Homework 1 - Problem 1

Write down a differential equation that describes the behavior of the RC circuit shown below. The equation should describe the voltage change (dV/dt) as a function of the applied current (I). Arrange the equation such that the term (dV/dt) is alone on the left-hand side of the equation and all of the other terms are on the right.


Define i_R as the current through the resistor, i_C as the current through the capacitor, I as the total current, and V as the voltage drop.

Current conservation dictates: I = i_R + i_C (1.1)
The resistive current is give by Ohm's law: I_R = V/R (1.2)
The capacitive current is give by: i_C = C (dV/dt) (1.3)
Substituting (1.2)-(1.3) into (1.1) I = (V/R) + C (dV/dt) (1.4)
Rearranging (1.4) to solve for dV/dt: dV/dt = -V/(RC) + I/C (1.5)

dV/dt = -V/(RC) + I/C

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