2. General Physics

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2.4 Problems

2.4.4  LRC Circuits

Problem 1:  Series RC Circuit

Part (a)  Solve for the charge q[t] for an arbitraty V[t].

eq1 = V[t] == R  q'[t]  +  q[t]/C

dsol = DSolve[eq1, q[t], t][[1]]

Part (b)  Evaluate this solution when V[t]=V0 Cos[ωt].

\(V[t_] = V0\ Cos[ω\ t]; 
q[t_] = q[t] /. dsol // Simplify\)

c1Sol = Solve[q[0] == q0, C[1]]

q[t_] = q[t] /. c1Sol[[1]] // Simplify

Part (c)  Plot q[t] vs. t

values1 = {q0 -> 1,  V0 -> 1,  C -> 1,  R -> 1,  ω -> 6}

\(\(Plot[Evaluate[q[t] /. values1]
, {t, 0, 6}
, 
      AxesLabel -> {"Time (t)", "Charge (q[t])"}
, 
      PlotLabel -> FontForm["RC Loop", {"Times", 14}]]; \)\)

values2 = {q0 -> 1,  V0 -> 1,  R -> 1,  ω -> 6}

\(\(Plot3D[Evaluate[q[t] /. values2]
, {t, 0, 6}, {C, 1/100, 25/100}
, 
      AxesLabel -> {"Time", "Capacitance", "Charge"}
, 
      FaceGrids -> {{0, \(-1\), 0}}
, ViewPoint -> { .3, \(-1\),  .3}
, 
      PlotPoints -> 35]; \)\)