7.3 Motional emf

Definition of motional emf

When a conductor moves through a magnetic field, the magnetic force pushes charge carriers inside it to one end, creating a voltage across the conductor. That voltage is the motional emf. It's one of the most direct examples of converting mechanical energy into electrical energy, and it's the core idea behind how generators work.

Electromagnetic induction basics

Electromagnetic induction is the broader principle: whenever the magnetic flux through a circuit changes, a voltage is induced. Faraday discovered this in 1831. Motional emf is a specific case of induction where the flux changes because the conductor itself is moving (rather than the magnetic field changing in time).

• Induced emf depends on the rate of change of magnetic flux through a circuit
• This principle underlies transformers, inductors, and generators

Moving conductor in magnetic field

Picture a metal bar sliding along a pair of rails in a uniform magnetic field. The free charges inside the bar are moving with it, so they experience a magnetic force ($F = qv \times B$). That force pushes positive charges toward one end of the bar and negative charges toward the other, building up a potential difference.

• The direction of the induced emf is determined by the right-hand rule: point your fingers in the direction of the velocity, curl them toward $B$, and your thumb points in the direction of the force on positive charges.
• The magnitude depends on three things: the conductor's speed $v$, the field strength $B$, and the length of the conductor $L$.

Faraday's law of induction

Faraday's law gives you the quantitative relationship between changing magnetic flux and induced emf. It's the single most important equation in this unit.

Mathematical expression

$\varepsilon = -N\frac{d\Phi_B}{dt}$

• $\varepsilon$ is the induced emf (in volts)
• $N$ is the number of turns in a coil (for a single conductor, $N = 1$)
• $\frac{d\Phi_B}{dt}$ is the rate of change of magnetic flux
• The negative sign comes from Lenz's law: the induced emf acts in the direction that opposes the flux change

Flux change vs time

Magnetic flux is defined as:

$\Phi_B = BA\cos\theta$

where $B$ is the magnetic field strength, $A$ is the area of the loop, and $\theta$ is the angle between the field and the area's normal vector.

Flux can change if any of those three quantities change:

• $B$ increases or decreases
• $A$ grows or shrinks (like a sliding bar expanding the loop area)
• $\theta$ changes (like a rotating coil)

Faster changes produce larger induced emfs. A slow, gentle change in flux gives a small voltage; a sudden change gives a large one.

Lenz's law

Lenz's law tells you the direction of the induced current. The rule: the induced current always flows in the direction that creates a magnetic field opposing the change that caused it.

Direction of induced current

Here's how to apply Lenz's law step by step:

1. Determine whether the magnetic flux through the loop is increasing or decreasing.
2. The induced current will create a magnetic field that opposes that change. If flux is increasing, the induced field points opposite to the external field. If flux is decreasing, the induced field points in the same direction as the external field.
3. Use the right-hand rule to find which direction the current must flow to produce that opposing field: curl your right hand so your fingers follow the current, and your thumb points in the direction of the induced magnetic field.

Conservation of energy

Lenz's law is really a statement about energy conservation. If the induced current aided the flux change instead of opposing it, you'd get a runaway effect that creates energy from nothing.

• The opposing force means you have to do work to keep the conductor moving.
• That mechanical work is what gets converted into electrical energy.
• In a generator, for example, the engine must continuously push against the magnetic braking force on the current-carrying coils.

Motional emf formula

The motional emf for a straight conductor moving perpendicular to a uniform magnetic field is:

$\varepsilon = vBL$

This is the workhorse equation for motional emf problems.

Derivation from first principles

1. A charge $q$ inside the conductor moves with velocity $v$ through field $B$, so it feels a magnetic force: $F = qvB$
2. This force acts like an electric force, pushing charges along the length of the conductor. The equivalent electric field is: $E = vB$
3. The emf is the electric field times the length of the conductor over which it acts: $\varepsilon = EL = vBL$

This derivation assumes $v$, $B$, and $L$ are all mutually perpendicular. If they aren't, you need the cross product and only the perpendicular components contribute.

Key variables and units

A quick unit check: $\text{(m/s)} \times \text{T} \times \text{m} = \text{V}$. This works because 1 T = 1 kg/(A·s²), and the units reduce to volts.

Applications of motional emf

Electric generators

Generators are the most important application of motional emf. A coil of wire rotates inside a magnetic field, and the changing flux through the coil induces an emf.

• As the coil rotates, the angle $\theta$ in $\Phi_B = BA\cos\theta$ changes continuously, producing a sinusoidal (AC) voltage.
• The peak emf of a generator is $\varepsilon_0 = NBA\omega$, where $\omega$ is the angular velocity.
• Faraday's disk (homopolar generator) is a simpler design that produces DC by spinning a conducting disk in a magnetic field.

Electromagnetic braking

Eddy current brakes use motional emf to slow moving objects without physical contact.

• When a conductor moves through a magnetic field, induced eddy currents create forces that oppose the motion (Lenz's law in action).
• The braking force is proportional to speed: faster motion means larger induced currents and stronger opposition. The brake is self-regulating.
• Used in roller coasters, high-speed trains, and some exercise equipment. A major advantage is zero friction wear on brake pads.

Motional emf in different geometries

Linear motion

The simplest case: a straight bar sliding along parallel rails in a uniform field $B$ pointing perpendicular to the plane of the rails.

• The bar moves at velocity $v$, sweeping out new area at a rate $\frac{dA}{dt} = Lv$.
• The flux change is $\frac{d\Phi_B}{dt} = BLv$, giving $\varepsilon = BLv$.
• If the rails are connected by a resistor $R$, the induced current is $I = \frac{BLv}{R}$.

Rotational motion

For a coil rotating in a uniform magnetic field:

• The flux is $\Phi_B = BA\cos(\omega t)$, where $\omega$ is the angular frequency.
• Differentiating: $\varepsilon = NBA\omega\sin(\omega t)$.
• The emf is maximum when the coil plane is parallel to the field ($\sin(\omega t) = 1$), because that's when the flux is changing fastest.
• The emf is zero when the coil is perpendicular to the field, because at that instant the flux is at a maximum or minimum and momentarily not changing.

Induced electric fields

Motional emf can be understood through the force on charges in a moving conductor. But there's a deeper picture: changing magnetic fields create electric fields even in empty space.

Non-conservative nature

Induced electric fields are fundamentally different from the electrostatic fields you studied earlier.

• Electrostatic fields are conservative: the work done moving a charge around any closed loop is zero.
• Induced electric fields are non-conservative: a charge moving around a closed loop can gain energy. That's exactly what drives current around a circuit.
• In differential form, this is expressed as $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$. A nonzero curl means the field is non-conservative.

Comparison with electrostatic fields

Eddy currents

Eddy currents are loops of current induced inside bulk conductors (not just wires) when they're exposed to changing magnetic fields.

Formation and effects

When a sheet of metal moves through a magnetic field, or when the field through a stationary sheet changes, currents swirl in closed loops within the metal. These are called eddy currents because they resemble eddies in flowing water.

• By Lenz's law, eddy currents produce magnetic fields that oppose the change causing them.
• The currents flow through the resistance of the metal, dissipating energy as heat (Joule heating: $P = I^2R$).
• This energy dissipation is why eddy currents act as a braking mechanism.

Industrial applications

• Induction heating: Rapidly alternating magnetic fields induce large eddy currents in metal, heating it. Used in metalworking and induction cooktops.
• Electromagnetic damping: Eddy currents in the metal frame of a galvanometer quickly bring the needle to rest without oscillation.
• Non-destructive testing: Eddy current probes detect cracks or defects in metal parts by sensing changes in the induced current patterns.
• Magnetic levitation: A changing magnetic field induces eddy currents in a conductor, and the repulsive force can levitate objects.

Motional emf in conductors

Solid conductors vs fluids

Most textbook problems deal with solid conductors (metal bars, wire coils), where the geometry is fixed and the analysis is straightforward. But motional emf also occurs in conducting fluids like liquid metals, saltwater, and plasmas.

• Magnetohydrodynamics (MHD) is the field that studies the interaction of magnetic fields with conducting fluids.
• MHD generators pass a high-velocity conducting gas through a magnetic field to generate electricity directly, with no moving mechanical parts.
• In astrophysics, motional emf in stellar plasmas drives large-scale magnetic field generation (the dynamo effect).

Hall effect relationship

The Hall effect is closely related to motional emf. When current flows through a conductor in a magnetic field, the charge carriers are deflected sideways, creating a voltage perpendicular to both the current and the field.

• The Hall voltage is $V_H = \frac{IB}{nqt}$, where $n$ is the charge carrier density and $t$ is the conductor thickness.
• Hall probes are used to measure magnetic field strength with high precision.
• In a moving conductor with no external current, the motional emf is the Hall-type voltage: the magnetic force separates charges just as it does in the Hall effect.

Measurement techniques

Experimental setups

• Sliding bar on rails: The classic setup. A bar slides along conducting rails in a known magnetic field. Measure the emf with a voltmeter across the rails.
• Rotating coil: Spin a coil at known angular velocity in a uniform field. The sinusoidal output voltage confirms $\varepsilon = NBA\omega\sin(\omega t)$.
• Faraday disk: A conducting disk rotates in a magnetic field, producing a steady DC voltage between the center and the rim.

Error sources and mitigation

• Contact resistance at sliding or rotating connections can reduce the measured voltage. Use clean, low-friction contacts.
• Thermal emfs from temperature differences at junctions between dissimilar metals. Minimize by using the same material for all connections.
• Stray magnetic fields from nearby equipment. Shield the experiment or measure and subtract the background field.
• Instrument loading: A voltmeter with insufficient input impedance draws current and reduces the measured emf. Use a high-impedance voltmeter.

Motional emf in Earth's magnetic field

Earth's magnetic field (roughly $5 \times 10^{-5}$ T at the surface) is weak, but it can produce measurable effects over large distances because $\varepsilon = vBL$ grows with conductor length.

Geomagnetic induction

• A commercial airplane with a 60 m wingspan flying at 250 m/s through Earth's field develops a motional emf of roughly $\varepsilon = (250)(5 \times 10^{-5})(60) \approx 0.75$ V across its wings. Small, but real.
• Long power lines and pipelines (hundreds of km) can develop significant induced voltages during geomagnetic disturbances.
• Ocean currents moving through Earth's field generate tiny electric fields that oceanographers use to measure flow rates.

Space weather effects

• During solar storms, rapid changes in Earth's magnetosphere induce large electric fields at the surface.
• These geomagnetically induced currents (GICs) can flow through power grids, overheating transformers. The 1989 Quebec blackout was caused by a geomagnetic storm.
• Satellites in orbit move through Earth's field at ~7.5 km/s, and long conducting tethers in space can generate substantial voltages (NASA's TSS-1R experiment produced over 3,500 V across a 20 km tether).

Advanced concepts

Relativistic motional emf

Motional emf reveals something deep about the connection between electric and magnetic fields. In special relativity, what one observer sees as a magnetic force on a moving charge, another observer (moving with the charge) sees as a purely electric force.

• The Lorentz transformation shows that electric and magnetic fields mix into each other when you change reference frames.
• Motional emf in one frame becomes an induced electric field in another. The physical result (the voltage, the current) is the same in both frames.
• This equivalence was one of the key insights that led Einstein to develop special relativity.

Quantum Hall effect

At very low temperatures and in strong magnetic fields, the Hall resistance of a two-dimensional electron system becomes quantized, taking only discrete values:

$R_H = \frac{h}{ne^2}$

where $h$ is Planck's constant, $e$ is the electron charge, and $n$ is an integer. This quantum Hall effect is so precise that it's used as a standard for electrical resistance. While it goes well beyond the classical motional emf covered in this unit, it shows how the same magnetic-force-on-moving-charges physics extends into quantum mechanics.