2. *#Coupled RLC circuit. Consider the circuit shown in the bottom of the page. (a) Use Kirchoff's junction law to express the current that flows through the resis- tor in terms of /, and 12. (b) Use Kirchoff's loop law to show that the coupled equations that describe the currents that flow in this circuit can be written Lhi + R(h - 12) + 7 = 0 Liz + R(12 - h) +12 = I C. I L R(c) Assume a solution cat to the equations and find the frequencies of the normal modes. Here, o = A + iw is a complex number but A and w are real. Assume the resistor is small enough that (/1/LC > R/L. (d) In part (c), you should have found that one mode is undamped and the other is damped. (i) Find the current flow pattern for the undamped mode. (ii) Is is a symmetric mode or an anti-symmetric mode? (iii) Explain its physics: why is it undamped? (iv) Use equivalent circuits for inductors and capacitors to explain its frequency. (e) Without calculation, what do you think the current pattern is for the damped mode? Physically, why is it damped