E.m.f & Internal Resistance

Conservation of Energy: The total energy supplied by the source (e.m.f) must equal the sum of the energy used in the external circuit (terminal p.d.) and the energy wasted inside the source (lost volts).

The E.m.f Equation: Mathematically, this is expressed as $\epsilon = V + v$. Substituting Ohm's Law ($V=IR$ and $v=Ir$), we get the fundamental relationship: $\epsilon = I(R + r)$ where $I$ is the total current, $R$ is the external resistance, and $r$ is the internal resistance.

Voltage-Current Relationship: Rearranging the equation to $V = \epsilon - Ir$ shows that the terminal potential difference is a linear function of the current. As the current increases, the terminal p.d. decreases because more volts are 'lost' internally.

Determining E.m.f and r Graphically: By varying an external variable resistor and measuring the resulting current ($I$) and terminal p.d. ($V$), one can plot a graph of $V$ against $I$.

Linear Analysis: The equation $V = -rI + \epsilon$ follows the form $y = mx + c$. In this context, the y-intercept represents the e.m.f ($\epsilon$), and the gradient of the line represents the negative internal resistance ($-r$).

Open Circuit Measurement: If a high-resistance voltmeter is connected across a cell when no current flows ($I=0$), the reading on the voltmeter is equal to the e.m.f, as there are no lost volts ($Ir = 0$).

Identify 'Ideal' Sources: If a question mentions a 'battery of negligible internal resistance,' assume $r = 0$ and $\epsilon = V$. Do not waste time calculating lost volts unless $r$ is specified.

Graph Interpretation: When given a $V-I$ graph, always check the axes. If the line hits the x-axis, that point represents the short-circuit current, where $V=0$ and $I = \epsilon / r$.

Unit Consistency: Ensure all resistances are in Ohms ($\Omega$) and currents in Amperes (A) before using the e.m.f equation. Common traps include providing internal resistance in $m\Omega$.

Sanity Check: The terminal p.d. ($V$) should always be less than or equal to the e.m.f ($\epsilon$). If your calculated $V$ is higher than $\epsilon$, you have likely made an algebraic error.

The 'Force' Misconception: Despite its name, e.m.f is not a force (measured in Newtons) but a potential (measured in Volts). It describes energy transfer, not mechanical push.

Constant Terminal Voltage: Students often mistakenly assume the voltage of a battery is constant. In reality, the terminal p.d. 'sags' as more components are added in parallel, because the total current increases, leading to higher lost volts.

Internal Resistance Location: Remember that internal resistance is physically inside the battery. You cannot 'remove' it; it is an inseparable part of the power supply's behavior.