Answer to Question #136553 in Mechanics | Relativity for Bawe Bilanyuy

Question

he intensity, I, of an electromagnetic wave is related to the electric field strength E and magnetic induction B of the fields producing it by the equation I = E.B / 20 where 0 is the permeability of free space. Show that this equation is homogeneous.

Answer 1

An equation is called homogeneous, if it satisfies the following property:

$$f(\alpha x,\beta y) = \alpha \beta f(x,y)$$

where $\alpha$ and $\beta$ are some constants. In our case we have the function:

$$f(x,y) = f(\mathbf{E},\mathbf{B}) = \dfrac{\mathbf{E}\cdot \mathbf{B}}{\varepsilon_0}$$

where $\cdot$ denotes a dot product. As far, as the dot product is linear, we can write:

$$f(\alpha \mathbf{E},\beta\mathbf{B}) = \dfrac{\alpha\mathbf{E}\cdot \beta\mathbf{B}}{\varepsilon_0} = \alpha\beta\dfrac{\mathbf{E}\cdot \mathbf{B}}{\varepsilon_0} = \alpha\beta f(\mathbf{E},\mathbf{B})$$

thus, we have proved that equation is homogeneous.

Answer. Q.E.D.