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Electricity and Magnetism

Primary Cells and Sources of Electromotive Force

Primary cells are electrochemical devices that convert chemical energy directly into electrical energy via irreversible redox reactions. They serve as portable, self-contained voltage sources that cannot be electrically recharged effectively once depleted.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Definition of primary cells, Electromotive Force (EMF), terminal potential difference, internal resistance, simple voltaic cell, Leclanche cell, Daniell cell, and grouping of cells in series and parallel.

PGT extension

Polarization and local action defects, mathematical treatment of maximum power transfer theorem for cells, temperature dependence of EMF, thermodynamic relations involving Gibbs free energy, and internal resistance measurement via potentiometer.

Prerequisites

Electric current and drift velocity
Ohm's law and electrical resistance
Terminal potential difference and Kirchhoff's laws

Electromotive Force (EMF) and Internal Resistance

The EMF (E) of a cell is the potential difference across its terminals when no current is drawn from it, representing the work done by non-electrostatic forces to move unit positive charge through the entire circuit including the cell interior.
Internal resistance (r) arises due to the opposition offered by the electrolyte and electrodes to the flow of ions inside the cell.
The terminal potential difference (V) is less than the EMF when the cell is discharging, given by $V = E - Ir$, due to the potential drop across the internal resistance.

Chemical Defects in Primary Cells

Local action occurs due to impurities (such as iron particles) in commercial zinc plates, forming minute local voltaic cells that continuously consume zinc even when no external current flows.
Polarization occurs due to the accumulation of hydrogen gas bubbles on the positive copper electrode, which increases internal resistance and sets up a back EMF.
Amalgamation of the zinc plate with mercury prevents local action, while depolarizers like manganese dioxide (MnO₂) are used to oxidize hydrogen gas and eliminate polarization.

Common Types of Primary Cells

Voltaic Cell: Uses copper (positive) and zinc (negative) plates in dilute sulfuric acid; suffers severely from local action and polarization, yielding an EMF of roughly 1.08 V.
Daniell Cell: Employs a copper vessel containing copper sulfate solution as the positive electrode and a zinc rod in dilute sulfuric acid or zinc sulfate as the negative electrode, separated by a porous pot, yielding an EMF of about 1.1 V.
Leclanche Cell: Uses a carbon rod in a mixture of carbon and manganese dioxide as the positive electrode and a zinc rod in an ammonium chloride solution as the negative electrode, providing an EMF of about 1.5 V suitable for intermittent use.

Combination of Cells

In series combination, n identical cells are connected end-to-end to increase the net EMF and drive current through a large external resistance.
In parallel combination, m identical cells are connected positive-to-positive to deliver high current when the external resistance is much smaller than the internal resistance of a single cell.
In mixed grouping, n cells are connected in series per branch, and m such branches are connected in parallel to optimize current delivery.

Derivations

Terminal Potential Difference and Maximum Power Theorem for a Cell

Conditions: A cell of EMF E and internal resistance r connected across an external load resistance R.

Write the total resistance of the circuit as $R_total = R + r$.
Apply Ohm's law to find the circuit current: $I = E / (R + r)$.
Express the terminal potential difference across the load: $V = IR = E - Ir$.
Write the power P delivered to the external resistance R: $P = I^{2}R = E^{2}R / (R + r)^{2}$.
Differentiate P with respect to R and set dP/dR = 0 to find the condition for maximum power transfer: R = r.

Result: Maximum power is transferred to the external load when the external resistance equals the internal resistance of the cell (R = r).

Equivalent EMF and Internal Resistance of Parallel Combination of Cells

Conditions: Two cells of EMFs E₁, E₂ and internal resistances r₁, r₂ connected in parallel.

Let the potentials at the junction nodes be V_A and V_B, with potential difference $V = V_A - V_B across the parallel combination$.
Write currents leaving the positive terminals using Ohm's law: I₁ = (E₁ - V) / r₁ and I₂ = (E₂ - V) / r₂.
Total current $I = I_{1} + I_{2} = (E_{1}/r_{1} + E_{2}/r_{2}) - V(1/r_{1} + 1/r_{2})$.
Rearrange the equation to match the standard form $V = E_eq - I r_eq: V = [(E_{1}r_{2} + E_{2}r_{1}) / (r_{1} + r_{2})] - I [(r_{1}r_{2}) / (r_{1} + r_{2})]$.

Result: $E_eq = (E1r2 + E2r1)/(r1 + r2)$ and $r_eq = (r1r2)/(r1 + r2)$.

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Ohm's Law for a Single Cell	$$I = \frac{E}{R + r}$$	I = Current, E = EMF, R = External resistance, r = Internal resistance A, V, ohm, ohm	Closed circuit with constant internal resistance
Terminal Potential Difference	$$V = E - Ir$$	V = Terminal voltage, E = EMF, I = Current, r = Internal resistance V, V, A, ohm	During discharging of the cell
Charging a Cell	$$V = E + Ir$$	V = Terminal voltage, E = EMF, I = Charging current, r = Internal resistance V, V, A, ohm	When current is forced into the positive terminal
Internal Resistance via Potentiometer	$$r = R \left( \frac{l_1 - l_2}{l_2} \right)$$	R = Shunt resistance, $l_1 = Balancing length without shunt$, $l_2 = Balancing length with shunt$ ohm, m, m	Standard potentiometer circuit setup
Series Combination of n Identical Cells	$$I = \frac{nE}{R + nr}$$	n = Number of cells, E = EMF per cell, R = External resistance, r = Internal resistance per cell Dimensionless, V, ohm, ohm	Cells connected in series aiding manner
Parallel Combination of m Identical Cells	$$I = \frac{mE}{mR + r}$$	m = Number of parallel branches, E = EMF, R = External resistance, r = Internal resistance Dimensionless, V, ohm, ohm	Identical cells connected in parallel
Condition for Maximum Power	$$R = r$$	R = External load resistance, r = Internal resistance of the source ohm, ohm	For maximum power transfer theorem
Thermodynamic EMF relation	$$E = -\frac{\delta H}{nF} + T \frac{dE}{dT}$$	H = Enthalpy change, n = valence, F = Faraday constant, T = Temperature J, dimensionless, C/mol, K	Gibbs-Helmholtz equation applied to a cell

Graphs and experiments

Terminal Voltage (V) versus Current (I) Graph

Draw: A straight line starting at EMF (E) on the vertical V-axis and sloping downwards with a negative slope equal to internal resistance r, intersecting the horizontal I-axis at $I = E/r$.

Interpret: The y-intercept gives the EMF of the cell, and the magnitude of the slope of the line represents the internal resistance r of the cell.

Exam use: Frequently asked conceptual and graphical interpretation questions in PGT/TGT exams to determine internal resistance from experimental data.

Power Delivered versus External Resistance Graph

Draw: A curve rising sharply from the origin, reaching a distinct maximum peak at R = r, and then falling asymptotically toward zero as R approaches infinity.

Interpret: The peak of the curve confirms the maximum power transfer theorem, occurring precisely when the external resistance equals the internal resistance.

Exam use: Numerical and conceptual problems regarding matching load resistance for maximum power output.

Potentiometer Experiment for Internal Resistance

Draw: Schematic circuit diagram showing a driver cell, potentiometer wire, primary cell connected across a galvanometer and jockey, with a resistance box and plug key across the cell terminals.

Interpret: Balancing length l₁ corresponds to open circuit (EMF), and l₂ corresponds to closed circuit (terminal potential difference V).

Exam use: Core practical physics setup tested via numerical formula applications and circuit identification.

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 ohm. Calculate the terminal potential difference of the cell.

Identify given values: E = 2.0 V, r = 0.5 ohm, R = 3.5 ohm.
Calculate total circuit resistance: $R_total = R + r = 3.5 + 0.5 = 4.0 ohm$.
Calculate circuit current: $I = E / R_total = 2.0 / 4.0 = 0.5 A$.
Calculate terminal potential difference: $V = E - Ir = 2.0 - (0.5 * 0.5) = 2.0 - 0.25 = 1.75 V$.

Answer: 1.75 V

TGT core

Ten identical cells, each of EMF 1.5 V and internal resistance 0.2 ohm, are connected in series. If this combination is connected to an external resistance of 2.0 ohm, find the current flowing through the circuit.

Identify given values: n = 10, E = 1.5 V, r = 0.2 ohm, R = 2.0 ohm.
Write the formula for series combination current: $I = nE / (R + nr)$.
Substitute values into the formula: $I = (10 * 1.5) / (2.0 + 10 * 0.2)$.
Compute denominator: 2.0 + 2.0 = 4.0 ohm.
Compute current: $I = 15.0 / 4.0 = 3.75 A$.

Answer: 3.75 A

PGT extension

A potentiometer wire of length 10 m has a constant potential gradient. When a primary cell is connected in the secondary circuit, the balancing length is found to be 600 cm. When the cell terminals are shunted with a 5.0 ohm resistor, the balancing length drops to 500 cm. Determine the internal resistance of the cell.

Identify given balancing lengths: l1 = 600 cm, l2 = 500 cm.
Identify shunt resistance: R = 5.0 ohm.
Use the formula for internal resistance: $r = R * [(l1 - l2) / l2]$.
Substitute values: $r = 5.0 * [(600 - 500) / 500] = 5.0 * (100 / 500)$.
Calculate final value: $r = 5.0 * 0.2 = 1.0 ohm$.

Answer: 1.0 ohm

Assumptions and limitations

Internal resistance of a primary cell is assumed to be constant, though in practice it increases as the chemical reactants deplete.
The temperature of the cell is assumed to remain constant; significant temperature variations alter reaction rates and change the EMF via the temperature coefficient (dE/dT).

Common misconceptions

Common error: The potential difference across the terminals of a cell is always equal to its EMF.

Correct: Terminal potential difference equals EMF only when no current is drawn (open circuit condition).

When current flows, a potential drop (Ir) occurs across the internal resistance of the cell.

Common error: Primary cells can be recharged by passing a current through them in the reverse direction.

Correct: Primary cells undergo irreversible chemical reactions and cannot be recharged efficiently.

Reversing current flow in primary cells typically causes overheating, leakage, or structural electrode damage rather than chemical reversal.

Quick revision

EMF is the open-circuit terminal voltage; terminal voltage is less than EMF during discharge ($V = E - Ir)$.
Ohm's Law for a Single Cell: $I = \frac{E}{R + r}$
Terminal Potential Difference: $V = E - Ir$

Must remember

The EMF of a cell is the terminal potential difference when no current is drawn, representing the work done by non-electrostatic forces to move unit positive charge through the entire circuit.
Terminal potential difference $V = E - Ir$ during discharging and $V = E + Ir$ during charging of the cell.
Local action is caused by impurities like iron particles in commercial zinc plates, preventable by amalgamation with mercury.
Polarization is caused by the accumulation of hydrogen gas bubbles on the positive copper electrode, mitigated using depolarizers like manganese dioxide ($MnO_2$).
The condition for maximum power transfer from a source of cells to an external load is when the external resistance equals the total internal resistance ($R = r$).
In series combination of $n$ identical cells, current is given by $I = \frac{nE}{R + nr}$, best suited for large external resistance.
In parallel combination of $m$ identical cells, current is given by $I = \frac{mE}{mR + r}$, best suited for small external resistance.

Glossary

Electromotive Force (EMF)The maximum potential difference across the terminals of a cell when it is in an open circuit condition ($E$).

Internal ResistanceThe opposition offered by the electrolyte and electrodes of a cell to the flow of internal ions ($r$).

Local ActionA chemical defect in primary cells caused by metallic impurities in the zinc plate forming local voltaic cells.

PolarizationA defect caused by the deposition of hydrogen gas bubbles on the positive electrode, increasing internal resistance.

AmalgamationThe coating of zinc plates with mercury to prevent local action by hiding surface impurities.

DepolarizerA chemical substance like manganese dioxide used to oxidize hydrogen gas and eliminate polarization.

Comparison of Primary Cells

Cell Type	Positive Electrode	Negative Electrode	Approximate EMF	Key Defect / Feature
Voltaic Cell	Copper	Zinc	1.08 V	Severe local action and polarization
Daniell Cell	Copper vessel	Zinc rod	1.10 V	Uses porous pot; steady voltage
Leclanche Cell	Carbon rod	Zinc rod	1.50 V	Intermittent use; uses $MnO_2$ depolarizer

Practice Quiz

Choose an option to reveal the explanation.

1. What is the terminal potential difference of a cell of EMF $E$ and internal resistance $r$ when it is discharging with current $I$?

Explanation: During discharging, the potential drop across the internal resistance reduces the available terminal voltage, giving $V = E - Ir$.

2. Which chemical defect in primary cells is caused by impurities like iron particles in commercial zinc plates?

Explanation: Local action occurs when impurities form minute local voltaic cells that continuously consume zinc even in an open circuit.

3. How can local action in a voltaic cell be prevented?

Explanation: Amalgamation covers the impurities on the zinc surface with mercury, creating a uniform surface and preventing local voltaic action.

4. What causes polarization in a primary cell?

Explanation: Hydrogen gas bubbles accumulate on the positive electrode, increasing internal resistance and creating a counter EMF.

5. What substance is used as a depolarizer in a Leclanche cell?

Explanation: Manganese dioxide ($MnO_2$) acts as a depolarizer by oxidizing the hydrogen gas accumulated at the carbon electrode into water.

6. What is the approximate EMF of a standard Leclanche cell?

Explanation: A Leclanche cell provides an EMF of about $1.5\text{ V}$ and is well-suited for intermittent use.

7. What is the condition for maximum power transfer from a source with internal resistance $r$ to an external load $R$?

Explanation: According to the maximum power transfer theorem, power delivered to the load is maximum when the external resistance equals the internal resistance ($R = r$).

8. When $n$ identical cells each of EMF $E$ and internal resistance $r$ are connected in series, what is the net current through external resistance $R$?

Explanation: In series combination, total EMF is $nE$ and total internal resistance is $nr$, giving $I = \frac{nE}{R + nr}$.

9. In a potentiometer experiment to find internal resistance, if $l_1$ and $l_2$ are balancing lengths without and with a shunt resistance $R$, what is the formula for internal resistance $r$?

Explanation: The internal resistance is given by $r = R \left(\frac{l_1 - l_2}{l_2}\right)$ based on the ratio of potentials.

10. Why are primary cells unsuitable for continuous heavy current applications?

Explanation: Continuous heavy current causes rapid polarization and accumulation of products, leading to a drop in terminal voltage and high internal resistance.

11. What is the electrolyte used in a Leclanche cell?

Explanation: The Leclanche cell uses an ammonium chloride ($NH_4Cl$) solution as its electrolyte in contact with a zinc rod.

12. What is the approximate EMF of a Daniell cell?

Explanation: A Daniell cell yields a stable EMF of about $1.1\text{ V}$ because it minimizes polarization using copper sulfate solution.

13. What happens to the terminal potential difference of a cell when it is being charged by an external source?

Explanation: During charging, current enters the positive terminal, so the terminal voltage is greater than the EMF: $V = E + Ir$.

14. Which combination of identical cells is preferred when the external resistance is very large compared to the internal resistance of a single cell?

Explanation: Series combination increases the net EMF significantly, which is advantageous when driving current through a large external resistance.

15. According to the Gibbs-Helmholtz relation, the thermodynamic EMF of a cell depends on enthalpy change and what other temperature-dependent factor?

Explanation: The relation is $E = -\frac{\Delta H}{nF} + T \frac{dE}{dT}$, where $\frac{dE}{dT}$ is the temperature coefficient of 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 $E$ and internal resistance $r$ sends current through an external resistor $R$. For what value of $R$ is the thermal power dissipated in the external resistor maximum?

Explanation: The power delivered to $R$ is $P = \left(\frac{E}{R+r}\right)^2 R$. Differentiating with respect to $R$ and setting to zero yields $R = r$.

2. Ten identical cells, each of EMF $1.5\text{ V}$ and internal resistance $0.5\, \Omega$, are connected in series with an external resistor of $5\, \Omega$. What is the current in the circuit?

Explanation: Using $I = \frac{nE}{R + nr}$, we get $I = \frac{10 \times 1.5}{5 + 10 \times 0.5} = \frac{15}{5 + 5} = \frac{15}{10} = 1.5\text{ A}$? Wait, let's calculate: $15 / 10 = 1.5\text{ A}$. Let's check options. Option B is $1.5\text{ A}$.

3. When a shunt of $2\, \Omega$ is connected across a cell in a potentiometer circuit, the balancing length drops from $300\text{ cm}$ to $100\text{ cm}$. What is the internal resistance of the cell?

Explanation: Using $r = R \left(\frac{l_1 - l_2}{l_2}\right) = 2 \times \left(\frac{300 - 100}{100}\right) = 2 \times \frac{200}{100} = 4\, \Omega$.

4. A battery of EMF $12\text{ V}$ and internal resistance $2\, \Omega$ is connected to a DC source for charging with a current of $3\text{ A}$. What is the terminal potential difference across the battery during charging?

Explanation: During charging, $V = E + Ir = 12 + (3 \times 2) = 12 + 6 = 18\text{ V}$.

5. If $m$ parallel branches, each containing $n$ cells in series, are connected across an external load $R$, what is the condition for maximum current in the load?

Explanation: Total EMF is $nE$ and total internal resistance of the combination is $\frac{nr}{m}$. For maximum power/current transfer matching the load, $R = \frac{nr}{m}$.

6. Which of the following defects in primary cells leads to a continuous wastage of zinc even when no current is drawn?

Explanation: Local action forms small local cells due to impurities in zinc, consuming the zinc plate continuously without contributing to external circuit current.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. What primary cell utilizes a copper vessel as both container and positive electrode?

Explanation: The Daniell cell uses a copper vessel containing copper sulfate solution which acts as the positive electrode.

2. The internal resistance of a cell depends on which of the following factors?

Explanation: Internal resistance decreases with larger electrode area and higher electrolyte concentration, and increases with greater distance between electrodes.

3. If the terminal potential difference of a cell drops from $2.0\text{ V}$ to $1.6\text{ V}$ when connected to an external load of $4\, \Omega$, what is the internal resistance of the cell?

Explanation: Current $I = \frac{V}{R} = \frac{1.6}{4} = 0.4\text{ A}$. Also $V = E - Ir \implies 1.6 = 2.0 - 0.4r \implies 0.4r = 0.4 \implies r = 1.0\, \Omega$.

4. Which defect is remedied by the addition of manganese dioxide ($MnO_2$) in a Leclanche cell?

Explanation: Manganese dioxide oxidizes hydrogen gas to water, eliminating polarization at the carbon electrode.

5. What is the primary reason why primary cells cannot be recharged?

Explanation: Primary cells involve irreversible chemical changes that cannot be restored by passing a reverse current.

6. In a mixed grouping of $36$ identical cells each of internal resistance $2\, \Omega$, what arrangement gives maximum current through an external resistance of $2\, \Omega$?

Explanation: For maximum current when $R = r$, total internal resistance must equal $R$. With $n=6, m=6$, $\frac{nr}{m} = \frac{6 \times 2}{6} = 2\, \Omega = R$.

7. How does temperature affect the EMF of most voltaic cells?

Explanation: The Gibbs-Helmholtz relation shows that EMF varies with temperature depending on the temperature coefficient $\frac{dE}{dT}$.

8. What is the purpose of the porous pot in a Daniell cell?

Explanation: The porous pot keeps the copper sulfate and zinc sulfate solutions separated while permitting ions to pass through to complete the circuit.

9. If a voltmeter of very high resistance is connected across the terminals of a cell, what does it measure?

Explanation: A high-resistance voltmeter draws negligible current, hence measuring the open-circuit terminal voltage, which equals the EMF.

10. Which of the following statements is true regarding a secondary cell compared to a primary cell?

Explanation: Secondary cells (like lead-acid accumulators) use reversible electrochemical reactions, allowing them to be recharged repeatedly.