5.1 Electromotive force

Definition of electromotive force

Electromotive force (EMF) is the energy per unit charge that a source supplies to an electrical circuit. Despite the name, it's not actually a force. It's a measure of how much energy a source (like a battery or generator) gives to each coulomb of charge that passes through it. EMF is what drives current through a circuit.

Voltage vs electromotive force

These two terms get confused constantly, so here's the distinction:

• EMF is the maximum potential difference a source can provide, measured when no current is flowing. It's a property of the source itself.
• Voltage (potential difference) is measured between any two points in a circuit and changes depending on circuit conditions.

When current flows, voltage drops occur across resistors and other components. EMF stays constant as a characteristic of the source. Think of EMF as what the battery can provide, and terminal voltage as what it actually delivers once internal losses are accounted for.

Units of measurement

EMF is measured in volts (V), the same unit as voltage. One volt equals one joule of energy per coulomb of charge:

$1 \text{ V} = 1 \text{ J/C}$

EMF can also be written in terms of work done per unit charge:

$\text{EMF} = \frac{W}{q}$

For smaller values, millivolts (mV) are common; for larger values, kilovolts (kV).

Sources of electromotive force

Different EMF sources convert different forms of energy into electrical energy. The type of source determines its voltage output, efficiency, and best applications.

Batteries and chemical cells

Batteries convert chemical energy into electrical energy through redox (reduction-oxidation) reactions. They consist of two electrodes (an anode and a cathode) immersed in an electrolyte.

• A standard zinc-carbon dry cell provides about $1.5 \text{ V}$
• Lithium-ion batteries offer higher energy density and are rechargeable
• A car's lead-acid battery produces $12 \text{ V}$ by connecting six $2 \text{ V}$ cells in series

Generators and dynamos

Generators convert mechanical energy into electrical energy through electromagnetic induction. A coil rotates inside a magnetic field, and the changing magnetic flux induces an EMF.

• AC generators (alternators) produce a sinusoidal EMF waveform
• DC generators use a commutator to convert the AC output into DC
• The EMF magnitude depends on magnetic field strength, number of coil turns, and rotation speed

Solar cells and photovoltaics

Solar cells convert light energy into electrical energy using the photovoltaic effect. They're made of semiconductor materials (typically silicon) arranged in p-n junctions.

• A single silicon solar cell typically produces $0.5 \text{–} 0.6 \text{ V}$
• Cells are connected in series to increase the total EMF output
• They generate DC and work with sunlight or artificial light

Thermoelectric devices

These convert temperature differences into electrical energy using the Seebeck effect. Two dissimilar conductors are joined at two junctions kept at different temperatures.

• The EMF generated is proportional to the temperature difference between the junctions
• Thermocouples use this principle for temperature measurement
• Peltier devices can work in reverse: apply current to create a temperature difference

Electromotive force in circuits

Once you place an EMF source in a circuit, its behavior depends on internal resistance, load resistance, and current draw. This is where the ideal picture of a "perfect" voltage source breaks down.

EMF vs terminal voltage

The terminal voltage is what you actually measure across a source's terminals when current is flowing. It's always less than or equal to the EMF because of internal resistance losses.

The relationship is:

$V_t = \text{EMF} - Ir$

where $r$ is the internal resistance and $I$ is the current. As you draw more current, the terminal voltage drops further below the EMF.

Internal resistance effects

Every real EMF source has some internal resistance $r$, which causes a voltage drop inside the source itself.

• This reduces the voltage available to the external circuit
• It limits the maximum current the source can deliver
• It causes power to be wasted as heat inside the source
• Battery internal resistance increases with age, which is why old batteries perform worse

Load resistance considerations

The load resistance $R_L$ determines how much current the source delivers and how power is distributed.

• Short circuit ($R_L \approx 0$): maximum current flows, but almost all power is lost to internal resistance
• Open circuit ($R_L \to \infty$): no current flows, and terminal voltage equals EMF
• Maximum power transfer occurs when $R_L = r$ (load resistance equals internal resistance)

The power delivered to the load is:

$P = \frac{\text{EMF}^2 \, R_L}{(R_L + r)^2}$

Kirchhoff's voltage law

Kirchhoff's Voltage Law (KVL) states that the sum of all voltage changes around any closed loop in a circuit equals zero. This is a direct consequence of energy conservation: a charge that travels around a complete loop returns to its starting potential.

Application to EMF sources

When writing KVL equations, EMF sources contribute a positive voltage rise (if you traverse them from $-$ to $+$). Key points:

• Multiple EMF sources can appear in a single loop
• Opposing EMF sources partially or fully cancel each other
• KVL lets you solve for unknown currents and voltages in circuits with multiple sources

Loop equations in circuits

To apply KVL systematically:

1. Identify all independent loops in the circuit.
2. Assign a current direction for each loop (if your guess is wrong, the answer will just come out negative).
3. Write one KVL equation per loop, summing voltage rises and drops to zero.
4. Solve the resulting system of simultaneous equations.

For example, a loop with two EMF sources and two resistors might give:

$\text{EMF}_1 - IR_1 - IR_2 - \text{EMF}_2 = 0$

Each independent loop yields one equation, and you need as many equations as you have unknowns.

Electromagnetic induction

Electromagnetic induction is the process of generating an EMF through a changing magnetic field. It's the principle behind generators, transformers, and many sensors.

Faraday's law of induction

Faraday's law states that the induced EMF is proportional to the rate of change of magnetic flux through a loop:

$\text{EMF} = -N\frac{d\Phi_B}{dt}$

where $N$ is the number of turns and $\Phi_B$ is the magnetic flux. The negative sign reflects Lenz's law (the induced EMF opposes the change that caused it).

Flux can change by varying the magnetic field strength, the area of the loop, or the angle between the field and the loop's surface.

Lenz's law and EMF

Lenz's law tells you the direction of the induced EMF: it always acts to oppose the change in flux that produced it. This is really just conservation of energy at work. If the induced current helped the change instead of opposing it, you'd get energy from nothing.

Lenz's law also explains eddy currents, which are loops of current induced in bulk conductors moving through magnetic fields.

Motional EMF

When a straight conductor of length $l$ moves with velocity $v$ through a uniform magnetic field $B$, the induced EMF is:

$\text{EMF} = Blv$

Use the right-hand rule to find the direction: point your thumb along the velocity, curl your fingers along the field, and your palm indicates the direction of the force on positive charges (which determines the EMF polarity).

Motional EMF is really a special case of Faraday's law where the flux changes because the conductor sweeps out new area.

EMF in transformers

Transformers use electromagnetic induction to step voltage up or down in AC circuits. They consist of two coils (primary and secondary) wound around a shared magnetic core.

Ideal transformer equations

For an ideal transformer (perfect coupling, no losses):

• Voltage ratio equals turns ratio: $\frac{\text{EMF}_s}{\text{EMF}_p} = \frac{N_s}{N_p}$
• Current ratio is the inverse: $\frac{I_s}{I_p} = \frac{N_p}{N_s}$
• Power is conserved: $\text{EMF}_p \, I_p = \text{EMF}_s \, I_s$
• Impedance transforms as the square of the turns ratio: $\frac{Z_s}{Z_p} = \left(\frac{N_s}{N_p}\right)^2$

Step-up vs step-down transformers

• Step-up ($N_s > N_p$): increases voltage, decreases current. Used in power transmission to reduce line losses (since $P_{\text{loss}} = I^2 R$, lower current means less wasted power).
• Step-down ($N_s < N_p$): decreases voltage, increases current. Used in consumer electronics and household outlets.
• Autotransformers use a single tapped coil instead of two separate windings.

Back EMF in motors

When a motor spins, its rotating armature acts like a generator and produces an EMF that opposes the applied voltage. This is called back EMF (or counter-EMF).

Counter-electromotive force

Back EMF increases as the motor speeds up:

$\text{EMF}_{\text{back}} = K\omega$

where $K$ is the motor constant and $\omega$ is the angular velocity. This is actually a useful self-regulating feature. As the motor speeds up, back EMF rises, which reduces the net voltage across the windings and limits the current. Without back EMF, the motor would draw excessive current and burn out.

At startup ($\omega = 0$), there's no back EMF, so the initial current surge is very high. That's why motors often need startup resistors or soft-start circuits.

Motor speed control

• Back EMF naturally regulates speed: if the motor is loaded and slows down, back EMF decreases, more current flows, and torque increases to compensate.
• Speed can be controlled by adjusting the applied voltage or using pulse-width modulation (PWM).
• Regenerative braking in electric vehicles takes advantage of back EMF: when the motor acts as a generator during deceleration, it converts kinetic energy back into electrical energy.

Measurement techniques

Measuring EMF accurately requires care, because the act of measurement itself can affect the result.

Voltmeters and EMF

A voltmeter connected across a source measures terminal voltage, not necessarily the true EMF. To get close to the actual EMF:

• Use a digital voltmeter with very high input impedance, so it draws negligible current
• Analog voltmeters have lower input impedance and can "load" the source, giving a reading below the true EMF
• The truest EMF reading comes under open-circuit conditions (no current flowing)

Potentiometer method

The potentiometer method measures EMF without drawing any current from the source, which eliminates the effect of internal resistance entirely.

1. Connect the unknown EMF source and a standard reference cell to a potentiometer wire.
2. Slide the contact along the wire until the galvanometer reads zero (null condition).
3. At null, no current flows from the unknown source, so you're measuring its true EMF.
4. Calculate the unknown EMF from the ratio of wire lengths.

This null-balance approach gives very precise results and is a classic Physics II lab technique.

Applications of EMF

Power generation systems

Nearly all large-scale electricity generation relies on EMF from electromagnetic induction. Hydroelectric dams, wind turbines, nuclear plants, and fossil fuel plants all use turbines to spin generators. The energy source differs, but the EMF-producing mechanism is the same: rotating coils in magnetic fields.

Energy conversion devices

Several less common devices also produce EMF:

• Fuel cells convert chemical energy (e.g., hydrogen + oxygen) directly into electrical EMF
• Thermoelectric generators produce EMF from temperature gradients (used in spacecraft)
• Piezoelectric materials generate EMF when mechanically stressed (used in sensors and igniters)
• Magnetohydrodynamic (MHD) generators produce EMF by passing conducting fluids through magnetic fields

Electrochemical processes

Applied EMF can also drive non-spontaneous reactions:

• Electrolysis uses EMF to decompose compounds (e.g., splitting water into hydrogen and oxygen)
• Electroplating deposits metal coatings onto objects
• Electrorefining purifies metals in electrolytic cells
• Galvanic corrosion protection systems use generated EMF to protect metal structures

EMF in electrochemistry

EMF connects electrical potential to chemical equilibria. The voltage of an electrochemical cell tells you whether a reaction is spontaneous and how much energy it can deliver.

Standard electrode potential

Each half-reaction has a standard electrode potential, measured relative to the standard hydrogen electrode (SHE), which is defined as $0 \text{ V}$.

• More positive values indicate stronger oxidizing agents (greater tendency to be reduced)
• The cell EMF equals the difference between the cathode and anode standard potentials:

$E^\circ_{\text{cell}} = E^\circ_{\text{cathode}} - E^\circ_{\text{anode}}$

A positive cell EMF means the reaction is spontaneous under standard conditions.

Nernst equation

The Nernst equation adjusts the cell EMF for non-standard concentrations:

$E = E^\circ - \frac{RT}{nF}\ln Q$

where $E^\circ$ is the standard cell potential, $R$ is the gas constant, $T$ is temperature in Kelvin, $n$ is the number of moles of electrons transferred, $F$ is Faraday's constant, and $Q$ is the reaction quotient.

This equation explains why battery voltage changes as the battery discharges (the concentrations of reactants and products shift, changing $Q$).

Limitations and non-idealities

Real EMF sources never behave perfectly. Recognizing these deviations helps you make more accurate predictions about circuit behavior.

EMF losses in real systems

• Internal resistance causes voltage drops and heat dissipation inside the source
• Polarization effects in batteries reduce available EMF during heavy discharge
• Eddy current losses in transformers and generators waste energy as heat in the core
• Hysteresis losses in magnetic materials dissipate energy during each AC cycle
• Unwanted contact potentials and thermoelectric effects can introduce small stray EMFs

Efficiency considerations

No EMF source is 100% efficient. Typical conversion efficiencies vary widely:

• Solar cells: roughly 15–25%
• Thermoelectric generators: about 5–8%
• Large mechanical generators: over 90%
• The gap between EMF and useful output always involves some combination of resistive, magnetic, and thermal losses