1. The document discusses Maxwell's first law and the Poisson equation, which relates charge distribution to electric potential. 2. It then focuses on the two-dimensional Laplace equation and provides an example of using boundary conditions to solve for the general solution of electric potential. 3. Finally, it briefly discusses the three-dimensional Laplace equation and provides the general solution for electric potential inside a microwave oven, with one face held at a constant potential.

1. Camilo Chaves
Electrical Engineer and Physicist
The E-Field inside a Microwave Oven
Laplace Equation Theory
1a Maxwell law in its differential form :
(1.1)
We hope that the Electric Force that acts upon a charge, generated by the other charges, is the
variation of the electric potential energy, cause by the movement of the charge q since the reference
until the position r, close to the other charges.
(1.2)
(1.3)
(1.4)
Hence, the electric field is the gradient of the potential!
Substituting (1.4) em (1.1)
(1.5)
This is the Poisson Equation, which establishes the relation between a certain charge distribution and
the electric potential generated by it. It might occur that in certain regions the charge density is null.
In this case, the contour regions of the problem are given by fixating certain potentials in the surface

2. of some conductors, such as the microwave oven.
Bi-dimensional Laplace
(1.1.1)
(1.1.2)
(1.1.3)
(1.1.4)
(1.1.5)
(1.1.6)
(1.1.7)
If K=0:
(1.1.8)
(1.1.9)
General solution:
(1.1.10)
Now, just create contour regions to solve the general solution!
Example

3. Suppose the following contour regions:
(1.1.1.1)
(1.1.1.2)
_C6 e _C2 must be 0.
(1.1.1.3)
(1.1.1.4)
(1.1.1.5)
To be true:
(1.1.1.6)
(1.1.1.7)
Because of condition , if , V(x,y) will diverge. So _C3 must be zero as well !
(1.1.1.8)
(1.1.1.9)
(1.1.1.10)
The general solution is the sum of all solutions o V(x,y) for every n

4. (1.1.1.11)
The last condition states that: If y=0, V(x,0)=
(1.1.1.12)
This is the expansion of the function
obtained by the equation:
in a Fourier series of sin. The coefficient can be
(1.1.1.13)
If , a constant
(1.1.1.14)
(1.1.1.15)
(1.1.1.16)
In some text books you can find 1.1.1.16 rewritten as
Tridimensional Laplace
(1.2.1)
We can solve the equation straight using PDSOLVE command from Maple.
general solution:

5. (1.2.2)
To solve for the general solution, let's suppose a box with sides , which delimits a closed
volume in a space free of charges. This is a tipical example of the microwave oven.
Substituting these initial conditions on the general solutions one can get the equation for the
potential below:

6. (1.2.3)
The equation above is the electric potential inside the box when the z face is subjected to the
constant potential . The electric field is the gradient of this equation.
To visualise the field we must enter some values for the potential over time and the size of the
box. The frequency of the potential is 2.34GHz and its maximum value is 10000 volts.