My question is: in electromagnetic waves, if we consider the electric field as a sine function, the magnetic field will be also a sine function, but I am confused why that is this way.
If I look at Maxwell's equation, the changing magnetic field generates the electric field and the changing electric field generates the magnetic field, so according to my opinion if the accelerating electron generates a sine electric field change, then its magnetic field should be a cosine function because $\frac{d(\sin x)}{dx}=\cos x$.
Best Answer
The Maxwell equations that relate electric and magnetic fields to each other read (in vacuum, in SI units) as \begin{align} \nabla \times \mathbf E & = -\frac{\partial\mathbf B}{\partial t} \\ \nabla \times \mathbf B & = \frac{1}{c^2} \frac{\partial\mathbf E}{\partial t}, \end{align} where the notation $\nabla \times{\cdot}$ is a spatial derivative (the curl). This means that both sides have derivatives, and if you're applying them to a function like $\cos(kx-\omega t)$, then they will both change the cosine into a sine. This is what locks the phase of both waves to equal values.