What does Faraday's law of electromagnetic induction state and what is its point form?

In one line

Faraday's law states that the emf induced in a closed loop equals the negative rate of change of magnetic flux through it; in point form, \(\nabla\times\vec{E} = -\partial\vec{B}/\partial t\).

Explained simply

Move a magnet into a coil and a galvanometer needle jumps; hold it still and the needle returns to zero. Only a changing magnetic flux produces a voltage. The minus sign (Lenz's law) means the induced current opposes the change, like the coil "resisting" the magnet. This is how generators, transformers and induction cooktops work.

How it works

1. Magnetic flux through a surface: \(\Phi_B = \int_S\vec{B}\cdot d\vec{S}\).
2. Faraday-Lenz law: \(e = -\dfrac{d\Phi_B}{dt}\); for \(N\) turns, \(e = -N\dfrac{d\Phi_B}{dt}\).
3. The emf is the work done per unit charge around the loop: \(e = \oint_C\vec{E}\cdot d\vec{l}\).
4. Integral form: \(\oint_C\vec{E}\cdot d\vec{l} = -\int_S\dfrac{\partial\vec{B}}{\partial t}\cdot d\vec{S}\) (for a fixed loop).
5. Apply Stokes' theorem: \(\int_S(\nabla\times\vec{E})\cdot d\vec{S} = -\int_S\dfrac{\partial\vec{B}}{\partial t}\cdot d\vec{S}\).
6. Point form: \(\nabla\times\vec{E} = -\dfrac{\partial\vec{B}}{\partial t}\). A changing magnetic field creates a circulating (non-conservative) electric field.

Key formula

\(e = -N\dfrac{d\Phi_B}{dt}\); \(\nabla\times\vec{E} = -\dfrac{\partial\vec{B}}{\partial t}\). \(e\): emf (V), \(N\): number of turns, \(\Phi_B\): magnetic flux (Wb = T m²), \(t\): time (s), \(\vec{E}\): electric field (V/m), \(\vec{B}\): flux density (T).

Quick example

A coil of 100 turns has its flux changed uniformly from 0.02 Wb to 0 in 0.1 s. Magnitude of emf \(= N\Delta\Phi/\Delta t = 100\times0.02/0.1 = 20\) V. The induced current flows in a direction that tries to keep the flux from falling.

Common mistakes

• Dropping the minus sign, which carries Lenz's law.
• Thinking a steady (constant) magnetic field induces emf in a stationary coil.
• Writing \(\nabla\times\vec{E} = 0\) in time-varying situations; that is true only in electrostatics.

How MU asks it

Asked as "State Faraday's law and derive its differential form" (5 marks), usually inside "Derive Maxwell's equations". State the law with Lenz's sign, write flux and emf as integrals, use Stokes' theorem, and box \(\nabla\times\vec{E} = -\partial\vec{B}/\partial t\).