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6. A resistor, R, is in parallel with an inductor, L, and the two of them, together, are in series with a capacitor, C. a) Calculate the complex impedance of the circuit, as a function of the angular frequency, omega. b) For what value of frequency, f, does the magnitude of the impedance reach its maximum value?
[ f --- > infinity]
7. An inverting amplifier can be made from an op amp with an input resistor R1 and a feedback resistor R2. The amplification will equal A=-R2/R1. If either of these resistors is replaced with a capacitor, inductor, or some other combination of R, L, or C, the resistances in this expression should be replaced with impedances, Z. a) Which resistor should be replaced by a capacitor to make a low-pass filter, and which should be replaced by a capacitor to make a high-pass filter? What is the amplification for each? b) Calculate the frequency-dependence of the amplification if the input impedance is a 10kW resistor, and the feedback impedance is a chose that behaves like a 50mH inductor in series with a 100ohm resistor. At what frequency is the magnitude of A equal to 0.01?
CHAP. 10: Electric fields in matter -- Purcell, Ch. 4
1. Two identical electric dipoles, p, sit along the x-axis, one at the origin, the other at x=+d. The dipole at the origin, p1, points at an angle of 45o above the positive x-axis, while p2 points at a direction of 135o. a) Calculate the electric field at p2 due to p1, in terms of Er and Etheta. b) Calculate the potential energy of the system of charges.
[1.414p/d3, 1.414p/2d3, -p2/2d3]
2. A water molecule is an electric dipole with dipole moment p=1.8410-18esu.cm. Consider a microwave oven which deposits 500W into a 1L=(10cm) 3 cube of water. a) If the power is uniformly deposited on the surface of the cube, what is Erms at the surface? b) What is the average energy needed to flip a water molecule from an average random orientation U=0, to full alignment with the field? c) If the whole 500J is absorbed by the water in one second in producing spin flips, with that energy quickly dissipated by collisions between molecules, and if the liter of water contains (1000/18)NA, where NA=61023 molecules, then how many spin flips, on average, does each molecule undergo in one second?
[0.059esu/cm2, -1.110-26J, 1360/s]
3. Two identical electric dipoles, p, are a distance d apart, with the following orientations: p1 is on the left and points up; p2 on the right, points right. a) Calculate the torque on p2 due to p1. b) Is this torque clockwise or counterclockwise? c) Is the torque on p1 due to p2 clockwise or counterclockwise?
[ p2/d3, clockwise, ditto]
4. The index of refraction in air at room temperature is about 1.00027 for visible light. a) Claculate the corresponding dielectric constant. b) Calculate the electric susceptibility. c) The susceptibility is just the polarizability per volume. What is the polarizability of each molecule in air if one mole occupies approximately 22L? The units of polarizability should be units of volume, and should approximate the volume of the individual molecule.
5. Estimate the capacitance of a 'Leyden jar', a bottle which is covered on the outside by foil and which contains water. The water acts as one of the two electrodes. Make some reasonable assumptions about the bottle's dimensions. The dielectric constant of glass can be assumed to be about 4. Take the thickness of the glass to be about 2mm.
[1nF]
6. Consider two electric dipoles sitting along the x-axis, separated by r=3A. If the dipoles begin pointing in parallel, how much energy does it take to flip one of them, ifa) they start out pointing in the x-direction? b) they start out pointing in the y-direction? Calculate the dipole moment necessary for the energy in part (a) to equal kT=4.010-14erg. If the thermal energy is smaller than the energy above, thermal agitation will be modest, and the dipoles can stay aligned.
[4p2/r3, -2p2/r3, 5.210-19esucm]
7. Calculate the force on a dipole near a point charge, +Q, located at the origin. The dipole is located on the x-axis, is poining in the positive x-direction, and has a magnitude p. (Be sure you can convince yourself that the form of the answer -- which is not zero -- makes sense.)
CHAP. 11: Magnetic fields in matter, Maxwells Eqs. in matter -- Purcell, Ch. 6, 9
1. Magnets are often characterized by their 'lift', the amount of force they can pull. Assuming that you are using a magnet to pull some iron, and assuming that you induce a dipole moment in the iron equivalent to the dipole moment of the magnet, derive an expression for the lift of a spherical magnet of radius r, in terms of its magnetization, M.
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[kc=sqrt(me)w, Bo=sqrt(me)Eo]
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