Simplification of Maxwell equations assuming that the induced fields are much weaker than the applied fields In cases where induced fields are much smaller than the applied fields, are there any terms in the Maxwell equations that can be neglected? I tried to do a simple scaling analysis to come up with a solution, but did not succeed. Can anyone help? 
Thanks!
 A: If I understand your question, the answer is really very simple. These are Maxwell's equations:
$∇^2E=k∂^2E/∂t^2$, 
$∇^2B=k∂^2B/∂t^2$,
$∇⋅E=4πρ$, and
$∇⋅B=0$.
If you set the right-hand sides of the top two equations to zero, the only solutions will have no time dependence and therefore no induced field.  You could make it a bit more complicated: start with the full set of equations, set $E = E_o(x) + w F(t)$, and find the limit as $w ->0$, but you would end up with essentially the same result.
A: The general representation of Maxwell's equations in vacuum is given by:
$$
\begin{align}
  \nabla \cdot \mathbf{E} & = \frac{\rho}{\varepsilon_{o}} \tag{0} \\
  \nabla \cdot \mathbf{B} & = 0 \tag{1} \\
  \nabla \times \mathbf{E} & = - \frac{\partial \mathbf{B}}{ \partial t } \tag{2} \\
  \nabla \times \mathbf{B} & = \mu_{o} \ \mathbf{j} + \frac{1}{c^{2}} \ \frac{\partial \mathbf{E}}{ \partial t } \tag{3}
\end{align}
$$
where $c^{-2} = \mu_{o} \ \varepsilon_{o}$ is the speed of light defined by the product of the permitivity and permeability of free space.  We can take the curl of Equations 2 and 3 to find:
$$
\begin{align}
  \nabla^{2} \mathbf{E} - \frac{1}{c^{2}} \ \frac{\partial^{2} \mathbf{E}}{ \partial t^{2} } & = \frac{1}{\varepsilon_{o}} \left( \nabla \rho + \frac{1}{c^{2}} \ \frac{\partial \mathbf{j}}{ \partial t } \right) \tag{4} \\
  \nabla^{2} \mathbf{B} - \frac{1}{c^{2}} \ \frac{\partial^{2} \mathbf{B}}{ \partial t^{2} } & = - \mu_{o} \ \nabla \times \mathbf{j} \tag{5}
\end{align}
$$
where these are referred to as the wave equations for $\mathbf{E}$ and $\mathbf{B}$.

In cases where induced fields are much smaller than the applied fields, are there any terms in the Maxwell equations that can be neglected?

By "induced fields" I am assuming you are referring to electromagnetic induction, correct?  If so, then you can take the terms $\frac{\partial}{ \partial t } \rightarrow 0$, which will reduce the problem to an electrostatic or magnetostatic one.
Note that you can still induce electrical charges under the electrostatic approximation, i.e., called electrostatic induction.  It's really just a redistribution of charges.  If you neglect this effect as well, then let $\rho \rightarrow 0$.
