Electromotive Force

• A power source acts as a 'charge pump' that converts non-electrical energy (chemical in batteries, mechanical in generators, or thermal in thermocouples) into electrical potential energy.

• This energy is then used to overcome the resistance of the external circuit and the internal resistance of the source itself as the charge circulates.

• According to the Law of Conservation of Energy, the total energy supplied by the source must equal the sum of the energy dissipated in the external load and the energy lost within the source.

• Every real-world power supply possesses some degree of internal resistance ($r$), which is the inherent resistance to the flow of current within the source's materials.

• When current ($I$) flows, some of the EMF is 'consumed' internally to move the charge through the source's own resistance. This value, $Ir$, is known as the lost volts.

• These lost volts are dissipated as heat within the battery, which is why batteries often feel warm during heavy use or rapid charging.

• The relationship between EMF, terminal voltage, and internal resistance is derived from Ohm's Law and energy conservation.

• The Terminal Potential Difference ($V$) is the voltage actually available to the external circuit (the load).

Mathematical Model

• The total EMF is the sum of the terminal voltage and the lost volts:

$\mathcal{E} = V + Ir$

• Substituting $V = IR$ (where $R$ is the external load resistance), we get the full circuit equation:

$\mathcal{E} = I(R + r)$

• This formula allows for the calculation of any single variable if the others are known, provided the circuit is in a steady state.

• While both are measured in Volts, they represent different stages of energy transfer in a circuit.

• Identify Open Circuits: If a question states a voltmeter is connected across a battery in an 'open circuit', the reading is the EMF. If the circuit is 'closed', the reading is the terminal voltage.

• Graphing V vs. I: In a graph of terminal voltage ($V$) on the y-axis and current ($I$) on the x-axis, the equation $V = -rI + \mathcal{E}$ shows that the y-intercept is the EMF and the negative slope is the internal resistance ($r$).

• Sanity Check: The terminal voltage ($V$) should always be less than or equal to the EMF ($\mathcal{E}$) for a discharging battery. If your calculated $V$ is higher than $\mathcal{E}$, check if the battery is being charged by an external source.

• The 'Force' Misnomer: Students often mistakenly treat EMF as a force in Newtons. Always remember it is an energy-per-charge quantity (Potential).

• Ignoring Internal Resistance: In many basic physics problems, batteries are assumed to be 'ideal' ($r = 0$). However, in advanced problems, you must always check if an internal resistance value is provided.

• Constant Terminal Voltage: A common error is assuming the terminal voltage of a battery is constant. In reality, as the current drawn increases, the 'lost volts' ($Ir$) increase, causing the terminal voltage ($V$) to drop.