Batteries and internal resistance Knowledge-Based Study

Study notes

1. Electromotive Force (emf)

   An ideal battery maintains a constant potential difference between its terminals regardless of the current drawn by an external circuit. This work done per unit charge in moving charge carriers from the lower to the higher potential terminal inside the source is known as the electromotive force, symbolized by $\mathcal{E}$.

   Although named a force historically, electromotive force is an electrical potential measured in volts (V), where 1 V = 1 J/C.

   Why Real Sources Differ

   In any real physical cell or battery, electrical current passes through internal components such as chemical electrolytes, metal electrodes, or carbon plates. These materials inherently resist the motion of charges. As a result, a real battery is modeled as an ideal source of emf $\mathcal{E}$ connected in series with an inherent internal resistance, denoted by $r$.

   Terminal Voltage under Load

   When a battery delivers current to an external load, charges must overcome the internal resistance $r$. This results in an internal potential drop, often referred to as the lost volts, equal to $I r$.

   The potential difference measured across the battery's external terminals is the terminal potential difference (or terminal voltage), $V_{\text{terminal}}$:

   $$V_{\text{terminal}} = \mathcal{E} - I r$$

   Key Operational Cases

   • Open Circuit (Zero Current): When no external circuit is connected, $I = 0$. Therefore, $V_{\text{terminal}} = \mathcal{E}$. An ideal voltmeter placed across an isolated battery measures its open-circuit emf directly.
   • Discharging (Delivering Current): When the cell supplies current $I > 0$ to a load, $V_{\text{terminal}} < \mathcal{E}$ due to the internal drop $Ir$.
   • Recharging (Current Driven Backward): If an external generator drives current into the positive terminal of the cell, the potential equation becomes $V_{\text{terminal}} = \mathcal{E} + I r$.

   Applying Ohm's Law to a Complete Circuit

   Consider a real battery of emf $\mathcal{E}$ and internal resistance $r$ connected across a single external load resistor of resistance $R$. By Kirchhoff's loop rule, the sum of potential changes around the closed loop must equal zero:

   $$\mathcal{E} - I r - I R = 0$$

   Solving for the circuit current $I$ gives:

   $$I = \frac{\mathcal{E}}{R + r}$$

   Short-Circuit Current

   If the terminals of a battery are directly connected with negligible external resistance ($R \to 0$), the resulting current is termed the short-circuit current:

   $$I_{\text{sc}} = \frac{\mathcal{E}}{r}$$

   Because $r$ is typically small, $I_{\text{sc}}$ can be extremely large, leading to rapid overheating and potential hazard.

   Power Distribution in the Circuit

   Multiplying the loop potential equation by current $I$ yields the conservation of energy per unit time:

   $$\mathcal{E} I = I^2 R + I^2 r$$

   • Total power produced by the source: $P_{\text{total}} = \mathcal{E} I$
   • Useful power delivered to the load: $P_{\text{load}} = I^2 R = \frac{\mathcal{E}^2 R}{(R + r)^2}$
   • Power dissipated internally as heat: $P_{\text{internal}} = I^2 r$

   Maximum Power Transfer Theorem

   For a fixed real battery with constant $\mathcal{E}$ and $r$, the power delivered to a variable external load $R$ reaches its maximum value when the load resistance equals the internal resistance:

   $$R = r$$

   At this matching condition, the maximum power transferred is:

   $$P_{\text{max}} = \frac{\mathcal{E}^2}{4r}$$

   Note that while power transfer is maximized when $R = r$, the electrical efficiency under this condition is only 50%, since half of the total chemical energy is converted to thermal energy within the battery itself.