JEE Main & Advanced Physics Electro Magnetic Induction Induced Electric Field

It is non-conservative and non-electrostatic in nature. Its field lines are concentric circular closed curves.

A time varying magnetic field \[\frac{dB}{dt}\] always produced induced electric field in all space surrounding it.

Induced electric field \[({{E}_{in}})\] is directly proportional to induced emf so \[e=\oint{{{{\vec{E}}}_{in}}\cdot d\vec{l}}\]  ...(i)

From Faraday's second laws  \[e=-\frac{d\varphi }{dt}\]          ...(ii)

From (i) and (ii) \[e=\oint{{{{\vec{E}}}_{in}}.d\vec{l}}=-\frac{d\varphi }{dt}\]  This is known as integral form of Faraday's laws of EMI.

A uniform but time varying magnetic field B(t) exists in a circular region of radius 'a' and is directed into the plane of the paper as shown, the magnitude of the induced electric field \[({{E}_{in}})\] at point P lies at a distance r from the centre of the circular region is calculated as follows.

So \[\oint{{{{\vec{E}}}_{in}}d\vec{l}}=e=\frac{d\varphi }{dt}=A\frac{dB}{dt}\]  i.e.  \[E(2\pi r)=\pi {{a}^{2}}\frac{dB}{dt}\]   

where  \[r\ge a\] or \[E=\frac{{{a}^{2}}}{2r}\frac{dB}{dt}\];  \[{{E}_{\mathbf{in}}}\propto \frac{1}{r}\]