Animated Solution for Physics - Current Electricity: Two sources of equal emf are connected to an external resistance $R$. The internal resistances of the two sources are $R_1$ and $R_2$ ($R_2>R_1$). If the potential difference across the source having internal resistance $R_2$ is zero, then

The problem presents a classic circuit scenario involving two cells connected in series with an external resistor. Our goal is to find the value of the external resistance $R$ given a specific condition: the potential difference across the second cell is zero.

Imagine you are tracing the path of the current. We have two cells, each with an electromotive force (EMF) of $E$. They are connected in series, meaning their EMFs add up to drive the current in the same direction. The total EMF of the circuit is $2E$.

The circuit also contains resistance. The first cell has an internal resistance $R_1$, the second has $R_2$, and there is an external resistor $R$. Since all these components are in a single loop, they are in series. The equivalent resistance of the entire circuit is simply the sum of all individual resistances:

With the total EMF and total resistance known, we can use Ohm's law to find the current $I$ flowing through the circuit:

Now, let's focus on the crucial condition given in the problem: the potential difference across the second cell is zero. When a cell is discharging (supplying current), the potential difference $V$ across its terminals is less than its EMF due to the voltage drop across its internal resistance. The formula is:

For our second cell, the EMF is $E$ and the internal resistance is $R_2$. Therefore, the potential difference $V_2$ across it is:

The problem states that $V_2=0$. Setting our equation to zero gives:

Now, we substitute the expression for the current $I$ that we found earlier:

Notice how the EMF $E$ appears on both sides of the equation. We can safely cancel it out, simplifying our work significantly:

Cross-multiplying to solve for $R$, we get:

Rearranging the terms to isolate $R$:

This is our final answer. The external resistance must be exactly the difference between the two internal resistances.

Notice the condition $R_2>R_1$ provided in the question. This ensures that our calculated resistance $R$ is a positive value, which is a physical requirement for standard resistors. If $R_1$ were greater than $R_2$, the required external resistance would be negative, meaning the condition of zero potential difference across the second cell could never be met in a simple passive circuit.