distinguish between e.m.f. and potential difference (p.d.) in terms of energy considerations

Practical Circuits – e.m.f. vs Potential Difference (p.d.)

1. Learning Objectives

• Define electromotive force (e.m.f.) ℰ and potential difference V in terms of work done per coulomb.
• Explain the energy‑based distinction between ℰ (inside the source) and V (outside the source).
• Apply the terminal‑voltage equation \(V=\mathcal{E}-Ir\) and determine internal resistance r.
• Use the correct sign‑convention and recognise the effect of the voltmeter’s internal resistance.
• Apply Kirchhoff’s laws and the potential‑divider principle in the context of the energy view.

2. Definitions (Syllabus 10.1)

Term	Definition (energy view)
Electromotive force (e.m.f.) ℰ	Work done **by the internal chemical, photovoltaic, etc. processes** of a source on a charge q while the charge moves **inside** the source from the negative to the positive terminal. \[ \boxed{\mathcal{E}= \frac{W_{\text{inside}}}{q}\quad\left[\text{J C}^{-1}= \text{V}\right]} \]
Potential difference (p.d.) V	Work done **by the external electric field** on a charge q as it moves **outside** the source between any two points of the circuit (e.g. across a resistor, lamp or the terminals under load). \[ \boxed{V= \frac{W_{\text{outside}}}{q}\quad\left[\text{J C}^{-1}= \text{V}\right]} \]

2.1 Sign‑convention

• For a source, the positive direction is from the negative terminal → positive terminal; ℰ is taken as positive in this direction.
• For a p.d. across a passive element, the conventional current flows from higher to lower potential; V is positive in the direction of current flow.

2.2 Internal resistance model

3. Terminal‑Voltage Equation (Energy Perspective)

When a current I flows, the source must supply energy both to the external circuit and to overcome the voltage drop across its internal resistance.

\[ \underbrace{V_{\text{terminal}}}_{\text{p.d. outside}} \;=\; \underbrace{\mathcal{E}}_{\text{e.m.f. inside}} \;-\; I\,r \]

• Open‑circuit (no current, I = 0): \(V_{\text{terminal}}=\mathcal{E}\). The voltmeter reads the e.m.f.
• Closed‑circuit (I ≠ 0): The terminal p.d. is reduced by the internal drop \(I r\). The energy lost per coulomb inside the source is \(I r\) J C⁻¹ (converted to heat in r).

3.1 Voltmeter loading effect (AO3)

A real voltmeter has a finite internal resistance \(R_{\text{vm}}\). When it is connected across the source:

\[ I_{\text{vm}} = \frac{V_{\text{terminal}}}{R_{\text{vm}}} \]

This small current adds to the circuit current, increasing the internal drop \(I_{\text{total}}r\) and causing the measured voltage to be slightly lower than the true terminal p.d. The effect is negligible only if \(R_{\text{vm}}\gg R_{\text{load}}\).

4. Energy Interpretation

For a charge \(q\) that traverses the whole circuit:

• Energy supplied by the source (inside) \(W_{\text{source}} = q\,\mathcal{E}\).
• Energy delivered to the external circuit \(W_{\text{ext}} = q\,V_{\text{terminal}} = q(\mathcal{E}-Ir)\).
• Energy dissipated as heat in the internal resistance \(W_{r}=q\,Ir\).

5. Comparison of e.m.f. and Potential Difference

Aspect	e.m.f. ℰ	Potential Difference V
Physical meaning	Work done per coulomb **by internal processes** of the source (inside the source).	Work done per coulomb **by the external electric field** (outside the source).
Location in circuit	Across the ideal source; represented by the internal‑resistance model.	Across any component or the terminals when the source supplies current.
Sign convention	Positive from negative → positive terminal of the source.	Positive in the direction of conventional current (high → low potential).
Energy transfer per coulomb	Supplies energy \(W = q\mathcal{E}\).	Consumes (or stores) energy \(W = qV\) (e.g., heat in a resistor).
Measurement condition	Measured with a voltmeter **under open‑circuit** (no load current).	Measured with a voltmeter **while the circuit is operating** (load present).
Relation to internal resistance	Maximum possible voltage of the source.	Terminal voltage \(V = \mathcal{E} - Ir\).
Effect of voltmeter loading	Negligible if \(R_{\text{vm}}\gg r\); otherwise the reading is slightly low.	Same loading effect applies; must be considered when high accuracy is required.

6. Practical Determination of ℰ and r (Syllabus 10.1)

1. **Open‑circuit measurement** – Connect a voltmeter (high \(R_{\text{vm}}\)) across the battery terminals with no external load. The reading is the e.m.f. ℰ.
2. **Load measurement** – Attach a known resistor \(R\) across the terminals, allow a steady current \(I\) to flow, and read the terminal voltage again. This is the p.d. \(V_{\text{terminal}}\).
3. **Calculate internal resistance** \[ r = \frac{\mathcal{E} - V_{\text{terminal}}}{I}. \]

Uncertainty note (AO3): Record the voltmeter’s internal resistance and the tolerance of R; propagate these uncertainties to obtain an uncertainty for r.

7. Worked Example (Energy View)

Given a 12 V battery with internal resistance \(r = 0.5\;\Omega\) delivering a current \(I = 2\;\text{A}\) to an external resistor.

\[ \begin{aligned} \mathcal{E} &= 12\ \text{V} \\ V_{\text{terminal}} &= \mathcal{E} - I r = 12 - (2)(0.5) = 11\ \text{V} \\ \text{Energy supplied per coulomb} &= q\mathcal{E}=12\ \text{J C}^{-1} \\ \text{Energy delivered to external circuit per coulomb} &= qV_{\text{terminal}}=11\ \text{J C}^{-1} \\ \text{Energy lost in internal resistance per coulomb} &= qIr = 1\ \text{J C}^{-1}. \end{aligned} \]

The 1 V (or 1 J C⁻¹) difference represents the heat generated inside the battery.

8. Kirchhoff’s Laws (Syllabus 10.2)

8.1 Kirchhoff’s Current Law (KCL)

At any junction, the algebraic sum of currents entering equals the sum leaving.

\[ \sum I_{\text{in}} = \sum I_{\text{out}}. \]

8.2 Kirchhoff’s Voltage Law (KVL) – Energy Form

For any closed loop, the sum of the e.m.f.s and the algebraic sum of potential differences is zero. Using the energy view, each term represents work per coulomb.

\[ \sum \mathcal{E} - \sum V = 0. \]

8.3 Example Circuit

Applying KCL at the node where the current splits:

\[ I = I_{1}+I_{2}. \]

KVL around the left loop (ℰ → r → R₁ → R₃):

\[ \mathcal{E} - I r - I_{1}R_{1} - I_{3}R_{3}=0. \]

KVL around the right loop (ℰ → r → R₂ → R₃):

\[ \mathcal{E} - I r - I_{2}R_{2} - I_{3}R_{3}=0. \]

These equations, together with KCL, allow calculation of all branch currents, illustrating how the energy‑based voltages naturally satisfy Kirchhoff’s laws.

9. Potential Divider (Syllabus 10.3)

9.1 Principle (energy view)

Two series resistors \(R_{1}\) and \(R_{2}\) across a source of e.m.f. ℰ share the total voltage in proportion to their resistances because the same current flows through each. The work per coulomb (voltage) across each resistor is:

\[ V_{R_{1}} = \mathcal{E}\,\frac{R_{1}}{R_{1}+R_{2}},\qquad V_{R_{2}} = \mathcal{E}\,\frac{R_{2}}{R_{1}+R_{2}}. \]

9.2 Null (galvanometer) method

• Connect a galvanometer in opposition to the voltage across \(R_{2}\).
• Adjust a known standard resistor until the galvanometer reads zero (null). At null the galvanometer voltage equals the divided voltage, giving a precise ratio \(\displaystyle\frac{R_{2}}{R_{1}+R_{2}}\).

9.3 Example – Thermistor as a Variable Resistor

1. Place a thermistor \(R_{T}\) in series with a fixed resistor \(R_{F}\) across a 5 V battery.
2. The output voltage across the thermistor is \[ V_{T}=5\ \text{V}\times\frac{R_{T}}{R_{F}+R_{T}}. \]
3. As temperature changes, \(R_{T}\) changes, so \(V_{T}\) varies proportionally – the basis of many temperature‑sensing circuits.

10. Integrated Practical Diagram

11. Key Take‑aways

• e.m.f. ℰ – maximum work per coulomb supplied by the source’s internal processes; measured under open‑circuit conditions.
• Potential difference V – work per coulomb done by the external field between two points; depends on the current flowing and is measured under load.
• The terminal‑voltage relation \(V = \mathcal{E} - Ir\) links the two concepts and introduces the internal resistance of real sources.
• Correct sign‑convention and awareness of voltmeter loading are essential for accurate measurements (AO3).
• Kirchhoff’s laws are simply the algebraic expression of energy conservation around nodes (KCL) and loops (KVL) using ℰ and V.
• The potential‑divider exploits the linear relationship between series resistances and voltage; a galvanometer null method provides high‑precision measurements.