DC Circuits: Kirchhoff's Laws, RC Circuits & Analysis | PHYSICS HRK

House of Physics September 30, 2025
DC Circuits: Complete Guide to Kirchhoff's Laws, RC Circuits & Multiloop Analysis

Chapter 33: DC Circuits

Complete Guide to Kirchhoff's Laws, RC Circuits & Multiloop Analysis - B.Sc. Physics Edition 2015-16
DC Circuits Kirchhoff's Laws RC Circuits Electromotive Force Reading Time: 30 min

📋 Table of Contents

Introduction to DC Circuits

⚡ What are DC Circuits?

DC (Direct Current) circuits are electrical circuits in which the direction of current doesn't change with time. In DC circuits containing only batteries and resistors, the magnitude of current doesn't vary with time. However, in DC circuits containing capacitors, the magnitude of current may be time-dependent.

🔬 Key Characteristics of DC Circuits

  • Constant direction: Current flows in one direction only
  • Steady state: Current magnitude remains constant in resistive circuits
  • Time-dependent behavior: Current may vary in circuits with capacitors
  • Energy sources: Typically batteries or DC power supplies

Electromotive Force (EMF)

🔋 Definition of Electromotive Force

The amount of energy supplied per unit charge in order to move it in a circuit is called electromotive force (EMF). Despite its name, EMF is not actually a force - we don't measure it in newtons. The unit of EMF is joule/coulomb, which is volt:

\[ 1 \, \text{volt} = 1 \, \text{joule/coulomb} \]

📏 Important Note About EMF

The term "electromotive force" is historical and somewhat misleading, as EMF is not a force but rather an energy per unit charge. It represents the maximum potential difference between the terminals of a source when no current is flowing.

Sources of Electromotive Force

🔌 What are EMF Sources?

Devices that provide EMF in electrical circuits to sustain steady current flow are called sources of EMF. These sources provide energy to charge carriers to make the steady flow of current possible. Sources of EMF convert some non-electrical energy (chemical, mechanical, heat, or solar energy) to electrical energy.

How EMF Sources Work

A source of electromotive force maintains one terminal at a high potential and its other terminal at low potential. The EMF causes positive charge carriers to move in the external circuit from the positive to the negative terminal.

Internal Mechanism

In its interior, the source of EMF acts to move positive charges from the point of low potential to the point of high potential. The charges then move through the external circuit, dissipating energy in the process, and return to the negative terminal, from which the EMF raises them to the positive terminal again.

🔋 Types of EMF Sources

  • Chemical: Batteries, fuel cells
  • Mechanical: Generators, dynamos
  • Thermal: Thermocouples
  • Solar: Photovoltaic cells
  • Electromagnetic: Transformers

Current in Single Loop Using Energy Conservation

⚖️ Energy Conservation Principle

The energy supplied by the source of EMF must equal the energy dissipated in the resistor. This principle allows us to determine the current in a single loop circuit.

Energy Supplied by EMF Source

In time dt, a charge dq (= I dt) moves through the source of EMF. The energy supplied is:

\[ \text{Energy Supplied} = \mathcal{E} \cdot dq \]
\[ = \mathcal{E} \cdot I dt \]

Energy Dissipated in Resistor

The energy dissipated in the resistor R is:

\[ \text{Energy Dissipated} = I^2 R dt \]

Apply Energy Conservation

From conservation of energy:

\[ \mathcal{E} I dt = I^2 R dt \]

Solve for Current

\[ \mathcal{E} I = I^2 R \]
\[ \mathcal{E} = I R \]
\[ I = \frac{\mathcal{E}}{R} \]

📐 Single Loop Current Formula

For a single loop circuit with EMF source \(\mathcal{E}\) and resistor R:

\[ I = \frac{\mathcal{E}}{R} \]

Kirchhoff's Voltage Rule

⚖️ Statement of Kirchhoff's Voltage Rule

Kirchhoff's Voltage Rule is a particular way of stating the law of conservation of energy for a charge carrier traveling in a closed circuit:

"The algebraic sum of the changes in potential encountered in a complete traversal of any closed circuit is zero."

🔁 Understanding the Rule

When a charge completes a loop in a circuit, it returns to its starting point with the same potential energy it had initially. Therefore, the net change in potential around any closed loop must be zero.

💡 Applying Kirchhoff's Voltage Rule

  • Choose a direction (clockwise or counterclockwise) to traverse the loop
  • When going through a battery from negative to positive terminal: +EMF
  • When going through a battery from positive to negative terminal: -EMF
  • When going through a resistor in the direction of current: -IR
  • When going through a resistor opposite to current direction: +IR

Current in Single Loop Using Kirchhoff Rule

🔍 Applying Kirchhoff's Rule to Single Loop

Consider a single loop circuit with one source of EMF \(\mathcal{E}\) and one resistor R. Let a steady current I flow through the loop.

Apply Kirchhoff's Voltage Rule

Starting from any point and traversing the loop:

\[ \mathcal{E} - IR = 0 \]

Solve for Current

\[ \mathcal{E} = IR \]
\[ I = \frac{\mathcal{E}}{R} \]

📐 Verification

This result matches the one we obtained using energy conservation, confirming the consistency of both approaches.

Internal Resistance of EMF Sources

🔋 Real EMF Sources Have Internal Resistance

All sources of EMF have an intrinsic internal resistance, which can't be removed because it is an inherent part of the device. This internal resistance affects the current that the EMF source can supply to the external circuit.

Circuit with Internal Resistance

Consider a single loop circuit with EMF source \(\mathcal{E}\) having internal resistance r and external resistor R.

Apply Kirchhoff's Voltage Rule

\[ \mathcal{E} - Ir - IR = 0 \]

Solve for Current

\[ \mathcal{E} = I(r + R) \]
\[ I = \frac{\mathcal{E}}{R + r} \]

📐 Current with Internal Resistance

For a single loop circuit with EMF source \(\mathcal{E}\), internal resistance r, and external resistance R:

\[ I = \frac{\mathcal{E}}{R + r} \]

The internal resistance r reduces the current that the EMF source can supply to the external circuit.

Potential Differences in Circuits

🔌 Calculating Potential Differences

To find the potential difference between two points in a circuit, we need to consider the path taken between those points and account for all potential changes along that path.

Circuit Setup

Consider a circuit with resistor R and EMF source \(\mathcal{E}\) with internal resistance r. We want to find the potential difference \(V_{ab}\) between points a and b.

Potential Difference Across Resistor

\[ V_b + IR = V_a \]
\[ V_{ab} = V_a - V_b = IR \]

Substitute Current Expression

\[ I = \frac{\mathcal{E}}{R + r} \]
\[ V_{ab} = \left( \frac{\mathcal{E}}{R + r} \right) R \]
\[ V_{ab} = \mathcal{E} \frac{R}{R + r} \]

📐 Potential Difference Formula

The potential difference across a resistor R in a circuit with EMF source \(\mathcal{E}\) and internal resistance r is:

\[ V_{ab} = \mathcal{E} \frac{R}{R + r} \]

Equivalent Resistance in Parallel

🔗 Parallel Resistor Configuration

In parallel arrangement, resistors are connected side by side with their ends joined together at common points. The potential difference across each resistor is the same.

Current in Parallel Circuit

The total current is shared among the branches:

\[ I = I_1 + I_2 \]

Current Through Each Resistor

\[ I_1 = \frac{V}{R_1} \]
\[ I_2 = \frac{V}{R_2} \]

Total Current Expression

\[ I = \frac{V}{R_1} + \frac{V}{R_2} \]
\[ I = V \left( \frac{1}{R_1} + \frac{1}{R_2} \right) \]

Equivalent Resistance

For equivalent resistance \(R_{eq}\):

\[ I = \frac{V}{R_{eq}} \]
\[ \frac{V}{R_{eq}} = V \left( \frac{1}{R_1} + \frac{1}{R_2} \right) \]
\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} \]

📐 General Formula for Parallel Resistors

For any number of resistors in parallel:

\[ \frac{1}{R_{eq}} = \sum_n \frac{1}{R_n} \]

Resistors Connected in Series

🔗 Series Resistor Configuration

When resistors are connected end to end such that the same current passes through all of them, they are said to be connected in series.

Potential Differences in Series

The potential difference across each resistor is:

\[ V_1 = IR_1 \]
\[ V_2 = IR_2 \]

Total Potential Difference

The sum of potential drops equals the battery voltage:

\[ V = V_1 + V_2 \]

Equivalent Resistance

For equivalent resistance \(R_{eq}\):

\[ V = IR_{eq} \]
\[ IR_{eq} = IR_1 + IR_2 \]
\[ R_{eq} = R_1 + R_2 \]

📐 General Formula for Series Resistors

For any number of resistors in series:

\[ R_{eq} = \sum_n R_n \]

Multiloop Circuits

🔄 Analyzing Complex Circuits

Multiloop circuits contain multiple closed loops and junctions. To analyze such circuits, we need to apply both Kirchhoff's Voltage Rule and Kirchhoff's Current Rule.

Kirchhoff's Current Rule

Also known as the junction rule, it states that the algebraic sum of currents at any junction is zero:

\[ \sum I_{in} = \sum I_{out} \]

Kirchhoff's Voltage Rule

The algebraic sum of potential differences around any closed loop is zero:

\[ \sum V = 0 \]

Procedure for Solving Multiloop Circuits

  1. Assign currents to each branch (with directions)
  2. Apply Kirchhoff's Current Rule at junctions
  3. Apply Kirchhoff's Voltage Rule to each independent loop
  4. Solve the resulting system of equations

🔍 Example: Two-Loop Circuit

Consider a circuit with two EMF sources and three resistors. We would:

  • Assign currents I₁, I₂, I₃ to the three branches
  • Apply junction rule at one junction
  • Apply voltage rule to two independent loops
  • Solve the three equations for the three unknown currents

Growth of Charge in RC Circuits

⚡ RC Circuits with Capacitors

When a circuit contains both resistors and capacitors, the current and charge vary with time. These are called RC circuits.

Charging a Capacitor

Consider a circuit with EMF source \(\mathcal{E}\), resistor R, and capacitor C in series. Initially, the capacitor is uncharged (q=0 at t=0).

Apply Kirchhoff's Voltage Rule

\[ \mathcal{E} - IR - \frac{q}{C} = 0 \]

Express Current in Terms of Charge

\[ I = \frac{dq}{dt} \]
\[ \mathcal{E} - R\frac{dq}{dt} - \frac{q}{C} = 0 \]

Rearrange the Equation

\[ R\frac{dq}{dt} = \mathcal{E} - \frac{q}{C} \]
\[ \frac{dq}{dt} = \frac{\mathcal{E}}{R} - \frac{q}{RC} \]

Solve the Differential Equation

\[ \frac{dq}{dt} = \frac{\mathcal{E}C - q}{RC} \]
\[ \frac{dq}{\mathcal{E}C - q} = \frac{dt}{RC} \]

Integrate Both Sides

\[ \int_0^q \frac{dq'}{\mathcal{E}C - q'} = \frac{1}{RC} \int_0^t dt' \]
\[ -\ln(\mathcal{E}C - q') \Big|_0^q = \frac{t}{RC} \]
\[ -\ln(\mathcal{E}C - q) + \ln(\mathcal{E}C) = \frac{t}{RC} \]
\[ \ln\left( \frac{\mathcal{E}C}{\mathcal{E}C - q} \right) = \frac{t}{RC} \]

Exponentiate Both Sides

\[ \frac{\mathcal{E}C}{\mathcal{E}C - q} = e^{t/RC} \]
\[ \mathcal{E}C - q = \mathcal{E}C e^{-t/RC} \]
\[ q = \mathcal{E}C (1 - e^{-t/RC}) \]

📐 Charge Growth Formula

The charge on a capacitor in an RC circuit during charging is:

\[ q(t) = Q_f (1 - e^{-t/RC}) \]

where \(Q_f = \mathcal{E}C\) is the final charge on the capacitor.

Decay of Charge in RC Circuits

⚡ Discharging a Capacitor

When a charged capacitor is connected to a resistor (without an EMF source), the capacitor discharges through the resistor.

Circuit Setup

Consider a circuit with capacitor C and resistor R. Initially, the capacitor has charge \(Q_0\).

Apply Kirchhoff's Voltage Rule

\[ -IR - \frac{q}{C} = 0 \]

Express Current in Terms of Charge

\[ I = -\frac{dq}{dt} \]
\[ -(-R\frac{dq}{dt}) - \frac{q}{C} = 0 \]
\[ R\frac{dq}{dt} + \frac{q}{C} = 0 \]

Rearrange the Equation

\[ \frac{dq}{dt} = -\frac{q}{RC} \]

Solve the Differential Equation

\[ \frac{dq}{q} = -\frac{dt}{RC} \]
\[ \int_{Q_0}^q \frac{dq'}{q'} = -\frac{1}{RC} \int_0^t dt' \]
\[ \ln q' \Big|_{Q_0}^q = -\frac{t}{RC} \]
\[ \ln q - \ln Q_0 = -\frac{t}{RC} \]
\[ \ln\left( \frac{q}{Q_0} \right) = -\frac{t}{RC} \]

Exponentiate Both Sides

\[ \frac{q}{Q_0} = e^{-t/RC} \]
\[ q(t) = Q_0 e^{-t/RC} \]

📐 Charge Decay Formula

The charge on a capacitor in an RC circuit during discharging is:

\[ q(t) = Q_0 e^{-t/RC} \]

where \(Q_0\) is the initial charge on the capacitor.

⏱️ Time Constant in RC Circuits

The product RC is called the time constant (τ) of the circuit:

\[ \tau = RC \]

The time constant represents the time required for the charge to reach approximately 63.2% of its final value during charging, or to decay to approximately 36.8% of its initial value during discharging.

Frequently Asked Questions

❓ What is the difference between EMF and terminal voltage?

EMF is the maximum potential difference between the terminals of a source when no current is flowing. Terminal voltage is the actual potential difference between the terminals when current is flowing, which is less than the EMF due to internal resistance: \(V_{terminal} = \mathcal{E} - Ir\).

❓ Why do we need both Kirchhoff's rules to analyze multiloop circuits?

Kirchhoff's Current Rule (junction rule) ensures conservation of charge at junctions, while Kirchhoff's Voltage Rule (loop rule) ensures conservation of energy around closed loops. Both are needed to fully describe the behavior of complex circuits with multiple loops and junctions.

❓ What is the physical significance of the time constant in RC circuits?

The time constant τ = RC determines how quickly a capacitor charges or discharges in an RC circuit. After one time constant, the capacitor reaches about 63.2% of its final charge during charging, or decays to about 36.8% of its initial charge during discharging. After 5 time constants, the capacitor is considered fully charged or discharged (over 99%).

❓ How does internal resistance affect battery performance?

Internal resistance reduces the terminal voltage of a battery when current flows, limits the maximum current the battery can supply, and causes energy loss within the battery itself (manifested as heating). Batteries with lower internal resistance can deliver more power to external circuits.

❓ Why does current vary with time in RC circuits but not in purely resistive circuits?

In purely resistive circuits, current reaches its steady-state value almost instantly when voltage is applied. In RC circuits, capacitors take time to charge or discharge, causing current to vary exponentially with time until the capacitor is fully charged or discharged.

📚 Continue Your Physics Journey

Mastering DC Circuits is fundamental to understanding electrical engineering and advanced physics. These comprehensive notes provide a solid foundation for further studies in AC circuits, electronics, and electromagnetic theory.

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© 2025 Physics Education Initiative | Chapter 33: DC Circuits

These comprehensive notes are designed to help B.Sc. Physics students understand fundamental concepts of DC Circuits based on Halliday, Resnick and Krane

Author: Muhammad Ali Malik | Contact: +923016775811 | Email: aliphy2008@gmail.com

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