Time Varying Magnetic Field :

Consider a conducting loop of area ‘A’ in a uniform but time varying magnetic field. Rate of change of magnitude of magnetic field = dB/dt

For the loop , Flux linked with it = BA = φ, (say)

(taking area vector directed along B^{ →} )

Hence , Rate of change of flux φ = AdB/dt

Hence induced emf = − AdB/dt

For non-zero values of dB/dt , there could be a definite current in the loop, whose direction can be obtained using Lenz’s rule.

For example , if dB/dt > 0 i.e. B is increasing with time, magnetic field produced by induced current would oppose the existing magnetic field. Hence the induced current would be anticlockwise.

The current in the loop can be easily known if the resistance of the loop is known as I = ε/R.

In the case of motional emf, you learnt that the electric field caused due to drifting of electrons is responsible for the induced emf. Do we also have an electric field in present case also linked with the induced emf ? The answer is partly ‘yes’ and partly ‘No’.

Yes, as there is a definite field and the electric field that you know. The electric field that you learnt in electrostatics is conservative and the associated lines of force never form closed loops.

On the other hand the field associated with the induced emf in case of time-varying magnetic field is non-conservative then only we would have non-zero value for $\displaystyle \oint \vec{E_n}.\vec{dl} $

Here E_{n} denotes the induced field caused by time varying magnetic field.

For the path described by the loop,

$ \displaystyle \xi = -\frac{d\phi}{dt} = \oint \vec{E_n}.\vec{dl} $

Consider a magnetic field where B (magnitude of magnetic Field ) is a function of ‘r’.

For the circular path shown in the figure.

$ \displaystyle |\oint \vec{E_n}.\vec{dl}| = E_n(2 \pi r) $

$ \displaystyle E_n(2 \pi r) = \int_{0}^{r} \frac{dB}{dt}(2\pi r dr) $

Direction of can be easily obtained as it would be responsible for the induced current when a conducting loop is placed on the given path.

For example in the present case, for dB/dt > 0, path in anticlockwise sense.

Exercise: Find the magnitude of induced field ” $\vec{E}$ ” at a point r (>R) where a uniform but time varying magnetic field $ (\frac{dB}{dt} = b) $ exists in a region of radius R.