The gure shows a circuit containing an electromotive force, a capacitor with a

Chapter 7, Problem 27

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The gure shows a circuit containing an electromotive force, a capacitor with a capacitance of farads (F), and a resistor with a resistance of ohms ( ). The voltage drop across the capacitor is , where is the charge (in coulombs, C), so in this case Kirchhoffs Law givesRI Q C EtBut , so we haveR dQ dt1 C Q EtSuppose the resistance is , the capacitance is F, and a battery gives a constant voltage of 60 V. (a) Draw a direction eld for this differential equation. (b) What is the limiting value of the charge? (c) Is there an equilibrium solution? (d) If the initial charge is , use the direction eld to sketch the solution curve.(e) If the initial charge is , use Eulers method with step size 0.1 to estimate the charge after half a second.