Time-Varying Magnetic Field

Time – Varying magnetic field is one in which the magnetic induction B varies with time. Example: An alternating current sent through the winding of an electromagnet produced time-varying magnetic field between the poles.
To show that the electric field produced by a time-varying magnetic field is proportional to the rate of charge of magnetic induction.

To show that the electric field produced by a time-varying magnetic field is proportional to the rate of charge of magnetic induction.
Consider a circular loop of radius r placed in a time varying magnetic field B.  The field is changing at the rate dB/dt. The magnetic field is at right angles to the plane of the paper. Let B be the magnetic induction at any instant t.

The instantaneous magnetic flux enclosed by the loop is
        Φ = B.A = BA = πr²B
Where A (=πr2) is the area of the loop. According to Faraday’s Law,
Induced e.m.f., e = - dΦ/dt = - d/dt(πr²B) = - πr² dB/dt                            … (1)

Since the loop is stationary, the free electrons within it do not experience any magnetic force. Hence the induced e.m.f. cannot be explained using ‘Motional e.m.f.’ principle. But there is induced e.m.f. in the loop even when the magnetic force on electrons is zero.

This suggests that the changing magnetic flux sets up an induced electric field which exerts a force on the free electrons in the loop. This (electric) force causes the electrons to flow in the loop and is thus responsible for the induced current. We thus conclude that a time-varying magnetic field produces electric fields.

The electric field at every point on the loop is along the tangent to the loop.
Now, induced emf,             e = Φ E. dL                        … (2)
Where dI is a vector element of length of the loop.
From Eqs. (1) and (2), φ E. dl = -πr² dB/dt
Or            E (2πr) = - πr² dB/dt
    E = -1/2 r dB/dt
Thus the induced electric field depends on dB/dt.
The electric field produced by a time-varying magnetic field differs from that produced by static charges in one respect. Electric fields associated with stationary charges are conservative, i.e., for them φ E . dl = 0.

But, electric fields associated with changing magnetic fields are non-conservative because for them φ E. dl ≠ 0. We can cannot associate a scalar electric potential with this field. But still F = qE holds goods.