1. Concept Overview
Faraday’s second law tells me how much emf is induced when magnetic flux changes. The first law only says that an emf appears when flux changes, but this second law gives the actual relationship between the emf and the rate at which the flux changes.
The faster the magnetic flux changes, the greater the induced emf. If the flux changes slowly, the induced emf is small. This makes electromagnetic induction feel very intuitive — rapid changes create stronger responses.
1.1. One-line summary I always remember
Induced emf is directly proportional to the rate of change of magnetic flux.
2. Mathematical Statement of the Second Law
The law is expressed mathematically as:
\( \varepsilon = -\dfrac{d\Phi_B}{dt} \)
This equation tells me how the induced emf depends on the rate at which magnetic flux changes with time.
2.1. Meaning of each symbol
- \( \varepsilon \): induced emf
- \( \Phi_B \): magnetic flux
- \( \dfrac{d\Phi_B}{dt} \): rate of change of flux
- The minus sign: shows opposition according to Lenz’s law
2.2. Why the negative sign appears
The negative sign is not about the magnitude but about the direction. It represents that the induced emf tries to oppose the change in flux, which is exactly what Lenz’s law says.
Without this sign, the equation would not tell the correct direction of the induced current.
3. Understanding the Rate of Change of Flux
The core idea here is that induction depends on how fast the flux changes. Even a small change in flux can create a large induced emf if it happens very quickly. Similarly, a large change produces a small induced emf if it happens slowly.
3.1. Three ways flux rate can increase
- Faster motion of a magnet or coil → rapid flux change
- Quick rotation of a coil in a magnetic field → rapidly changing angle
- Rapid switching of magnetic field strength (as in AC systems)
3.2. Flux change without motion
Even if nothing moves, if the magnetic field itself changes with time (like in transformers), the rate of change of flux can be large and produce significant emf.
4. Relation Between Flux and Induced EMF
I like to see the law as a simple cause–effect pair:
- Cause: change in magnetic flux
- Effect: induced emf
And the strength of the effect depends on how quickly the cause happens.
4.1. If flux changes uniformly
If flux changes at a constant rate, then
\( \varepsilon = -\dfrac{\Delta \Phi_B}{\Delta t} \)
This is a simpler version useful for practical calculations.
4.2. If flux changes non-uniformly
When flux changes irregularly, the instantaneous emf is found using the derivative:
\( \varepsilon = -\dfrac{d\Phi_B}{dt} \)
5. Visualising the Law
To make the law feel real, I visualise a rectangular coil cutting magnetic field lines. If the coil moves slowly, only a few lines are cut per second → small emf. If the coil moves fast, many lines are cut per second → larger emf.
This picture helps me relate the formula to physical motion.
5.1. Example illustration
Imagine a magnet being pushed quickly into a coil:
- Fast push: flux changes rapidly → large galvanometer deflection
- Slow push: flux changes slowly → small deflection
- No movement: flux constant → no emf
6. Dependence on Number of Turns
The induced emf increases if the coil has more turns. The total emf is the sum of emf induced in each turn, so the relation becomes:
\( \varepsilon = -N \dfrac{d\Phi_B}{dt} \)
where \( N \) is the number of turns. This makes intuitive sense — more turns means the coil interacts more with the changing flux.
6.1. Why more turns create larger emf
Each loop experiences the same rate of change of flux. So, with N loops, the induced emf simply adds up N times.
7. Role of Lenz’s Law in the Second Law
Faraday’s second law gives the magnitude, but the negative sign connects it to the direction part, which comes from Lenz’s law. Together, both laws give a complete picture of electromagnetic induction.
7.1. Physical meaning of opposition
The induced current creates its own magnetic field that tries to cancel the flux change. This ensures energy conservation and prevents runaway induction.
8. Everyday Applications
Most electrical machines rely heavily on this law because they involve rapidly changing magnetic fields.
8.1. Transformers
Alternating current changes the magnetic flux in the primary coil rapidly, inducing emf in the secondary coil. Faraday’s second law explains why frequency affects transformer performance.
8.2. Generators
Rotating coils in magnetic fields cause continuous flux changes, producing alternating emf. Faster rotation means greater induced emf.
8.3. Induction motors and inductors
These devices rely on changing flux to induce currents, forces or magnetic effects that make them operate efficiently.