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PHY350 (2014) Assignment 4 Due Friday December 5, 2014 Here are the first five problems for your final assignment. The last one (on waves, hopefully) will be posted later. Do all 6 problems. They are of equal value. 1. A long thick cylindrical shell with inner radius a and outer radius b is made of conducting, linear magnetic material with permeability μ. The shell carries a slowly ! !
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varying current I(t) uniformly distributed though the material. Find B , H and E in all regions of space, in the quasi-­‐static approximation. 2. A capacitor with circular parallel plates of radius R and separation d << R is filled with material having dielectric constant ε r . A time-­‐varying potential difference
V = V0 cos ω t is applied to the plates: a) Find the electric field between the plates and the free surface charge density on the plates (ignoring magnetic and fringe-­‐field effects). b) Find the magnitude and direction of the magnetic field between the two plates as a function of the distance from the axis of the plates. c) Calculate the flux of the Poynting vector from the open edges of the capacitor. 3. Consider a solenoid (n turns/unit length, radius R) carrying a current that is increasing linearly with time I(t) = kt (where k is a constant with the appropriate units). Calculate the Poynting vector and use it to show that the flow of energy (per unit length) into volume occupied by the solenoid is given by dW
d ⎛1
⎞
= ⎜ LI 2 ⎟ ⎠
dt
dt ⎝ 2
where L is the self-­‐inductance per unit length of the solenoid. 4. Consider a long straight wire of radius a and electrical conductivity σ carrying a uniform current density J along it’s length. Find the magnitude and direction of the Poynting vector at the surface of the wire. [If you’re not sure how to approach this you may want to review some parts of Chapter 7]. PHY350 (2014) Assignment 4 Due Friday December 5, 2014 5. Consider the system illustrated below, consisting of a inner sphere of radius a, and a concentric outer sphere of radius b, with a small hole (which you can otherwise ignore) to allow a wire to pass through to charge up (and suspend) the inner sphere. The inner sphere is made of non-­‐conducting magnetic media with a uniforms polarization in the direction indicated. It is coated with a thin conducting layer. The inner and outer spheres thus form a spherical capacitor. If there is a charge of +Q on the inner conductor and –Q on the outer conductor: a) calculate the momentum stored in the fields; b) calculate the angular momentum stored in the fields; c) describe (both qualitatively and quantitatively) what happens as the charge is allowed to drain away.