Alexander Fufaev

Formula: Wave Equation for E-Field

Electromagnetic wave (EM wave)

Electric field

Electric field specifies the force that would act on an electric charge if it were placed in a location where the electric field exists.

Solving the vectorial wave equation with the respective boundary conditions yields the electric field. For example, a simple solution of the wave equation yields the E-field in the form of plane waves.

Vacuum Permittivity

The vacuum permittivity is a physical constant that appears in equations involving electromagnetic fields. It has the following experimentally determined value: $$ \varepsilon_0 ~\approx~ 8.854 \, 187 \, 8128 ~\cdot~ 10^{-12} \, \frac{\mathrm{As}}{\mathrm{Vm}} $$

Magnetic field constant

It is a natural constant and occurs whenever electromagnetic fields are involved. It has the value \( \mu_0 = 4\pi \cdot 10^{-7} \, \frac{ \text{N} }{ \text{A}^2 } \).

Nabla operator

The operator \(\nabla^2\) is applied to the electric field to differentiate the components of the E-field with respect to the spatial coordinates.

Applying \(\nabla^2\) to the E-field yields a vector quantity. The first component of this vector quantity is: \[ \frac{\partial^2 E_x}{\partial x^2} + \frac{\partial^2 E_x}{\partial y^2} + \frac{\partial^2 E_x}{\partial z^2} ~=~ \mu_0 \, \varepsilon_0 \, \frac{\partial^2 E_x}{\partial t^2} \]