It's all the same question, please answer parts (a-d) I will upvote if so!

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It's all the same question, please answer parts (a-d) I will
upvote if so!

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R С L w 1. Consider the circuit in the diagram to the right. A voltage source of peak voltage En is connected in series with a resistor of re sistance R, inductor of inductance L, and capacitor of capacitance C. The circuit is driven at a frequency w. En cos (wt) V (a) Using the Kirchhoff loop rule, write down the equation describing the circuit. Write your answer in terms of the charge, obtaining an equation Q + Ag + Bq = cos (wt), and determine the constants A, B, and C. 9

(b) Recall the Euler relation for complex numbers, Cos (0) + i sin (0) such that the real part of the complex exponential, Re [C] = cos(6), while the imaginary part, Im [ccm] = sin(0). So, we can write cos (wt) = Re [ew]. Because the equation that you derived in part (a) is linear, we can then write the solution q as the real part of some complex function, z; that is, we can write q = Re [z]. Thus, you can transform your equation in part (a) to read z + Aż + B2 = Ceint, solve the equation for z(t), and then take the real part of this solution to get a(t). As a guess (more fancily called an ansalz), try 2 (t) = zoetwat where zo is a complex constant, independent of t. Plug in this guess and show that all the exponentials cancel. You have transformed a differential equation into an algebraic equation!

(c) Now, for a general complex mumber zo =1+iy, you can always write zo = |zolets, where (20) = x2 + y2, and tan (8) = y/t. Use this to determine zo, and show that z(t) = |zoleiwt+6) (d) Finally, take the real part of your solution for 2 (t) to determine 90). Compare your answer to the results we found in class. Do they agree?