Faraday’s Law of Induction describes how a time-varying magnetic field produces an electric field. It is one of Maxwell’s four equations and forms the foundation of electromagnetic induction — the principle behind electric generators, transformers, and many astrophysical phenomena involving dynamic magnetic fields.
It also provides a profound link between electricity and magnetism, showing that changing magnetic environments can create electric effects, even in vacuum.
Let C be a closed loop enclosing a surface S. The integral form is:
∮CE⋅dℓ=−dtd∫SB⋅dA
Where:
E is the electric field,
dℓ is a differential length element along the closed loop,
B is the magnetic field,
dA is an oriented area element on surface S.
This tells us that the electromotive force (emf) around a closed loop is equal to the negative time derivative of magnetic flux through the enclosed surface.
Applying Stokes' theorem, we convert the integral form into the differential form:
∇×E=−∂t∂B
This says that a non-conservative electric field is generated wherever the magnetic field changes in time. Unlike electrostatic fields, these electric fields form closed loops.
Faraday’s Law introduces several key physical concepts:
Induced electric fields: Unlike electrostatic fields, which begin and end on charges, the electric fields generated by changing Bform closed loops and are not associated with charges directly.
Non-conservative behavior: The work done around a closed path by the induced electric field is non-zero — this violates the usual assumption that ∮E⋅dℓ=0 in electrostatics.
Causal mechanism: Changing magnetic fields cause electric fields. This is a cornerstone of classical field theory and underlies electromagnetic wave propagation.
In vacuum, these effects persist — no material medium is required for changing B to induce E.
⚡ Example: Induced Electric Field from a Time-Varying Toroidal Magnetic Field Around an Accretion Disk
Problem:
Consider a simplified model of an accretion disk with a toroidal magnetic fieldBϕ(t) threading a circular loop of radius R. The toroidal magnetic field varies with time as:
Bϕ(t)=B0cos(ωt)
Assume the loop lies in the poloidal plane, and we want to find the induced electric field E(t) around the loop due to the changing magnetic field.
The time-varying toroidal magnetic field induces an azimuthal electric field around the loop.
The induced E can accelerate charged particles in the disk corona or jet base.
This mechanism contributes to energy extraction and jet launching in accretion systems around black holes.
The magnitude scales linearly with loop radius R and oscillation frequency ω.
Faraday’s Law bridges magnetism and electricity by showing how time-varying magnetic fields induce electric fields. It explains the origin of electromagnetic waves (when combined with the Maxwell–Ampère Law), underlies all electrical machinery, and appears in diverse astrophysical processes.
Its integral form emphasizes the global relationship between magnetic flux and emf, while the differential form expresses a local, field-theoretic version — electric field curls where magnetic fields vary in time.
Understanding Faraday’s Law is key to interpreting field dynamics in both vacuum and plasma environments.