Problem 1[10 points] A parallel plate capacitor is made from two circular conducting plates of radius r = 5 cm separated by a distance d = 10 cm. The top half of the region between the plates is filled with air (κ = 1.00) and the bottom half is filled with mineral oil. In the mineral oil, the electric field is reduced from that outside the oil by a factor 1/κ = 1/2.1 where κ is the dielectric constant. The plates are connected to a 24 V voltage source. (a) Find the electric field strength in units of V/m in the air and mineral oil. (b) Use Gauss’s law to argue that there must be charge on the top surface of the mineral oil. Find that charge density in units of C/m2 . (c) Find the capacitance in units of farad of this arrangement. 24V Problem 2[10 points] A cosmic-ray proton of kinetic energy K = 10 MeV is trapped inside the galaxy by the inter-galatic magnetic field. Suppose this proton’s orbit has a radius of r = 6×1010 m (roughly the radius of Mercury’s orbit). What is the magnitude of the galatic field in this region of space? 2 Problem 3[10 points] When traveling outside the U.S. you may encounter electrical outlets that supply RMS voltage of 120 V or 220 V. Suppose you want to design a travel hair-dryer capable of generating 1000 W of RMS drying power using a resistive heating coil. (a) What resistance is required of the coil for each voltage? (b) Design a circuit which uses a single heating coil, a switch, and a resistor which would allow you get the same performance (1000 W of drying power from the coil) from either voltage source by adjusting the position of the switch. Note: While the power is supplied as alternating current, in this problem we have only resistive devices and we are using the RMS voltage and power and can therefore can solve the problem completely by treating it as a direct-current circuit. Problem 4[10 points] A toroid is formed from N turns of a wire. The toroid has a square cross-section of side S and inner radius R. (a) If a current I is passed through the toroid there will be a field B1 at the inner edge, r = R, and B2 at the outer edge r = R + S. what is the ratio of the field strength B1/B2? (b) Compute the magnetic flux through the square S × S cross-section of toroid. (c) From this show that this toroid has an inductance of L = N2S 2π0c 2 ln(1 + S/R) 3 Problem 5[10 points] A radio receiver is 8.0 km due West of transmitter, which transmits at a constant amplitude at a frequency of 91.5 MHz. The receiver determines that at t = 0 the wave has its maximum electric field amplitude of 98.0 millivolts/meter and points upward (call that the +z–direction and West the +x–direction.). (a) What is the magnetic field (magnitude and direction) at the receiver at t = 0? (b) What are is magnitude of the Poynting vector at the receiver vector when t = 0? (c) Make an estimate of the total average radio power emitted by the transmitter. Problem 6[10 points] An optical device consists of a rectangular of thickness d that rotates, glass has an index of refraction of 1.62. Show that this device shifts a perpendicular beam of light laterally with respect to its initial path. How large is the shift as a function of the thickness d? and angle of rotation θ?