Internal Resistance

Conservation of Energy: The total energy supplied by the source must equal the sum of the energy used in the external circuit and the energy lost internally. This is expressed by the fundamental equation $\epsilon = V + Ir$.

Ohm's Law Application: The terminal voltage $V$ is the product of the current $I$ and the external resistance $R$ ($V = IR$). Substituting this into the energy equation gives $\epsilon = I(R + r)$.

Thermal Dissipation: When current flows through the internal resistance, electrical energy is converted into thermal energy. This explains why batteries often feel warm to the touch during heavy use or rapid charging.

Determining Internal Resistance Graphically: By varying the external resistance $R$ and measuring the terminal voltage $V$ and current $I$, one can plot a graph of $V$ against $I$. The relationship $V = \epsilon - Ir$ shows that this graph is a straight line with a y-intercept of $\epsilon$ and a gradient of $-r$.

Open-Circuit Measurement: To find the EMF of a cell, use a high-resistance voltmeter to measure the voltage across the terminals when no other components are connected. Because the current is effectively zero, the $Ir$ term vanishes, and the reading equals the EMF.

Short-Circuit Analysis: In a theoretical short-circuit where $R = 0$, the current is limited only by the internal resistance ($I_{max} = \frac{\epsilon}{r}$). This represents the maximum possible current the source can provide, though it often damages the source.

Identify the Intercepts: On a $V$ vs. $I$ graph, the point where the line crosses the vertical axis is the EMF. The point where it crosses the horizontal axis represents the short-circuit current.

Check the Gradient: Always remember that the gradient of a $V-I$ graph for a power source is negative. The absolute value of this gradient is the internal resistance $r$.

Power Calculations: Be prepared to calculate the power dissipated internally using $P = I^2 r$. This is often used in questions involving efficiency or thermal limits of a battery.

Sanity Check: The terminal voltage $V$ should never exceed the EMF $\epsilon$ in a simple discharging circuit. If your calculation shows $V > \epsilon$, check for errors in your algebraic signs or current direction.

Confusing $V$ and $\epsilon$: Students often use the EMF value as the voltage in $V = IR$ calculations. You must subtract the 'lost volts' ($Ir$) from the EMF to find the correct voltage acting across the external resistor.

Assuming $r$ is Constant: While often treated as a constant in introductory physics, internal resistance can change with temperature, age, and the state of charge of a battery. In advanced problems, look for clues that $r$ might vary.

Ignoring Internal Resistance in Series: When multiple cells are in series, their internal resistances add up just like their EMFs. Forgetting to sum the $r$ values will lead to an overestimation of the circuit current.