Electromotive Force (EMF)

Electromotive Force (EMF), denoted by $\varepsilon$ (epsilon), is the electrical energy per unit charge generated by a source, converted from chemical, mechanical, or other forms of energy.

Definition

EMF is the energy supplied by a source per unit charge:

$$\varepsilon = \frac{W}{q} = \frac{P}{I}$$

where:

  • $\varepsilon$ = EMF (Volts, V)
  • $W$ = work done (Joules, J)
  • $q$ = charge (Coulombs, C)
  • $P$ = power (Watts, W)
  • $I$ = current (Amperes, A)

Key Characteristics

EMF as Open-Circuit Voltage

EMF is the potential difference across a source when NO current is being drawn from or delivered to it:

$$V_{\text{open circuit}} = \varepsilon$$

Function of EMF

  • Generated by a source (battery, generator, solar cell)
  • Provides the "push" that makes charge flow from one terminal to another
  • Converts non-electrical energy into electrical energy:
    • Batteries: Chemical → Electrical
    • Generators: Mechanical → Electrical
    • Solar cells: Light → Electrical
    • Fuel cells: Chemical → Electrical

EMF vs Potential Difference

Aspect EMF Potential Difference (Voltage)
Definition Energy supplied per unit charge by the source Energy dissipated per unit charge in the circuit
Condition Exists even with no current (open circuit) Exists only when current flows
Measurement Across source terminals with open circuit Across any component or source in a closed circuit
Value Maximum possible voltage of the source Always less than or equal to EMF
Cause Energy conversion within the source Energy dissipation in circuit components

Important Distinction

$$\varepsilon \geq V$$

  • EMF ($\varepsilon$): The "ideal" voltage the source can provide
  • Terminal Voltage ($V$): The "actual" voltage available at the terminals when current flows, reduced by internal resistance effects

Formula Relationships

Complete Circuit Equation

$$\varepsilon = IR + Ir = V + Ir$$

where:

  • $IR = V$ = terminal voltage (voltage across external load)
  • $Ir$ = voltage drop across internal resistance
  • $r$ = internal resistance of the source

Terminal Voltage

$$V = \varepsilon - Ir$$

This shows that the terminal voltage is always less than EMF when the source is delivering current (discharging).

Sources of EMF

Common EMF Sources

Source EMF Generation Typical EMF Values
AA Alkaline Battery Chemical reaction 1.5 V
Car Battery (Lead-acid) Chemical reaction 12 V
Lithium-ion Battery Chemical reaction 3.7 V
Solar Cell Photovoltaic effect 0.5 V per cell
Generator Electromagnetic induction Variable (110-240 V AC)
Fuel Cell Chemical reaction 0.7-1.2 V per cell

Internal Resistance of Sources

All real sources have some internal resistance ($r$):

  • Fresh AA battery: $r \approx 0.1 - 0.5$ Ω
  • Car battery: $r \approx 0.005 - 0.02$ Ω
  • Power supply: $r \approx 0.001 - 0.1$ Ω

Worked Examples

Example 1: Basic EMF Calculation

Problem: A battery delivers 12 W of power at 2 A current. What is its EMF?

Solution:

$$\varepsilon = \frac{P}{I} = \frac{12}{2} = 6 \text{ V}$$

Example 2: Finding EMF from Terminal Voltage

Problem: A battery has terminal voltage 11.8 V when delivering 5 A current. Its internal resistance is 0.04 Ω. What is its EMF?

Solution:

$$\varepsilon = V + Ir = 11.8 + (5)(0.04) = 11.8 + 0.2 = 12.0 \text{ V}$$

Example 3: Maximum Current (Short Circuit)

Problem: A 9 V battery has internal resistance 2 Ω. What is the maximum current it can theoretically deliver?

Solution:

When short-circuited ($R = 0$):

$$I_{\text{max}} = \frac{\varepsilon}{r} = \frac{9}{2} = 4.5 \text{ A}$$

Mermaid Diagram: EMF in a Circuit

flowchart LR
    subgraph Source["EMF Source (Battery)"]
        E[("EMF ε")]
        Rint[("Internal Resistance r")]
    end
    
    subgraph Circuit["External Circuit"]
        Rext[("Load R")]
    end
    
    E -->|"ε = V + Ir"| Rint
    Rint --> A["Terminal +"]
    A --> Rext
    Rext --> B["Terminal -"]
    B --> E
    
    style E fill:#90EE90
    style Rint fill:#FFB6C1
    style Rext fill:#87CEEB

Related Concepts

Key Takeaways

  1. EMF is not a force — despite the name, EMF is energy per unit charge (voltage)
  2. EMF is constant for a given source (under constant conditions)
  3. Terminal voltage varies with current due to internal resistance
  4. EMF = Terminal Voltage only when no current flows (open circuit)
  5. The unit of EMF is the Volt (V), same as potential difference