# Boundary condition for Maxwell equations

It's obvious that if Maxwell equations are fulfilled by some $$E(x,y,z,t)$$
and $$B(x,y,z,t)$$, they are also fullfiled by $$E(x,y,z,t)+ E_0$$
and $$B(x,y,z,t)+B_0$$, where $$E_0$$ and $$B_0$$
are constants. This freedom has physical significance as it changes the Lorentz force
which act on a charge. It implies that together with Maxwell equation we should
give some boundary condition. But unfortunately I can't find any book where they are
explicitly given (Dirichlet and Neumann boundary condition are most often introduced for
equation for potential not field).

## Answers and Replies

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ZapperZ
Staff Emeritus
Science Advisor
Education Advisor
But the E and B fields CAN be transformed into their respective potential fields. That's why we only use potential and the Dirichlet/Neumann boundary conditions. In fact, in many instances, one applies the Green's function with the boundary conditions. That, in principle, should completely define the field problem.

Edit: er.. never mind. I just realized that you have a time-dependence field.

Zz.

Born2bwire
Science Advisor
Gold Member
It depends on the problem. If it is free-space, then there is an implicit requirement that the fields must go to zero at infinity for uniqueness to be guaranteed (Sommerfeld Radiation Condition). If you have a discontinuous permeability and/or permittivity then you can deduce the boundary conditions from the partial differential equations themselves.

Weng Cho Chew's "Fields and Waves in Inhomogeneous Media" discusses both.