# Formula: Wave Equation for E-Field

## Electric field

Unit
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

Unit
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

Unit
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

Unit
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: $\frac{\partial^2 E_x}{\partial x^2} + \frac{\partial^2 E_x}{\partial y^2} + \frac{\partial^2 E_x}{\partial z^2} ~=~ \mu_0 \, \varepsilon_0 \, \frac{\partial^2 E_x}{\partial t^2}$

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