[NOTES/EM-09003]-Understanding Electromagnetic Induction-I

  When a loop is placed in time varying magnetic field, an e.m.f. is induced in the loop. This phenomenon is requires a modification of the Maxwell's second equation as follows. \begin{equation} \nabla \times \vec{E}=0 \longrightarrow \nabla \times \vec{E} = -\pp[\vec{B}]{t} \end{equation}

1. Conductor time varying magnetic field 

Assume that a conducting loop lies in a region having time varying field $\vec{B}$  $$ \frac{\overrightarrow{dB}}{dt}~\ne~0 $$ In this case there is an induced e.m.f. in the loop and a current will flow. The mathematical formulation of Faraday's law of induction modifies the Maxwell equation to the time varying magnetic field as follows. \begin{equation} \nabla \times \vec{E}=0 \longrightarrow \nabla \times \vec{E} = -\pp[\vec{B}]{t} \end{equation}\hspace{5mm} This induced electric field curls around the magnetic field and explains the induced e.m.f.

\begin{align}\nonumber \mathcal E = & \oint_\gamma \vec{E}\cdot\overrightarrow{dl}\quad = \int(\bar{\nabla}\times\bar{E})\cdot\overrightarrow{dS}\\ =& -\frac{\partial}{\partial t}~ \int_S \vec{B}\cdot \overrightarrow{dS}\quad =-\frac{\partial}{\partial t}~\Phi_S \label{EQ02} \end{align} $\Phi_S=$ Flux of magnetic field through the circuit. The direction of the induced current is given by the red arrow in Fig23-10 In case of a planar loop $S$ is a area enclosed by $\gamma$ as boundary. For a general loop in three dimensions, any surface \(S\) with \(\gamma\) as boundary will do. This is because, as a consequence of \(\nabla\cdot\vec{B}=0\), the flux of $\vec{B}$ through two surfaces $S_1$ and $S_2$, which have the same boundary, are equal Details? We want to have an idea about solution for $\vec{E}$ (induced electric field) Note the similarity between the two equations \begin{equation} \bar{\nabla} \times\bar{E} = - \frac{\partial \vec{B}}{\partial t} \qquad\qquad \bar{\nabla}\times\bar{B} = \mu_0\bar{j} \end{equation} Thus the direction of induced electric field is given by rules similar to those for the magnetic field due to a steady current. \\  \noindent If the magnetic field is changing with time, the $\vec{E}$ lines of force form loops. When a loop of a conducting material is present, the induced electric field gives rise to an e.m.f. that makes the electrons inside the loop go round in the loop and a current in flows in the loop. \noindent In all cases, the induced current gives rise to heating of the conductor and heat produced per sec is $$ \frac{e^2}{R},\quad \text{ where }\quad e = -\frac{d}{dt}\Phi_B $$ Therefore, if the resistance $R$ is small, or the magnetic field is changing rapidly the heat produced is large. When a circuit is immersed in time varying magnetic field, or moves in magnetic field, no current will be produced if the circuit is open but e.m.f. will be present as given by the flux rule.