the plate absorb the momentum flux you found in part C, momentum appears to disappear because inside the metal, the fields are all zero, hence  T is zero. Therefore the flux into the volume holding the charges (on the gap side) is just equal and opposite to the flux flowing out of the gap. What external force Fext per unit area is required to keep the total mechanical momentum of the charge equal to zero? Does your answer agree with that you found in part B? Q4. STRESS IN A MAGNETIC SOLENOID Now consider an infinite solenoid with constant current I and N loops per length l and radius a. Let the axis of the solenoid be the z-axis. A. Calculate all nine elements of the Maxwell stress tensor  T in the region inside the solenoid, and write out the results in a 3x3 matrix. Also determine the Poynting vector S in the region inside the solenoid. B. Find the pressure from the magnetic field on the curved walls of the solenoid using  T  da, as in part B of problem Q3. Is the force a pure pressure or are there shear forces as well? Use your knowledge of the force on currents to check the direction of the pressure. C. Now suppose that we cut off one end of the solenoid, so it is no longer infinite but rather semi-infinite. Using your knowledge of how the magnetic field behaves at the end of a solenoid, comment on how the elements of the stress tensor will change, and make a new matrix, but with only semi-quantitative information in it, i.e., describe which elements are big and which elements are small or zero. Do the elements depend on position? Indicate roughly how the elements depend on azimuthal angle around the solenoid. Use any available symmetries so that even if you don’t know what one element is, you can at least determine that it is the same as some other element in the matrix. D. Now, with your modified matrix for the end of the semi-infinite solenoid, consider once more the force per area on a piece of wall close to the end of the solenoid, on the +x side wall, that is, it has da = dzdy. Roughly, what will be the direction of this force? Is it a pure pressure or are there shear forces as