In the circuit shown, the potential difference between A and B is
(a) 3 V
(b) 1 V
(c) 6 V
(d) 2 V

Q: Why is the current in the $5\Omega$ and $10\Omega$ resistors zero?

Because terminals $A$ and $B$ are open, meaning they aren't connected to anything to form a closed loop. Without a closed path, electrons cannot flow, so the current is strictly zero.

Q: Can I use Kirchhoff's laws instead of Millman's Theorem?

Absolutely! You can assign loop currents and solve the simultaneous equations. However, Millman's Theorem is a powerful shortcut specifically designed for parallel branches, saving you valuable time during exams.

Q: What exactly is Millman's Theorem?

It's a theorem that simplifies a circuit with multiple parallel voltage sources into a single equivalent voltage source. It works by dividing the total short-circuit current of the branches by their total conductance.

Q: Why do we completely ignore the $5\Omega$ and $10\Omega$ resistors in the calculation?

Since no current flows through them ($I = 0$), the voltage drop across them is $0\text{ V}$ ($V = 0 \times R = 0$). Thus, they don't change the potential between the inner nodes ($D, C$) and the outer terminals ($A, B$).

Q: What if one of the batteries was connected in the opposite direction?

Great question! You would simply assign a negative sign to the EMF of that reversed battery when plugging it into the numerator of Millman's formula.

Question

In the circuit shown, the potential difference between A and B is

Options:

(A) 3 V

(B) 1 V

(C) 6 V

(D) 2 V

Sigma Percentile · Accepted Answer

Solving circuit problems often feels like untangling a complex web of wires, but with the right tools, it becomes an elegant puzzle. Let's dive into this fascinating problem and uncover the potential difference between two open terminals.

### Analyzing the Setup

Imagine you are looking at the circuit. We have two terminals, $A$ and $B$, which are completely open. What does this mean physically? An open circuit is like a broken bridge—electrons have no path to cross. Therefore, absolutely **zero current** flows through the $5\Omega$ and $10\Omega$ resistors connected to these terminals.

Since there is no current ($I = 0$), Ohm's Law ($V = IR$) tells us that the voltage drop across these outer resistors is exactly zero. This is a crucial insight! It means the potential at terminal $A$ is identical to the potential at node $D$ ($V_A = V_D$), and the potential at terminal $B$ is identical to the potential at node $C$ ($V_B = V_C$).

Consequently, finding the potential difference between $A$ and $B$ ($V_{AB}$) is exactly the same as finding the potential difference between $D$ and $C$ ($V_{DC}$).

### The Master Equation

Now, let's focus our attention on the core of the circuit: the three parallel branches between nodes $D$ and $C$. Each branch contains a voltage source and a resistor. When faced with multiple parallel batteries, **Millman's Theorem** is our ultimate weapon. It allows us to condense these parallel branches into a single equivalent voltage.

The theorem states:
$$V_{DC} = \frac{\sum \frac{E_i}{R_i}}{\sum \frac{1}{R_i}}$$

This elegant formula simply divides the sum of the short-circuit currents of each branch by the sum of their conductances.

### Final Calculation

Let's carefully substitute the values from our three branches into Millman's formula.

For the numerator (the sum of $\frac{E_i}{R_i}$):
- Top branch: $\frac{1	ext{ V}}{1\Omega} = 1	ext{ A}$
- Middle branch: $\frac{2	ext{ V}}{1\Omega} = 2	ext{ A}$
- Bottom branch: $\frac{3	ext{ V}}{1\Omega} = 3	ext{ A}$

For the denominator (the sum of conductances $\frac{1}{R_i}$):
- Each branch has a $1\Omega$ resistor, so we add $\frac{1}{1} + \frac{1}{1} + \frac{1}{1} = 3\Omega^{-1}$.

Putting it all together:
$$V_{DC} = \frac{1 + 2 + 3}{1 + 1 + 1}$$
$$V_{DC} = \frac{6}{3} = 2	ext{ V}$$

Since we established earlier that $V_{AB} = V_{DC}$, the potential difference between terminals $A$ and $B$ is exactly **$2	ext{ V}$**. The beauty of this problem lies in recognizing that the outer resistors are merely a distraction, leading us to a swift and satisfying solution!

Animated Solution for Physics - Current Electricity: In the circuit shown, the potential difference between A and B is

Select Answer:

Visualized Solution

Circuit Analysis

Zero Current in Outer Resistors

Millman's Theorem

Substituting Values

Calculating $V_{D C}$

Final Answer

The Way Forward

The Sigma Insight: Cells, EMF, and Internal Resistance

Analyzing the Setup

The Master Equation

Final Calculation

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