Comprehensive Study Guide on Internal Resistance and Circuit Dynamics

Fundamental Concepts of Electromotive Force and Internal Resistance

The Electro-Motive Force, abbreviated as EMF ($\epsilon$ or $E$), is defined as the maximum energy provided or the total work done by a battery per coulomb of charge that passes through it. It represents the maximum potential difference that a battery can supply and is exclusively measured across the terminals of a battery in an open circuit, where no current is flowing. When the circuit is closed and current begins to flow, the measured potential difference across the terminals decreases to a value lower than the EMF. This lower value is known as the terminal potential difference ($V_{external}$ or $V_{LOAD}$).

This discrepancy between the EMF and the terminal potential difference is caused by internal resistance ($r$) found within the battery itself. Real-world batteries are constructed from materials that possess inherent resistance. As a result, when current flows, the battery converts a portion of the electrical energy it produces into heat energy. In circuit modeling, a real battery is represented as a source of EMF ($\text{EMF}$) connected in series with an internal resistance ($r$). The reduction in potential difference observed when the current flows is formally referred to as "lost volts" ($V_{internal}$). Consequently, the internal resistance causes the terminal potential difference ($V$) to always be less than the EMF ($\epsilon$) whenever a current is active in the circuit.

Mathematical Principles of Circuit Analysis with Internal Resistance

In a closed circuit containing a battery with EMF ($\epsilon$) and internal resistance ($r$), the "lost volts" generated by the internal resistance is calculated using the formula $V_{internal} = Ir$. If the external circuit possesses a total resistance ($R$), a voltmeter placed in parallel with the battery will read the terminal potential difference, represented as $V_{external}$. This terminal potential difference is equivalent to the potential difference across the external components of the circuit, calculated as $V_{external} = IR$.

The EMF of the battery is the mathematical sum of the potential difference across the internal resistance and the potential difference across the external resistance. The relationship is expressed as $\text{EMF} = V_{external} + V_{internal}$, which expands to $\text{EMF} = IR + Ir$. Factoring out the current ($I$) provides the consolidated formula $\text{EMF} = I(R + r)$.

Changes in the external resistance ($R$) have a direct impact on the distribution of voltage. If the external resistance in a circuit is decreased, the total current ($I$) will increase (and vice versa). Because the "lost volts" ($V_{internal}$) is directly proportional to the current according to the relationship $V_{internal} = Ir$, an increase in current results in an increase in the volts lost within the battery. This means that as more current is drawn from the battery, the terminal potential difference available to the external circuit decreases.

Resistor Configurations: Series and Parallel

Resistors in a circuit can be arranged in series or parallel, each following distinct physical laws. In a series configuration, resistance is additive, meaning the equivalent resistance ($R_s$) is the sum of Individual resistors: $R_s = R_1 + R_2 + R_3 + \dots$. In these circuits, the total voltage ($V_{total}$) is split across the various resistors such that $V_{total} = V_1 + V_2 + V_3 + \dots$, while the current remains constant through every component in the series: $I_{total} = I_1 = I_2 = I_3 = \dots$.

In a parallel configuration, resistance adds inversely. The equivalent resistance ($R_p$) for resistors in parallel is calculated using the formula $\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots$. Unlike series circuits, the total voltage in a parallel circuit is the same across every branch: $V_{total} = V_1 = V_2 = V_3 = \dots$. However, the total current is split among the branches, where $I_{total} = I_1 + I_2 + I_3 + \dots$. If multiple resistors are connected in series within a single parallel branch, their resistances are first added together to be treated as a single resistor for the parallel calculation.

Electrical Power and Ohm’s Law

Ohm's Law states that for a resistor maintained at a constant temperature (known as an Ohmic resistor), the current flowing through it is directly proportional to the voltage across it, defined by the formula $V = IR$. Building upon this, electrical power ($P$) is defined as the rate at which electrical energy is converted within a circuit. The primary calculation for power is $P = IV$. By substituting Ohm's Law into the power equation, three equivalent formulas are derived to calculate power based on available variables:

1. $P = IV$

2. $P = I^2R$ (derived by substituting $V = IR$)

3. $P = \frac{V^2}{R}$ (derived by substituting $I = \frac{V}{R}$)

Worked Case Study: October/November 2019 Examination

In this scenario, a resistor $R$ with a resistance of $5.6\,\Omega$ is connected to a battery with an unknown internal resistance ($r$). A high-resistance voltmeter and an ammeter are included in the circuit. A graph of potential difference against time shows that before a switch is closed ($t_1$), the potential difference is $13\,V$. After the switch is closed, the potential difference drops to $10.5\,V$.

From this data, the EMF ($\epsilon$) is identified as the open-circuit voltage, which is $13\,V$. When the switch is closed, the calculation for current ($I$) through resistor $R$ is performed using Ohm’s Law: $I = \frac{V}{R} = \frac{10.5}{5.6} = 1.88\,A$. The power dissipated in resistor $R$ is then calculated as $P = VI = 10.5 \times 1.88 = 19.74\,W$. To find the internal resistance ($r$), the EMF formula is rearranged: $r = \frac{\epsilon}{I} - R = \frac{13}{1.88} - 5.6 = 1.31\,\Omega$.

In a follow-up modification, two identical resistors of resistance $X$ are added to the circuit and the ammeter reading increases to $4\,A$. When external resistance decreases (due to parallel additions), the current increases, which increases the internal "lost volts" ($V_{lost} = Ir$). Consequently, the voltmeter reading (terminal PD) decreases. To calculate the resistance $X$, we determine the new external resistance requirement: $R_{ext} = \frac{\epsilon}{I} - r = \frac{13}{4} - 1.31 = 1.94\,\Omega$. Using the parallel resistance formula $\frac{1}{R_p} = \frac{1}{R_{branch1}} + \frac{1}{R_{branch2}}$, we find $\frac{1}{1.94} = \frac{1}{5.6} + \frac{1}{2X}$. Solving for $2X$ yields $2.97\,\Omega$, resulting in $X = 1.49\,\Omega$.

Additional Exam Applications and Scenarios

In the October/November 2018 examination, a battery with an EMF of $12\,V$ and internal resistance of $0.5\,\Omega$ is used. If the ammeter reads $2\,A$ when the switch is open, this implies a specific circuit configuration affecting the voltmeter. Closing the switch typically causes the voltmeter reading to decrease because the total resistance of the circuit decreases, increasing the current and thereby increasing the "lost volts" inside the battery.

In the May/June 2018 examination, Ohm's law is defined as the principle that the current through a conductor between two points is directly proportional to the voltage across the two points, provided temperature remains constant. In a circuit with a battery internal resistance of $0.8\,\Omega$ and an ammeter reading of $0.6\,A$, specific measures of potential difference across various resistors and the total EMF ($\epsilon$) can be calculated. Additionally, the energy dissipated as heat inside the battery over a duration ($15\,s$) can be found using the formula $W = I^2rt$. This exam also introduces a practical windscreen wiper motor in a car connected to a $12\,V$ battery. If the variable resistor is decreased to increase the speed of the wiper, the current in the circuit increases, providing more power to the motor.

Finally, the February/March 2017 examination explores light bulb brightness in varied circuits. Identical bulbs $P$, $Q$, and $R$ are analyzed; bulbs in branches with lower resistance or higher current flow appear brighter. Adding a fourth bulb $T$ in parallel creates a short or a change in the resistance network, affecting previous bulbs' brightness. For more complex calculations involving an EMF of $20\,V$ and internal resistance of $1\,\Omega$ connected to resistors of $5\,\Omega$, $10\,\Omega$, and $8\,\Omega$, one must find the total equivalent resistance to solve for the main current, individual branch currents, and total power supplied by the battery ($P = VI$).