The Question is from neet-physics , and it was created by the author ABDUL JILANI BASHA

Question

Two batteries with EMFs $E_1$ and $E_2$ and internal resistances $r_1$ and $r_2$ respectively are connected in parallel across a resistor R.  The current through R is zero. Which of the following relations must hold true?

ABDUL JILANI BASHA · Accepted Answer

## CORE INFORMATION
✓ **Answer**: A
## CONCEPT FRAMEWORK
 **Primary Concept**: Kirchhoff's laws and parallel battery connections
 **Related Topics**: Circuit analysis, EMF, internal resistance
 **Prerequisites**: Understanding of Ohm's law, Kirchhoff's voltage and current laws.
## SOLUTION PATHWAY
**Given Information**:
  Two batteries with EMFs $E_1$ and $E_2$ and internal resistances $r_1$ and $r_2$ are connected in parallel across a resistor R.
  The current through R is zero.
**Method & Steps**:
 1. Apply Kirchhoff's voltage law (KVL) to the loop containing battery 1, its internal resistance, and resistor R.
  Reasoning: The sum of voltages around any closed loop in a circuit is zero.
  Formula/Rule: $\sum V = 0$
 2. Apply KVL to the loop containing battery 2, its internal resistance, and resistor R.
  Working: For the first loop, we have $E_1 - I_1r_1 - I_R R = 0$, where $I_1$ is the current through battery 1 and $I_R$ is the current through R. For the second loop, we have $E_2 - I_2r_2 - I_R R = 0$, where $I_2$ is the current through battery 2.
  Key Point: Since the current through R is zero ($I_R = 0$), the equations simplify to $E_1 - I_1r_1 = 0$ and $E_2 - I_2r_2 = 0$. This gives $I_1 = \frac{E_1}{r_1}$ and $I_2 = \frac{E_2}{r_2}$.
 3. Apply Kirchhoff's current law (KCL) at the junction where the batteries connect to R.
  Reasoning: The sum of currents entering a junction equals the sum of currents leaving the junction.
  Formula/Rule: $\sum I = 0$
  Result: KCL gives $I_1 + I_2 - I_R = 0$. Since $I_R = 0$, we have $I_1 = -I_2$. Substituting the expressions for $I_1$ and $I_2$, we get $\frac{E_1}{r_1} = -\frac{E_2}{r_2}$, which simplifies to $E_1r_2 = -E_2r_1$.  However, the currents are flowing in opposite directions, so we should consider the magnitudes. Thus we have $E_1r_2 = E_2r_1$.
  Verification: If this condition holds, the potential difference across the parallel combination of the batteries is zero which results in zero current through R.
 ## OPTION ANALYSIS
A ✓: $E_1r_2 = E_2r_1$: This is the correct relation derived from applying Kirchhoff's laws.
B :x: $E_1r_1 = E_2r_2$: This relation does not guarantee zero current through R.
C :x: $E_1 = E_2$: This is a special case; zero current through R can occur even if the EMFs are not equal.
D :x: $r_1 = r_2$:  This is also a special case; zero current through R can occur even if the internal resistances are not equal.
 ## LEARNING AIDS
 **Common Mistakes**:
  1.Ignoring the direction of currents when applying KCL:  Always consider the sign convention for current flow.
  2.Incorrectly applying KVL: Ensure you traverse the loop consistently (either clockwise or counterclockwise) and account for voltage drops across resistors and internal resistances.
 **Quick Tips**:
  1.Always draw a clear circuit diagram before applying Kirchhoff's laws.
  2.Remember that KVL deals with voltages around loops, and KCL deals with currents at junctions.

ABDUL JILANI BASHA · Answer

The correct Option is "$E_1r_2 = E_2r_1$"

ABDUL JILANI BASHA · Answer

$E_1r_1 = E_2r_2$

ABDUL JILANI BASHA · Answer

$E_1 = E_2$

ABDUL JILANI BASHA · Answer

$r_1 = r_2$

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