Program 1

Capacitor Charging / Discharging

Python 3.11 Ready

import numpy as np
import matplotlib.pyplot as plt

# Parameters
R = 5e3  # Resistance in ohms
C = 1e-6  # Capacitance in farads
V_in = 5  # Input voltage in volts
t_max = 0.05  # Time duration for simulation in seconds
dt = 1e-4  # Time step for Euler method in seconds

# Number of steps
n_steps = int(t_max / dt)

# Arrays to store time and voltage
time = np.linspace(0, t_max, n_steps)
voltage_charging = np.zeros(n_steps)
voltage_discharging = np.zeros(n_steps)

# Initial conditions
voltage_charging[0] = 0  # Initial voltage across capacitor for charging
voltage_discharging[0] = V_in  # Initial voltage across capacitor for discharging

# Euler method for charging
for i in range(1, n_steps):
    dV = (V_in - voltage_charging[i-1]) / (R * C)
    voltage_charging[i] = voltage_charging[i-1] + dV * dt

# Euler method for discharging
for i in range(1, n_steps):
    dV = -voltage_discharging[i-1] / (R * C)
    voltage_discharging[i] = voltage_discharging[i-1] + dV * dt

# Plotting the results
plt.plot(time, voltage_charging, label='Charging', color='blue')
plt.plot(time, voltage_discharging, label='Discharging', color='red')

plt.title('Capacitor Charging and Discharging')
plt.xlabel('Time (s)')
plt.ylabel('Voltage (V)')
plt.legend()
plt.grid(True)
plt.savefig('plot.png')
plt.show()

Click "Run Code" to execute script and render figures.

Mathematical Problem Formulation

ODE for Capacitor Charging and Discharging

The voltage across a capacitor V(t) in an RC circuit can be described by the following ODE:

Charging:
dV(t)/dt = (1/RC) * (V_in - V(t))

Discharging:
dV(t)/dt = -V(t)/(RC)

Here:

V(t) is the voltage across the capacitor at time t.
V_in is the input voltage (for charging).
R is the resistance in ohms.
C is the capacitance in farads.

Theoretical Principles & Computational Physics Guide

RC Circuit Transients: Capacitor Charging & Discharging Dynamics

Dr. Alorika Chatterjee & Dr. Anirban Shaw Science & Physics Learners Python 3 / NumPy / SciPy

1. Core Principles & Theoretical Framework

First-Order Relaxation: Governs exponential charge and voltage transients in resistive-capacitive networks.

Characteristic Time Constant: Relaxation timescale $\tau = RC$ determines the rate of charging and discharging.

Asymptotic Saturation: Charging capacitor reaches $\approx 63.2\%$ of EMF at $t = \tau$, and $> 99\%$ at $t = 5\tau$.

2. Mathematical Formulation & Algorithmic Pipeline

Kirchhoff's voltage law governing an RC series loop:

$$ R \frac{dq}{dt} + \frac{q}{C} = \mathcal{E} \implies R C \frac{dV_C}{dt} + V_C = \mathcal{E}, \quad V_C(t) = \mathcal{E}\left(1 - e^{-t/\tau}\right) $$

3. Numerical Mechanics & Implementation Logic

Time Discretization: Choose step size $h \ll RC$ (e.g. $h = 0.01 RC$) to ensure numerical stability.
Transient Update: Advance voltage $V_{n+1} = V_n + \frac{h}{RC}(\mathcal{E} - V_n)$.
Phase Switching: At specified discharge time $t_{\text{switch}}$, set driving EMF $\mathcal{E} = 0$ to observe decay.

4. Relevant Computational Practice Exercises

Exercise 1

Simulate capacitor charging from $t = 0$ to $5\tau$ and verify that $V_C(\tau) \approx 0.632\mathcal{E}$.

Exercise 2

Simulate an RC circuit driven by a square-wave source and plot voltage response demonstrating low-pass filtering.

Exercise 3

Compute instantaneous power $P(t) = I(t)^2 R$ and verify total energy conservation during complete charge-discharge cycles.

Standard Literature & References:

David J. Griffiths, "Introduction to Electrodynamics", Cambridge University Press (2017).

Ch. 6.3 Capacitor Charging / Discharging