You have five straight wires of length l = 2 in the xy plane, four of which make

Question

You have five straight wires of length l = 2 in the xy plane, four of which make a square centered at the origin with sides parallel to the axes of the coordinate system. The fifth wire bisects the square along the y axis. You now insert three inductors and two capacitors to make a special LC circuit. The inductors are located along the x axis at x = -1,0,1 and each has inductance L. The capacitors arc located along the line y = 1 at x values of plusminus 1/2 and each has capacitance C. If the direction of the current in the right side inductor is reversed, show that the angular frequency of the oscillating current is given by omega = 1/ .

Explanation / Answer

First, make sure the circuit itself is correct (the dots represent blank space - Yahoo deletes extra spaces): +-C--+--C--+ |........|.........| L.......L........L |........|.........| +-----+-------+ As you can see, the Ls are not in series and the Cs are not in parallel. Now you can write the equations for each loop and solve them. Hint: you have to look at question (a) to find out what they want in (b). EDIT: ************************ To apply Kirchoff, add the voltages around the loops: The left-hand loop current, i1, is clockwise and the right-hand loop current, i2, is counter-clockwise. Left hand loop: i1/C + (i1 + i2)"L + i1"L = 0 Right hand loop: i2/C + (i1 + i2)"L + i2"L = 0 Adding the equations: (i1 + i2)/C + 2(i1 + i2)"L + (i1 + i2)"L = 0 (i1 + i2)/C + 3(i1 + i2)"L = 0 If we replace (i1+i2) with the symbol I, we get: I/C + I"3L = 0 which has the form of a simple RC circuit with just two components, C and 3L, whose resonant frequency will be: 1/v(3LC) You can get an analogous result using complex impedances: Left hand loop: i1/j?C + (i1+i2)j?L + I1j?L = 0 Right hand loop: i2/j?C + (i1+i2)j?L + i2j?L = 0 Adding the equations: (i1+i2)/j?C + 2(i1+i2)j?L + (i1+i2)j?L = 0 (i1+i2)/j?C + 3(i1+i2)j?L = 0 If we replace (i1+i2) with the symbol I, we get: I/j?C + I(3j?L) = 0 which has the form of a simple RC circuit with just two components, C and 3L, whose resonant frequency will be: 1/v(3LC)

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