# Does Maxwell Tensor include Lorentz Forces ?

## Electromagnetic problem

An analytical case is presented in order to show that the airgap Maxwell Tensor method is taking into account the Lorentz forces (sometimes called Laplace forces).Let us have a straight electrical wire surrounded by the void and traversed by a current. The wire is oriented toward z-axis, as illustrated in the previous figure.

The magnetic flux generated by the wire can be expressed in the polar referential as:

This wire is immersed in an external homogeneous electromagnetic field

such that the total electromagnetic field is

which can be written as:

## Lorentz force

According to the Lorentz force acting on a section L of the wire is given by:

## Maxwell Tensor

According to the Maxwell Tensor method, the total magnetic force can be computed by integrating the Maxwell stress Tensor on a closed cylinder of height L and radius R:

such that the total force in the x-direction is:

Following the same method in the y-direction leads to:

## Conclusion

Then the Lorentz’s forces and the Maxwell forces are equal in this case:

It confirms that the integration of Maxwell stress tensor on a closed surface is more general than the Lorentz force method.