Calculating E.m.f

• The calculation of E.m.f is rooted in the 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 wasted internally.

• This relationship is expressed by the fundamental E.m.f equation:

$\epsilon = V + v$

• Where $V$ is the terminal potential difference and $v$ is the lost volts. Using Ohm's Law ($V=IR$), this can be expanded to account for the current $I$ flowing through both the external resistance $R$ and internal resistance $r$.

• The complete circuit equation becomes:

$\epsilon = I(R + r)$

• This shows that the total resistance of the circuit is the sum of the external and internal resistances.

• Calculating Current: To find the current in a circuit with internal resistance, rearrange the E.m.f equation to $I = \frac{\epsilon}{R + r}$. This requires knowing the total E.m.f and the sum of all resistances.

• Determining Lost Volts: Lost volts ($v$) can be calculated using the formula $v = I \cdot r$. This value represents the potential difference dropped across the internal resistance of the source.

• Finding Terminal P.D: The terminal potential difference can be found either by subtracting lost volts from E.m.f ($V = \epsilon - Ir$) or by applying Ohm's Law to the external load ($V = IR$).

• A common experimental method to find $\epsilon$ and $r$ involves varying the external resistance and measuring the resulting terminal P.D ($V$) and current ($I$).

• By rearranging $V = \epsilon - Ir$ into the linear form $y = mx + c$, we get:

$V = -rI + \epsilon$

• When $V$ is plotted on the y-axis and $I$ on the x-axis, the resulting graph is a straight line with a negative gradient.

• Y-intercept: The point where the line crosses the vertical axis represents the E.m.f ($\epsilon$), as this is the voltage when current is zero.

• Gradient: The magnitude of the gradient of the line is equal to the internal resistance ($r$).

• It is vital to distinguish between the theoretical maximum voltage and the practical voltage available during operation.

• Open Circuit vs. Closed Circuit: In an open circuit, no current flows, so $Ir = 0$ and $V = \epsilon$. In a closed circuit, current flows, causing a voltage drop across $r$, making $V < \epsilon$.

• Identify 'Negligible' Resistance: If a problem states the source has 'negligible internal resistance', you can assume $r = 0$ and therefore $V = \epsilon$. Always look for this keyword to simplify your calculations.

• Unit Consistency: Ensure all resistance values are in Ohms ($\Omega$) and currents are in Amperes (A) before using the E.m.f equation. Convert milliamperes ($mA$) by multiplying by $10^{-3}$.

• Sanity Check: The terminal potential difference $V$ should always be less than or equal to the E.m.f $\epsilon$. If your calculated $V$ is higher than $\epsilon$, re-check your algebraic steps or sign conventions.

• Graph Interpretation: In multiple-choice questions, remember that a steeper negative gradient on a $V-I$ plot indicates a higher internal resistance.