# EM Wave: Phase of the electric and magnetic waves?

PeterPeter
In a vacuum, the plane wave solutions to Maxwell's Equations are...
E=E0*cos(wt-kr)
B=B0*cos(wt-kr)
ie they are in phase. (See for example
Have you tried substituting them into Maxwell's equations to verify that they are indeed a solution? Make sure to use a suitable vector form for the solution. For a plane wave propagating in the z-direction, one such form is $$\vec E = \hat x E_0 \cos (\omega t - kz) \\ \vec B = \hat y B_0 \cos (\omega t - kz)$$ More explicitly in terms of components: $$E_x = E_0 \cos (\omega t - kz) \\ E_y = 0 \\ E_z = 0 \\ B_x = 0 \\ B_y = B_0 \cos (\omega t - kz) \\ B_z = 0$$ Consider for example the equation $$\nabla \times \vec E = - \frac {\partial \vec B}{\partial t}$$ On the left side you have first derivatives with respect to x, y, z, of components of ##\vec E##, which give you (for my example) either zeroes or sines. On the right side you have the first derivatives with respect to t, of components of ##\vec B##, which again give you either zeroes or sines.