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Deriving the Wave Equation from Maxwell’s Equations


 

Derivation of the Wave Equation from Maxwell’s Equations

 



 

Our aim is to start from the Maxwell’s Equations in order for us to obtain the Wave Equation for the field vectors \vec{E} and \vec{H}. Hence, we recall that the Maxwell’s Equations in free space is given by

(i) \vec{\nabla}\cdot\vec{E}=0     (ii) \vec{\nabla}\cdot\vec{B}=0      (iii) \vec{\nabla}\times\vec{E}=-\frac{\partial \vec{B}}{\partial t}      (iv) \vec{\nabla}\times\vec{H}=\frac{\partial \vec{D}}{\partial t}

We start from equation (ii) \vec{\nabla}\times\vec{E}=-\frac{\partial \vec{B}}{\partial t}

If we take the curl on both sides of the equation, we get

\vec{\nabla}\times ( \vec{\nabla}\times\vec{E})=-\vec{\nabla}\times\frac{\partial \vec{B}}{\partial t}                               [1]

But we recall the vector identity: \vec{\nabla} \times ( \vec{\nabla}\times\vec{A}) = \vec{\nabla} (\vec{\nabla} \cdot \vec{A}) - \vec{\nabla}^2 \vec{A}

Applying this identity,

\vec{\nabla}\times ( \vec{\nabla}\times\vec{E})=\vec{\nabla} (\vec{\nabla} \cdot \vec{E}) - \vec{\nabla}^2 \vec{E}                   [2]

\vec{\nabla} (\vec{\nabla} \cdot \vec{E}) - \vec{\nabla}^2 \vec{E} =-\vec{\nabla}\times\frac{\partial \vec{B}}{\partial t}                          [3]

                                  = - \frac{\partial}{\partial t} (\vec{\nabla} \times \vec{B})                  [4]

We remember also that equation (iii) can also be written as

\vec{\nabla}\times\vec{H}=\vec{\nabla}\times(\frac{1}{\mu_0} \vec{B})=\frac{\partial\vec{D}}{\partial t}                              [5]

\vec{\nabla} \times \vec{B} = \mu_0 \frac{\vec{\partial D}}{\partial t}                                                    [6]

And using expression [6] to substitute in [4], we have

\vec{\nabla} (\vec{\nabla} \cdot \vec{E}) - \vec{\nabla}^2 \vec{E} = - \frac{\partial}{\partial t} (\mu_0 \frac{\vec{\partial D}}{\partial t})                       [7]

Note that in free space, there is no charge density. As a result, we have equation (i) \vec{\nabla}\cdot\vec{E}=0.

Using this, expression [7] reduces to

- \vec{\nabla}^2 \vec{E} = - \mu_0\frac{\vec{\partial^2 D}}{\partial t^2}                                               [8]

Since we want the wave equation for \vec{E}, we  use that

\vec{D}=\varepsilon_0\vec{E} \rightarrow \vec{\nabla}^2 \vec{E} = \mu_0\frac{\partial^2}{\partial t^2}(\varepsilon_0\vec{E})

                 \vec{\nabla}^2 \vec{E} = \mu_0 \varepsilon_0\frac{\partial^2\vec{ E}}{\partial t^2}

Let \frac{1}{c^2}=\mu_0 \varepsilon_0, and finally we obtained the Wave Equation for the field vector \vec{E}.

\mathbf{\vec{\nabla}^2 \vec{E} = \frac{1}{c^2}\frac{\partial^2\vec{ E}}{\partial t^2}}

In a similar manner, we can obtain an expression for \vec{H} using (iii).

Following the same process, we will obtain the Wave Equation for the field vector \vec{H}.

\mathbf{\vec{\nabla}^2 \vec{H} = \frac{1}{c^2}\frac{\partial^2\vec{ H}}{\partial t^2}}

 


By: Joshua J. Ordeniza, MS Physics Student, MSU-IIT

 

 

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