Faraday's Law of Induction
Faraday's law of induction states that a changing magnetic field will induce an electric current in a conductor. This principle is used in the operation of devices such as transformers, inductors, and electric motors.
The Maxwell-Faraday equation, one of Maxwell's equations, states that a changing magnetic field creates an electric field, and Faraday's law states that a changing magnetic field induces an electromotive force (EMF) in a conductive loop. Faraday's law was discovered first, and later the aspect of it known as transformer EMF was formulated as the Maxwell-Faraday equation. The Maxwell-Faraday equation can be used to derive Faraday's law, which includes both transformer EMF and motional EMF. The integral form of the Maxwell-Faraday equation only describes transformer EMF.
Faraday's Law
Faraday's law States that:
The electromotive force around a closed path is equal to the negative of the time rate of change of the magnetic flux enclosed by the path.
Mathematically:
The magnetic flux, \(\Phi_B\), through a wire loop in a magnetic field is determined by the surface integral over any surface, \(\Sigma\) , whose boundary is the wire loop. The surface, \(\Sigma\), is time dependent since the loop may be moving, so it is represented as \(\Sigma\)(t). \[{\displaystyle \Phi _{B}=\iint _{\Sigma (t)}\mathbf {B} (t)\cdot \mathrm {d} \mathbf {A} \,,}\]
Where:
- \(\Phi_B\) is the magnetic flux
- \(\Sigma\) is the moving surface
- \(\mathbf{B}\) is the magnetic field
- \(d\mathbf{A}\) is an element of surface area
- \(\mathbf{B} \cdot d\mathbf{A}\) is a vector dot product representing the element of flux through \(d\mathbf{A}\)
Faraday's law of induction states that when the flux changes (because the magnetic field changes, the wire loop is moved or deformed, or both), an emf is generated in the wire loop. This emf is defined as the energy available from a unit charge that has traveled once around the wire loop and can also be thought of as the voltage that would be measured by cutting the wire to create an open circuit, and attaching a voltmeter to the leads.
Faraday's law states that the electromotive force (emf) is given by the rate of change of the magnetic flux: $$\mathcal{E} = -\frac{d\Phi_{B}}{dt}$$ where \(\mathcal{E}\) is the electromotive force and \(\Phi_{B}\) is the magnetic flux. The direction of the electromotive force is given by Lenz's law. It's worth noting that the laws of induction of electric currents in mathematical form were established by Franz Ernst Neumann in 1845. Additionally, Faraday's law contains information about the relationships between both the magnitudes and the directions of its variables, however, the relationships between the directions are not explicit and are hidden in the mathematical formula.