23.2 Faraday’s Law of Induction: Lenz’s Law

Faraday's Law of Induction and Lenz's Law

Faraday's Law of Induction and Lenz's Law explain how changing magnetic fields create electric currents. These two ideas are the foundation for technologies you encounter constantly: generators, transformers, induction cooktops, and even MRI machines. If you understand these concepts, you understand why those devices work.

Calculation of Induced EMF

Magnetic flux is the starting point. Before you can talk about induced voltage, you need to know what's changing. Magnetic flux ($\Phi_B$) measures how much magnetic field passes through a loop of wire:

$\Phi_B = BA\cos\theta$

• $B$ = magnetic field strength (in teslas, T)
• $A$ = area of the loop (in m²)
• $\theta$ = angle between the magnetic field and the line perpendicular (normal) to the loop surface
• Units of flux: webers (Wb), where 1 Wb = 1 T·m²

Notice that flux can change if any of those three quantities change: the field strength gets stronger or weaker, the loop area changes, or the loop rotates (changing $\theta$).

Faraday's Law then tells you the induced electromotive force (emf):

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

The faster the flux changes, the larger the induced emf. For a coil with $N$ turns, the equation becomes:

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

The negative sign isn't just a math detail; it encodes Lenz's Law, which tells you the direction of the induced emf (more on that below).

Once you have the induced emf, finding the induced current is just Ohm's Law:

$I = \frac{\varepsilon}{R}$

A larger emf or a smaller resistance means a larger induced current.

Direction Prediction with Lenz's Law

Lenz's Law says the induced current always flows in a direction that opposes the change in flux that caused it. Think of it as nature resisting change.

• If the magnetic flux through a loop is increasing, the induced current creates its own magnetic field that points opposite to the external field, fighting the increase.
• If the flux is decreasing, the induced current creates a field in the same direction as the external field, trying to maintain it.

This opposition is a direct consequence of conservation of energy. If the induced current helped the change instead of opposing it, you'd get runaway energy from nothing, which violates energy conservation.

Using the right-hand rule to find current direction:

1. Determine whether the flux through the loop is increasing or decreasing.
2. The induced current must create a magnetic field that opposes that change. Figure out which direction that opposing field needs to point.
3. Curl the fingers of your right hand in the direction the current would need to flow so that your thumb points in the direction of the opposing field. That curl is the current direction.

This lets you predict the direction of induced current without plugging into equations.

Applications of Electromagnetic Induction

Transformers change AC voltage levels using two coils wound around a shared iron core.

• The primary coil carries alternating current, which creates a continuously changing magnetic flux in the core.
• That changing flux passes through the secondary coil and induces an emf in it.
• The voltage ratio depends on the number of turns: $\frac{V_s}{V_p} = \frac{N_s}{N_p}$. More turns on the secondary coil means a higher output voltage (step-up); fewer turns means a lower output voltage (step-down).
• This is how power grids efficiently transmit electricity over long distances at high voltage, then step it down for household use.

Generators convert mechanical energy into electrical energy.

• A coil rotates inside a magnetic field, so the angle $\theta$ changes continuously, which changes the flux and induces an emf.
• The mechanical rotation can come from steam turbines, water turbines, or wind turbines.
• The output is alternating current (AC) because the flux increases and decreases periodically as the coil spins.

Induction cooktops use a coil beneath the cooking surface that carries high-frequency alternating current.

• The rapidly changing magnetic field induces eddy currents (loops of current) directly in the metal cookware.
• These eddy currents encounter resistance in the metal, which generates heat. The cookware heats up while the cooktop surface stays relatively cool.

MRI machines rely on induction to image the body.

• A strong magnetic field aligns hydrogen protons in body tissues.
• Radio-frequency pulses knock the protons out of alignment; as they realign, they emit signals.
• Induction coils surrounding the patient detect these tiny changing magnetic fields, and software reconstructs detailed images from the signals.

Electromagnetic Theory and Induction

Faraday's Law is one of Maxwell's four equations, which together describe all classical electromagnetic phenomena. In that broader framework, a changing magnetic field produces an electric field even in empty space, not just in a wire loop.

The Lorentz force ($F = qv \times B$) provides a complementary perspective: when a conductor moves through a magnetic field, the magnetic force pushes charges inside it, which is another way to understand why an emf is induced in a moving wire.

The magnetic dipole moment ($\mu = NIA$) describes the strength and orientation of a current loop acting as a small magnet. It determines the torque a loop experiences in an external field ($\tau = \mu B \sin\theta$) and connects Faraday's Law to the behavior of magnetic materials.