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The figure below shows a circuit containing an electromotive force, a capacitor with a capacitance of C farads (F), and a resistor with a resistance of R ohms (). The voltage drop across the capacitor is Q/C, where Q is the charge (in coulombs, C), so in this case Kirchhoff's law gives But I = dQ RI + dt -, so we have 2 = E(t). Q с R dQ dt + 11/1Q = E(t). C Suppose the resistance is 5, the capacitance is 0.05 F, and a battery gives a constant voltage of 100 V. C R (a) Draw a direction field for this differential equation. 511
(a) Draw a direction field for this differential equation. Q 10+ 8 6 4 01 0 1 2 3 4 5 Q 5 I 1 |||||I 3 4 5 Q 10 2 3 4
(b) What is the limiting value of the charge (in C)? (c) Is there an equilibrium solution? O Yes No (d) If the initial charge is Q(0) = 0 C, use the direction field to sketch the solution curve. Q 5 O 4 1 Q 5 1 2 3 4 5 6 Q 10 8 1 2 3 4 5 t Q 10 8 6 1 2 3 4 5
(e) If the initial charge is Q(0) = 0 C, use Euler's method with step size 0.1 to estimate the charge Q after half a second. (Round your answer to two decimal places.) Q(0.5) = с