[NOTES/EM-09009]-Microscopic Description of Induction

1. Microscopic description of induction

We would like to understand the following questions. Since the flux rule is not a fundamental rule and cannot be applied in all possible situations, we would like to understand:
(a) What is the correct explanation?
(b) How to correctly find the induced e.m.f.? There are two `different' microscopic explanations to appearance of e.m.f. in the two different types of set ups.

1. When a conductor moves in magnetic field the free electrons, inside the body of the conductor, experience a Lorentz force and start moving. This is just the induced current.
2. Next consider the conductor does not move but is placed in a magnetic field that changes with time. The time varying magnetic field produces electric field $\vec{E}$ which makes a current flow or generate e.m.f. along a path. Thus we see that in two different setups, there are two different origins of electromotive force that pushes the electrons in a loop. If \(\vec{F}\) denotes the electric force responsible for induction, then the e.m.f. in a circuit is defined by the line integral around the circuit: \begin{equation} \mathcal E = \oint \vec{F}\cdot \overrightarrow{d\ell}. \end{equation} In order that e.m.f. be non zero, the work done by the force field in a loop must be non zero. Thus the force \(\vec{F}\) must be non conservative.

We expand the above remarks and take up examples of two setups as mentioned above. We will obtain expressions for the induced e.m.f. in these examples.

2. Conductor moving in a magnetic field

Charge carriers in a conductor are free to move inside a conductor. They experience a force $qvB$ upwards from $A$ to $B$ \begin{align*} \epsilon = & \int \vec{F}~\overrightarrow{dl} = \int_{AB} \vec{F}~\overrightarrow{dl}\\ = & vBl = \frac{d}{dt}(xBl)\\ = & \text{rate of change of flux} \end{align*}

3.Conductor at rest in time varying magnetic field

The e.m.f. induced can be understood in terms of Lorentz force acting on a moving conductor. In the case of a conductor placed in a time varying magnetic field, the appearance of induced electric field explained by means of Maxwell's equation \begin{equation} \bar{\nabla}\times\bar{E} = -\frac{\partial\vec{B}}{\partial t} \end{equation} This equation tells us that an induced electric field, $\vec{E}$, is produced by time varying magnetic field and the electric field in turn is responsible for generation of e.m.f. in a closed path. Now consider, for example, a conducting planar loop placed in a time varying magnetic field. The induced electric field produces and e.m.f. in the loop given by the line integral, \begin{equation} \mathcal E = \oint_\Gamma \vec{E}\cdot \hat{n} d\ell = - \frac{d}{dt}\iint_S \vec{B}\cdot\hat{n} dS. \end{equation} where \(\hat{n}\) is normal to the loop and \(S\) is a surface area enclosed by the loop.The red arrow in Fig23-10 shows the direction of induced current.