In electromagnetic induction, line integral of induced field around a close path is __________, induced electric field is ___________.
Step1: Faraday's Law of Electromagnetic Induction
Step2: Induced EMF
If be the magnetic flux linked with a circuit at any time then the laws of electromagnetic induction can be expressed mathematically as
,
Where is the induced emf. Here it is important to point out that does not depend on how the flux is changed.
If the electric field in space is denoted by then, by definition, the emf around a closed path is
If is an open surface bounded by the curve placed in a magnetic field then the magnetic flux through the surface is
So equation can be written as
This is the integral form of Faraday's law.
If the circuit is rigid and stationary in our reference system then neither nor depend on and as a result the time derivative can be taken inside the integral, i.e.,
where partial derivative has been used beacuse is a function of position and time.
Using Stokes' theorem we can write,
Since this must be true for any arbitrary fixed surface , it follows that
This is the differential form of Faraday's law.
It relates the space derivatives of at a particular point to the time rate of change of at the same point.
Step3: Vector and Scalar Potential
For static electric field and we can introduce scalar potential by the relation .
But for a time-varying field and we cannot do so. The magnetic vector potential was defined by the fact that , which still holds for the time-varying fields. Hence the relation holds for time-varying situations.
Using it in Faraday's law in equation we get
or
Since the curl of the gradient of any scalar is always zero, we can write
or
From equation and we can find and provided that the potentials and are known. Equation suggests that for varying fields can be considered as the sum of the two terms and where
and .
is the field generated by charges; it is conservative and nonsolenoidal.
is the field generated by a varying magnetic field. It is non-conservative and solenoidal.
Hence, in electromagnetic induction, the line integral of the induced field around a close path is non-zero, the induced electric field is non-conservative.