Animated Solution for Physics - Current Electricity: Two batteries of different emfs and different internal resistances are connected as shown. The voltage across $AB$ in volt is

The problem asks us to find the voltage across the terminals $A$ and $B$ for a circuit containing two batteries connected in parallel.

There is no external resistor or load connected across them. This is a crucial observation! Because the circuit is open, no net current flows out of the parallel combination into any external circuit.

In such a scenario, the potential difference across the terminals $A$ and $B$, denoted as $V_{AB}$, is simply equal to the equivalent EMF ($E_{eq}$) of the parallel combination of the batteries.

To find the equivalent EMF of batteries connected in parallel, we use a very powerful and standard formula:

This formula elegantly combines the individual EMFs and internal resistances into a single equivalent battery. Before plugging in the values, we must always check the polarities. In our circuit, the positive terminals (the longer lines of the battery symbol) of both the $6\text{ V}$ and $3\text{ V}$ batteries are facing the same direction (towards terminal $A$). Therefore, both EMFs will be taken as positive.

Let's identify the parameters for each branch from the given diagram: - For the top branch: $E_1=6\text{ V}$ and $r_1=1\Omega$ - For the bottom branch: $E_2=3\text{ V}$ and $r_2=2\Omega$

Now, we carefully substitute these values into our master equation:

Let's simplify the numerator and the denominator step-by-step.

In the numerator:

In the denominator:

Now, put them back together:

Dividing $7.5$ by $1.5$ gives us exactly $5$.

Thus, the voltage across the terminals $A$ and $B$ is $5\text{ V}$.

If the $3\text{ V}$ battery had been connected in reverse (with its negative terminal facing $A$), we would have used $-3\text{ V}$ in the numerator, which would drastically change the final answer. Mastering this formula and being vigilant about sign conventions will save you a lot of time in exams!