Q.Two cells of the same emf $E$ but different internal resistances $r_1$ and $r_2$ are connected in series with each other and with an external resistor $R$, the first cell, the second cell and $R$ forming a single series loop. What should be the value of $R$ so that the potential difference across the terminals of the first cell (the one with internal resistance $r_1$) becomes zero?

Internal Resistance – The Battery That Fights Itself

Imagine you have a bucket of water with a tap at the bottom. When you open the tap fully, water flows out freely. But if the pipe is narrow or clogged, the flow is weaker even though the bucket is full. A battery behaves the same way: it has a "full bucket" of electrical energy (its emf, or electromotive force), but inside the battery, there is always some "clogged pipe" that resists the flow of charge. That clog is called internal resistance.

The Intuition

Every real battery is not a perfect source of voltage. If you connect a small bulb to a fresh battery, it glows brightly. But if you connect a powerful motor that draws a large current, the same battery might struggle — the voltage at its terminals drops, and the motor runs slower. Why? Because inside the battery, the chemical reactions and the materials themselves have a natural opposition to the flow of charge. This opposition is internal resistance, denoted by $r$ (or sometimes $R_{\text{int}}$).

Think of it this way: the battery has two "battles" to fight. First, it must push charge through the external circuit (the bulb, the motor, the wires). Second, it must push charge through its own internal structure. The harder it has to push (i.e., the larger the current), the more voltage it "loses" inside itself.

The Precise Statement

Internal resistance is the opposition to the flow of electric current offered by the materials and chemical processes inside a source of emf (like a battery, cell, or generator). It is measured in ohms ($\Omega$).

For a cell or battery, the relationship between the emf ($\mathcal{E}$), the terminal voltage ($V$), the current ($I$), and the internal resistance ($r$) is given by:

This is the single most important equation to remember.

Here:

• $\mathcal{E}$ is the electromotive force — the maximum voltage the battery can provide when no current flows (open circuit). Think of it as the "ideal" voltage.
• $V$ is the terminal voltage — the actual voltage you measure across the battery's terminals when it is delivering current.
• $I$ is the current flowing through the circuit (and therefore through the battery itself).
• $r$ is the internal resistance.

The term $I r$ is the voltage drop inside the battery. It is the "price" the battery pays for delivering current.

What Happens in Different Situations?

Why Does Internal Resistance Matter?

1. Power loss inside the battery: The power dissipated as heat inside the battery is $P_{\text{loss}} = I^2 r$. This is why batteries get warm when delivering large currents.
2. Maximum power transfer: There is a famous theorem (Maximum Power Transfer Theorem) that says a source delivers maximum power to an external load when the load resistance equals the internal resistance ($R_{\text{load}} = r$). But this is inefficient — half the power is wasted inside the source. …