DC Circuits – Ohm’s Law, Series/Parallel, Kirchhoff’s Rules, Wheatstone Bridge

What is a DC Circuit?

A DC (Direct Current) circuit is an electrical circuit where the current flows in one direction and has a constant magnitude. Unlike AC (Alternating Current), DC does not change direction over time.

Key Components of a DC Circuit:

1. Battery (or Cell): Provides the voltage (V) or electromotive force (emf) to drive the current.
2. Resistor: Opposes the flow of current (measured in ohms, Ω).
3. Conductors (Wires): Allow current to flow with negligible resistance.
4. Switch: Controls the flow of current (open = no current, closed = current flows).

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Ohm’s Law

Ohm’s law states that the current (I) flowing through a conductor is directly proportional to the voltage (V) across it and inversely proportional to its resistance (R).

Mathematically: $$V=I\cdot R$$ or $$I=\frac{V}{R}$$ or $$R=\frac{V}{I}$$

Worked Example 1: Calculating Current

A battery of 12 V is connected to a resistor of 4 Ω. Find the current flowing through the circuit. Solution: Using $I=\frac{V}{R}$: $$I=\frac{12\text{V}}{4\Omega }=3\text{A}$$ Answer: The current is 3 A.

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Resistance in Series and Parallel

Resistors can be connected in series or parallel, changing the equivalent resistance of the circuit.

1. Resistors in Series

• Resistors are connected end-to-end.
• The same current flows through all resistors.
• The total resistance (R_eq) is the sum of individual resistances: $$R_{\text{eq}}=R_1+R_2+R_3+\dots$$

2. Resistors in Parallel

• Resistors are connected across the same two points.
• The voltage across each resistor is the same.
• The total resistance (R_eq) is given by: $$\frac{1}{R_{\text{eq}}}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+\dots$$ or $$R_{\text{eq}}=\frac{1}{\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}}$$

Worked Example 2: Equivalent Resistance

Find the equivalent resistance of two resistors, 6 Ω and 3 Ω, connected:

1. In series
2. In parallel

Solution:

1. Series: $$R_{\text{eq}}=6\Omega +3\Omega =9\Omega$$
2. Parallel: $$\frac{1}{R_{\text{eq}}}=\frac{1}{6}+\frac{1}{3}=\frac{1}{6}+\frac{2}{6}=\frac{3}{6}=\frac{1}{2}$$ $$R_{\text{eq}}=2\Omega$$ Answer:

• Series: 9 Ω
• Parallel: 2 Ω

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Kirchhoff’s Laws

Kirchhoff’s laws help analyze complex circuits with multiple loops and branches.

1. Kirchhoff’s Current Law (KCL)

• The total current entering a junction is equal to the total current leaving it.
• Mathematically: $$\sum I_{\text{in}}=\sum I_{\text{out}}$$

2. Kirchhoff’s Voltage Law (KVL)

• The sum of all voltage drops in a closed loop is zero.
• Mathematically: $$\sum V=0$$

Worked Example 3: Applying Kirchhoff’s Laws

In the circuit below, find the currents $I_1$ and $I_2$.

Solution:

1. Apply KCL at the junction: $$I_1=I_2+I_3$$ (Here, $I_3$ is the current through the 6 Ω resistor.)

2. Apply KVL in the loop: $$10\text{V}-4I_1-6I_3=0$$ $$10-3I_2-6I_3=0\text{(since }{I}_1=I_2+I_3\text{)}$$

3. Solve the equations:

   • From $I_1=I_2+I_3$, and using Ohm’s law: $$I_2=\frac{V}{3},I_3=\frac{V}{6}$$
   • Substitute and solve: $$I_1=1\text{A},I_2=0.5\text{A}$$ Answer:

• $I_1=1\text{A}$
• $I_2=0.5\text{A}$

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Wheatstone Bridge

A Wheatstone bridge is used to measure unknown resistance by balancing two legs of a bridge circuit.

Principle:

• When the bridge is balanced, the voltage across the galvanometer (G) is zero.
• The condition for balance is: $$\frac{R_1}{R_2}=\frac{R_3}{R_x}$$ where $R_x$ is the unknown resistance.

Worked Example 4: Finding Unknown Resistance

In a Wheatstone bridge, $R_1=10\Omega$, $R_2=20\Omega$, $R_3=30\Omega$, and the bridge is balanced. Find $R_x$.

Solution: Using the balance condition: $$\frac{R_1}{R_2}=\frac{R_3}{R_x}$$ $$\frac{10}{20}=\frac{30}{R_x}$$ $$R_x=\frac{30\times 20}{10}=60\Omega$$ Answer: The unknown resistance $R_x=60\Omega$.

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Practical Applications of DC Circuits

1. Voltmeter: Measures voltage across a component. Connected in parallel.
   • Has high resistance to minimize current draw.
2. Ammeter: Measures current through a circuit. Connected in series.
   • Has low resistance to avoid affecting the circuit.
3. Potentiometer: A variable resistor used to adjust voltage or current in a circuit.
4. Battery Charging Circuits: Use resistors and diodes to regulate charging.

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Common Mistakes to Avoid

1. Mixing series and parallel rules: Always identify how resistors are connected.
2. Ignoring polarity in KVL: Voltage drops and rises must be accounted for with signs.
3. Assuming all circuits are simple: Complex circuits require Kirchhoff’s laws.
4. Forgetting units: Always include units (V, A, Ω) in answers.

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NEB Board-Style Questions

Short Answer Questions

1. State Ohm’s law and define resistance.
2. How are resistors connected in series? Give one advantage of series connection.
3. What is the condition for a Wheatstone bridge to be balanced?
4. Differentiate between an ammeter and a voltmeter.

Long Answer Questions

5. Derive the formula for equivalent resistance of two resistors connected in parallel. A 6 Ω and a 3 Ω resistor are connected in parallel to a 12 V battery. Calculate:

   • The equivalent resistance.
   • The current through each resistor.
   • The total current drawn from the battery.

6. In the circuit shown below, find the current through each resistor using Kirchhoff’s laws.

7. Explain the working principle of a Wheatstone bridge. How is it used to measure unknown resistance?

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Exam Tip

1. Always draw the circuit: Label all components and connections clearly.
2. Use Kirchhoff’s laws for complex circuits: Label currents and voltages in loops.
3. Check units: NEB often deducts marks for missing units.
4. Practice numerical problems: Most exam questions are numerical.
5. Memorize key formulas:
   • Ohm’s law: $V=IR$
   • Series resistance: $R_{\text{eq}}=R_1+R_2$
   • Parallel resistance: $\frac{1}{R_{\text{eq}}}=\frac{1}{R_1}+\frac{1}{R_2}$
   • Wheatstone bridge balance: $\frac{R_1}{R_2}=\frac{R_3}{R_x}$

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Practical application of resistance measurement (Image: Dabrams3, CC BY-SA 4.0, via Wikimedia Commons)