Radiation Pressure derivation Radiation pressures mathematical expression according to Wikipedia is,
$\frac{1}{\mu_0 c}\vec{E} × \vec{B}$
"Radiation pressure is the mechanical pressure(force/area) exerted upon any surface due to the exchange of momentum between the object and the electromagnetic field."
Maxwells equations tell us that the force per unit volume is
$f= \nabla \cdot \sigma -\epsilon_{0}\frac{\partial}{\partial t}\vec{E}×\vec{B}$
Through electromagnetic theory I can find the force per unit volume anywhere in space. How can I use this to derive the above expression for force per unit area. And what does it mean to have force per unit area in 3 dimensions (I guess we have to assume the surface has a charge density per unit area )
 A: A good pedagogical approach is found in Rothman and Boughn, American Journal of Physics 77, 122 (2009). Scroll to the last section, where they introduce a model of a light wave reflecting at normal incidence from a nearly perfect conductor.
Basically, you integrate the force in the object in one direction to convert the force per unit volume ($F/V$) to force per unit area ($F/A$). Suppose your mirror normal is in the z direction. You could calculate $F/V$ everywhere, then integrate the force density along the z-direction to get $F/A$. This force per area still has three components of direction, in general, so don't let that confuse you.
(Regarding the force density, there are actually several options to choose from, and they are not equivalent. For example, there's $F=-(\nabla\cdot P)E+J\times B$ and also $F=(P\cdot\nabla)E+J\times B$. As long as you only care about the total force on an object, you could use either of these or any other that conserves momentum. Just be sure to integrate the force density over the entire object, including the surface forces, and they will agree.)
Another option is to integrate the Maxwell stress tensor over a surface surrounding the object. See, e.g. Griffiths, Introduction to Electrodynamics, 3rd ed., Section 8.2.2 (or any other intro E&M textbook) for a derivation of the Maxwell stress tensor and why integrating it over a surface gives a force. Crucially, the surface must be in vacuum or you'd run into issues related to the different forms of the force density because you'd need to choose a form of the stress tensor in matter.
