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Electricity and Magnetism
Electromotive Force (EMF) and Internal Resistance
Electromotive force (EMF) is the work done by a non-electrostatic source in moving a unit positive charge completely around a closed circuit. It represents the potential difference across the terminals of a source of electrical energy when no current is drawn from it.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of EMF, terminal potential difference, internal resistance, combination of cells in series and parallel, and Ohm's law application for complete circuits.
PGT extension
Energy considerations in charging and discharging of secondary cells, maximum power transfer theorem, and terminal voltage variations in non-linear or multi-loop networks.
Prerequisites
Electric potential and potential difference
Ohm's Law in microscopic and macroscopic forms
Conservation of energy in electrical circuits
Definition and Physical Origin of EMF
EMF is not a force; it is a scalar quantity measured in volts (joules per coulomb) representing energy supplied per unit charge.
It originates from non-electrostatic forces (chemical, thermal, or mechanical) that drive charge carriers from a lower to a higher potential region inside the source.
The open-circuit terminal voltage of a real source equals its EMF because no internal voltage drop occurs when the current is zero.
Internal Resistance of a Source
Every real source of electrical energy possesses a finite internal resistance (r) due to the resistivity of its internal electrolytic or structural components.
When a current (I) flows through the source, an internal potential drop (Ir) reduces the available terminal voltage.
Internal resistance depends on temperature, electrolyte concentration, and the distance between electrodes in chemical cells.
Terminal Potential Difference
Terminal potential difference (V) is the actual potential difference available across the external load resistance (R) during current flow.
During discharge, $V = E - Ir$, showing that terminal voltage is less than EMF.
During charging (e.g., by an external generator), $V = E + Ir$, showing that terminal voltage exceeds EMF.
Power Transfer and Efficiency
The total power generated by a source of EMF is given by $P_total = EI$.
Useful power delivered to the external load is $P_load = I^2 R$, while power dissipated internally is $P_internal = I^2 r$.
Maximum power is transferred to the load when the load resistance equals the internal resistance (R = r), at which point efficiency is 50 percent.
Derivations
Terminal Potential Difference and Current in a Single-Loop Circuit
Conditions: A single cell of EMF E and internal resistance r connected across an external resistor R.
Apply Kirchhoff's loop rule to the single closed loop: E - Ir - IR = 0.
Rearrange the equation to express current: $I = E / (R + r)$.
Express the terminal potential difference across the external resistor: V = IR.
Substitute the expression for current into the potential difference equation: $V = E(R / (R + r)) or V = E - Ir$.
Result: $V = E - Ir$ and $I = E / (R + r)$
Maximum Power Transfer Theorem
Conditions: A cell of constant EMF E and internal resistance r connected to a variable load resistance R.
Write the power delivered to the load as a function of R: $P = I^2 R = (E / (R + r))^2 R$.
Differentiate the power with respect to R and set the derivative to zero for extrema: dP/d$R = E^2 [(R+r)^2 - 2R(R+r)] / (R+r)^4 = 0$.
Simplify the numerator: (R + r)(R + r - 2R) = 0, which yields r - R = 0.
Determine the condition for maximum power: R = r.
Result: Maximum power is transferred when R = r, and $P_max = E^2 / (4r)$.
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Current in a simple circuit
$$I = \frac{E}{R + r}$$
E = EMF, R = External resistance, r = Internal resistance, I = Current Amperes (A)
Single-loop circuit with constant parameters
Terminal potential difference (Discharge)
$$V = E - Ir$$
V = Terminal voltage, E = EMF, I = Current, r = Internal resistance Volts (V)
Cell discharging into a circuit
Terminal potential difference (Charging)
$$V = E + Ir$$
V = Terminal voltage, E = EMF, I = Current, r = Internal resistance Volts (V)
Cell being charged by a higher potential source
Equivalent EMF for series combination
$$E_{eq} = E_1 + E_2 + ... + E_n$$
$E_i = EMF of individual cells$ Volts (V)
Identical or non-identical cells connected aiding each other in series
Equivalent internal resistance for series combination
$$r_{eq} = r_1 + r_2 + ... + r_n$$
$r_i = Internal resistance of individual cells$ Ohms (Ω)
Cells in series
Equivalent EMF for parallel combination of two cells
Two cells in parallel with positive terminals connected together
Maximum power transfer condition
$$R = r$$
R = Load resistance, r = Internal resistance Ohms (Ω)
For maximum power output from a source
Efficiency of a cell
$$\eta = \frac{V}{E} = \frac{R}{R + r}$$
V = Terminal voltage, E = EMF, R = Load resistance, r = Internal resistance Dimensionless (fraction or percentage)
Linear DC circuit
Graphs and experiments
Terminal Voltage versus Current Graph
Draw: A straight line plot of V on the y-axis against I on the x-axis, having a negative slope.
Interpret: The y-intercept represents the EMF (E) of the cell, and the magnitude of the negative slope represents the internal resistance (r).
Exam use: Frequently tested in numerical and graphical MCQs to determine internal resistance from laboratory data.
Power versus Load Resistance Graph
Draw: A bell-shaped asymmetric curve of power P on the y-axis versus load resistance R on the x-axis, peaking at R = r.
Interpret: Power is zero at R = 0 (short circuit) and approaches zero as R approaches infinity (open circuit). The maximum occurs precisely at R = r.
Exam use: Used to test conceptual understanding of the maximum power transfer theorem.
Potentiometer Experiment for Internal Resistance
Draw: Circuit diagram showing a primary driving circuit and a secondary test circuit with a cell, resistance box, and galvanometer.
Interpret: The balancing length l_1 without the resistance box gives a measure of EMF (E), and length l_2 with shunt resistance R gives terminal voltage (V). Internal resistance is calculated as $r = R((l_1 / l_2) - 1)$.
Exam use: Standard practical-based theoretical question in PGT Physics and TGT Science.
Solved numericals
TGT core
A cell of EMF 2.0 V and internal resistance 0.5 ohm is connected across an external resistor of 3.5 ohms. Calculate the current in the circuit and the terminal potential difference across the resistor.
Identify given values: E = 2.0 V, r = 0.5 ohm, R = 3.5 ohms.
Calculate total circuit resistance: $R_total = R + r = 3.5 + 0.5 = 4.0 ohms$.
Calculate current using Ohm's law: $I = E / R_total = 2.0 / 4.0 = 0.5 A$.
Calculate terminal voltage: $V = I * R = 0.5 * 3.5 = 1.75 V$.
Answer: Current = 0.5 A, Terminal Potential Difference = 1.75 V
PGT extension
Two identical cells, each of EMF 1.5 V and internal resistance 1.0 ohm, are connected in parallel across an external load of 0.5 ohm. Find the current flowing through the external load.
Identify given values: $E_1 = E_2 = 1.5 V$, $r_1 = r_2 = 1.0 ohm$, R = 0.5 ohm.
Calculate equivalent EMF for parallel cells of equal EMF: $E_eq = E = 1.5 V$.
A battery delivers maximum power to an external resistor. If the EMF of the battery is 12 V and the maximum power transferred is 18 W, determine the internal resistance of the battery.
Recall the formula for maximum power: $P_max = E^2 / (4r)$.
Substitute the given values: $18 = (12)^2 / (4r)$.
Internal resistance is assumed to remain constant regardless of the magnitude of current flowing through the cell, which neglects polarization and thermal heating effects in high-current regimes.
Connecting wires are assumed to have zero resistance unless explicitly stated otherwise.
Common misconceptions
Common error: EMF is the maximum voltage a battery can ever provide under any condition.
Correct: EMF is the terminal voltage only when no current is drawn; charging a battery can raise its terminal voltage above its EMF.
During charging, the external source forces current backward against the internal EMF, making $V = E + Ir$.
Common error: Internal resistance is an external physical resistor attached inside the battery casing.
Correct: Internal resistance represents the total opposition offered by the electrolyte and electrodes to the flow of charge inside the source.
It is an inherent property of the energy source rather than a discrete component.
Quick revision
EMF is the open-circuit potential difference, given by $E = V + Ir$.
Current in a simple circuit: $I = \frac{E}{R + r}$
Terminal potential difference (Discharge): $V = E - Ir$
Must remember
EMF is not a force; it is the work done per unit charge in driving charge through the complete circuit, measured in volts.
The open-circuit terminal voltage of a cell equals its EMF because no internal voltage drop ($Ir$) occurs when current is zero.
During discharge, the terminal voltage is given by $V = E - Ir$, whereas during charging, it is $V = E + Ir$.
Maximum power is transferred to an external load when the load resistance equals the internal resistance ($R = r$), achieving an efficiency of $50\%$.
Internal resistance $r$ arises due to the resistivity of the internal electrolytic or structural components of the source.
Glossary
Electromotive Force (EMF)The energy supplied by a source per unit charge, representing the potential difference across terminals on open circuit.
Internal Resistance ($r$)The opposition offered by the internal material and electrolyte of a source to the flow of electric current.
Terminal Potential Difference ($V$)The actual potential difference across the external terminals of a source when current is flowing through the circuit.
Efficiency ($\eta$)The ratio of useful power delivered to the external load to the total power generated by the source, expressed as $\frac{R}{R+r}$.
EMF vs Terminal Potential Difference
Parameter
Electromotive Force (EMF)
Terminal Potential Difference ($V$)
Definition
Energy supplied per unit charge in open circuit
Actual potential difference across load during current flow
Dependence on Current
Independent of external circuit current
Depends on current and internal resistance ($V = E - Ir$)
Maximum Value
Always greater than or equal to terminal voltage
Less than EMF during discharge, greater during charging
Practice Quiz
Choose an option to reveal the explanation.
1. What is the physical nature of Electromotive Force (EMF)?
Explanation: EMF is not a force; it is a scalar quantity measured in volts (joules per coulomb) representing energy supplied per unit charge.
2. Under what condition does the terminal potential difference of a real source equal its EMF?
Explanation: In an open circuit, the current $I = 0$, meaning the internal voltage drop $Ir = 0$, so the terminal voltage equals the EMF ($V = E$).
3. What happens to the terminal potential difference ($V$) of a secondary cell when it is being charged by an external generator?
Explanation: During charging, current enters the positive terminal, opposing the internal EMF, resulting in a terminal voltage of $V = E + Ir$, which exceeds the EMF.
4. The internal resistance ($r$) of a chemical cell depends on which of the following factors?
Explanation: Internal resistance depends on temperature, electrolyte concentration, and the distance between electrodes in chemical cells.
5. What is the expression for the current $I$ flowing in a single-loop circuit with EMF $E$, internal resistance $r$, and load resistance $R$?
Explanation: According to Ohm's law for a complete circuit, total resistance is $R + r$, so $I = \frac{E}{R + r}$.
6. For maximum power transfer from a source of EMF with internal resistance $r$ to a load resistor $R$, what must be the relationship between $R$ and $r$?
Explanation: Maximum power is transferred to the load when the load resistance equals the internal resistance ($R = r$).
7. What is the efficiency of power transfer from a source when maximum power is delivered to the load ($R = r$)?
Explanation: Efficiency $\eta = \frac{R}{R + r}$. When $R = r$, $\eta = \frac{r}{r + r} = \frac{1}{2}$ or $50\%$.
8. Which of the following represents the equivalent EMF ($E_{eq}$) when $n$ identical cells of EMF $E$ are connected in series aiding each other?
Explanation: For cells in series aiding each other, equivalent EMF is the algebraic sum: $E_{eq} = E_1 + E_2 + ... + E_n = nE$.
9. What is the internal voltage drop across a source when a current $I$ flows through it and its internal resistance is $r$?
Explanation: By Ohm's law, the potential drop across the internal resistance is given by $Ir$.
10. If two cells of EMFs $E_1, E_2$ and internal resistances $r_1, r_2$ are connected in parallel with their positive terminals together, what is the formula for equivalent EMF $E_{eq}$?
Explanation: The standard formula for the parallel combination of two cells is $E_{eq} = \frac{\frac{E_1}{r_1} + \frac{E_2}{r_2}}{\frac{1}{r_1} + \frac{1}{r_2}}$.
11. What is the total power generated by a source of EMF delivering current $I$?
Explanation: The total power generated by a source of EMF is given by $P_{total} = EI$.
12. During the discharge of a battery, how does its terminal voltage $V$ relate to its EMF $E$?
Explanation: During discharge, $V = E - Ir$, meaning the terminal voltage is less than the EMF due to the internal potential drop.
13. Why is EMF measured in volts?
Explanation: EMF represents energy supplied per unit charge, expressed in joules per coulomb, which is defined as volts.
14. What is the equivalent internal resistance when $n$ identical cells each of internal resistance $r$ are connected in series?
Explanation: For series combination, internal resistances add up linearly: $r_{eq} = r_1 + r_2 + ... + r_n = nr$.
15. Which misconception is corrected by stating that a battery's terminal voltage can exceed its EMF?
Explanation: EMF is often mistaken as the absolute maximum voltage; however, during battery charging, terminal voltage becomes $V = E + Ir$, exceeding the EMF.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. A cell of EMF $2\text{ V}$ and internal resistance $0.5\ \Omega$ is connected across an external resistor of $4.5\ \Omega$. What is the terminal potential difference across the resistor?
Explanation: Current $I = \frac{E}{R + r} = \frac{2}{4.5 + 0.5} = \frac{2}{5} = 0.4\text{ A}$. Terminal voltage $V = E - Ir = 2 - (0.4 \times 0.5) = 2 - 0.2 = 1.8\text{ V}$.
2. When a resistance $R$ is connected across a cell of EMF $E$ and internal resistance $r$, the power consumed in $R$ is maximum. The relation between $R$ and $r$ is:
Explanation: According to the maximum power transfer theorem, power delivered to the load is maximum when load resistance equals internal resistance ($R = r$).
3. Two identical cells each of EMF $E$ and internal resistance $r$ are connected in parallel. The equivalent EMF of the combination is:
Explanation: For identical cells connected in parallel, the equivalent EMF is equal to the EMF of a single cell ($E_{eq} = E$).
4. A storage battery of EMF $12\text{ V}$ and internal resistance $0.05\ \Omega$ is being charged by a DC source of $120\text{ V}$ using a series resistor of $15.95\ \Omega$. What is the terminal voltage of the battery during charging?
5. What happens to the internal resistance of a chemical cell when the distance between its electrodes is increased?
Explanation: Internal resistance increases with an increase in the distance between the electrodes because the ionic path length increases, raising resistance.
6. The efficiency of a cell of internal resistance $r$ connected to an external resistance $R$ is $80\%$. The ratio of $R$ to $r$ is:
Attempt these questions under a ten-minute limit using a separate timer.
1. Why is EMF termed as work done per unit charge rather than a force?
Explanation: EMF represents energy per unit charge supplied by non-electrostatic forces (chemical, thermal, etc.), measured in volts (joules per coulomb), not a mechanical force.
2. A battery has an open-circuit voltage of $6\text{ V}$ and a terminal voltage of $5.2\text{ V}$ when connected to a $13\ \Omega$ load. What is the internal resistance of the battery?
Explanation: EMF $E = 6\text{ V}$, Terminal voltage $V = 5.2\text{ V}$, Load $R = 13\ \Omega$. Current $I = \frac{V}{R} = \frac{5.2}{13} = 0.4\text{ A}$. Since $V = E - Ir$, $5.2 = 6 - (0.4)r \implies 0.4r = 0.8 \implies r = 2\ \Omega$? Wait, $0.8 / 0.4 = 2\ \Omega$. Let's check options: Option B is $2\ \Omega$. Correct index is 1.
3. If $n$ identical cells each of EMF $E$ and internal resistance $r$ are connected in parallel, what is the current through an external resistor $R$ connected to the combination?
Explanation: For $n$ identical cells in parallel, equivalent EMF is $E$ and equivalent internal resistance is $\frac{r}{n}$. Current $I = \frac{E}{R + \frac{r}{n}}$.
4. In a laboratory experiment, a student measures the terminal potential difference $V$ of a cell for different currents $I$. The slope of the $V$ versus $I$ graph represents:
Explanation: Since $V = E - Ir$, plotting $V$ on the y-axis and $I$ on the x-axis gives a straight line equation $y = -r x + E$, where the slope is $-r$.
5. What is the power dissipated inside the source itself when a current $I$ flows through a source with internal resistance $r$?
Explanation: Power dissipated internally due to Joule heating within the source is $P_{internal} = I^2 r$.
6. Two cells of unequal EMFs $E_1$ and $E_2$ ($E_1 > E_2$) and internal resistances $r_1$ and $r_2$ are connected in parallel opposing each other. What is the condition for current to flow from the first cell to the second?
Explanation: The net EMF driving current in the loop is $E_1 - E_2$, which requires $E_1 > E_2$ for current to flow in the direction dictated by the stronger cell.
7. Which of the following parameters does NOT directly affect the internal resistance of a primary galvanic cell?
Explanation: Internal resistance is an intrinsic property of the source depending on temperature, electrolyte concentration, and electrode geometry, independent of external load resistance.
8. A cell supplies a current of $1.5\text{ A}$ through a $2\ \Omega$ resistor and $1.0\text{ A}$ through a $3.5\ \Omega$ resistor. What is the internal resistance of the cell?
9. What is the primary reason why terminal potential difference decreases as the current drawn from a cell increases during discharge?
Explanation: During discharge, $V = E - Ir$. As current $I$ increases, the internal voltage drop $Ir$ increases, directly reducing the terminal voltage $V$.
10. Under what circumstance can the useful power delivered to an external load resistance $R$ be maximized?
Explanation: According to the maximum power transfer theorem, power delivered to the load is maximized when the load resistance equals the internal resistance ($R = r$).