Derive equation of current for R-C circuit

Sure, let's derive the equation for the current in an RC (Resistor-Capacitor) circuit.

Step 1: Understand the Circuit An RC circuit consists of a resistor (R) and a capacitor (C) connected in series. When a voltage (V) is applied across the circuit, the capacitor starts charging up to the applied voltage.

Step 2: Apply Kirchhoff's Voltage Law According to Kirchhoff's voltage law, the sum of the potential differences (voltages) around any closed loop or mesh in a network is always equal to zero. This gives us the equation:

V = V_R + V_C

where V_R is the voltage across the resistor and V_C is the voltage across the capacitor.

Step 3: Express Voltages in terms of Current The voltage across the resistor can be expressed as V_R = I*R (Ohm's Law), and the voltage across the capacitor can be expressed as V_C = Q/C, where Q is the charge on the capacitor and I is the current flowing through the circuit.

Step 4: Express Charge in terms of Current The charge Q on the capacitor is the integral of the current I with respect to time (t), i.e., Q = ∫I dt.

Step 5: Substitute the Expressions Substitute the expressions from steps 3 and 4 into the equation from step 2:

V = I*R + ∫I dt / C

Step 6: Differentiate with respect to Time Differentiate the above equation with respect to time:

0 = R * dI/dt + I/C

Step 7: Rearrange the Equation Rearrange the equation to express it in terms of current:

dI/dt + I/(R*C) = 0

This is the differential equation for the current in an RC circuit. The solution to this equation gives the current as a function of time, which describes how the current changes over time in an RC circuit.