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Formula Wave Equation for E-Field

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

It is a natural constant and has the value \( \varepsilon_0 = 8.854 \cdot 10^{-12} \, \frac{\text{As}}{\text{Vm}} \).

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