Emf induced by a changing electric field Is there a law/formula that relates the rate of change of electric flux with the induced electromotive force? i.e. similar to what Faradey's law of induction states for the magnetic field.
 A: Let $L$ be a closed area with boundary $\partial L$. If it helps, you can imagine $\partial L$ to be a loop of wire. Now suppose we send a time-varying electric field through $L$. We seek to relate this electric field $\textbf{E}$ traveling through $L$ with the (induced) magnetic field $\textbf{B}$ in $\partial L$.
To do this, we recall the following Maxwell equation:
\begin{equation}                                                                                                
\nabla \times \textbf{B} = \mu_0 \: \textbf{J} +  \mu_0 \epsilon_0 \: \partial_t \textbf{E} \; \; \; \textrm{(i)}
\end{equation}
Assuming that there is no current traveling through the loop, we have that $\textbf{J} = \textbf{0}$. With this substitution, equation (i) becomes
\begin{equation}                                                                                                
\nabla \times \textbf{B} = \mu_0 \epsilon_0 \: \partial_t \textbf{E} \; \; \; \textrm{(ii)}
\end{equation} 
Now we take the surface integral over $L$ of each side of equation (ii):
\begin{equation}                                                                                                
\iint_L (\nabla \times \textbf{B}) \cdot d\textbf{A} = \mu_0 \epsilon_0 \iint_L \: \partial_t \textbf{E} \cdot d\textbf{A} = \mu_0 \epsilon_0 \frac{d}{dt} \iint_L \:  \textbf{E} \cdot d\textbf{A}  \; \; \; \textrm{(iii)}
\end{equation} 
Applying the Stokes Theorem to the LHS of equation (iii) and recognizing the RHS of equation (iii) as the rate of change of electric flux $\phi$ yields a "Law of Induction" for magnetic fields:
\begin{equation}                                                                                                
\oint_{\partial L} \textbf{B} \cdot d\textbf{L} =  \mu_0 \epsilon_0 \frac{d\phi}{dt}  \; \; \; \textrm{(iv)}
\end{equation}
