Continuing from where we left off:

Step 1: Apply Maxwell’s Equations to Derive the Helmholtz Equation

Maxwell’s equations for free space, where $\rho = 0$ and $\mathbf{J} = 0$, are:

$\nabla \cdot \mathbf{E} = 0, \quad \nabla \cdot \mathbf{B} = 0$ $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, \quad \nabla \times \mathbf{B} = \mu \epsilon \frac{\partial \mathbf{E}}{\partial t}$

Now, we will take the curl of both sides of the equation for $\mathbf{E}$ and apply the vector identity $\nabla \times (\nabla \times \mathbf{E}) = \nabla(\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E}$. Since $\nabla \cdot \mathbf{E} = 0$ (from Maxwell’s first equation), we get:

$\nabla \times (\nabla \times \mathbf{E}) = -\nabla^2 \mathbf{E}$

Using Maxwell’s equation $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$, we substitute this into the equation above:

$-\nabla^2 \mathbf{E} = -\mu \epsilon \frac{\partial^2 \mathbf{E}}{\partial t^2}$

Simplifying, we get the wave equation for $\mathbf{E}$:

$\nabla^2 \mathbf{E} = \mu \epsilon \frac{\partial^2 \mathbf{E}}{\partial t^2}$

Now, the Helmholtz equation for $\mathbf{E}$ in free space is:

$\nabla^2 \mathbf{E} + \beta^2 \mathbf{E} = 0$

where $\beta = \omega \sqrt{\mu \epsilon} = \frac{2\pi}{\lambda}$ is the phase constant.

Thus, the electric field satisfies the Helmholtz equation:

$\nabla^2 \mathbf{E} + \beta^2 \mathbf{E} = 0$

Step 2: Repeat the Process for the Magnetic Field

Similarly, we can derive the Helmholtz equation for the magnetic field. Starting with Maxwell’s equation $\nabla \times \mathbf{B} = \mu \epsilon \frac{\partial \mathbf{E}}{\partial t}$, we take the curl of both sides:

$\nabla \times (\nabla \times \mathbf{B}) = \nabla(\nabla \cdot \mathbf{B}) - \nabla^2 \mathbf{B}$

Since $\nabla \cdot \mathbf{B} = 0$ (from Maxwell’s second equation), we have:

$\nabla^2 \mathbf{B} = \mu \epsilon \frac{\partial^2 \mathbf{B}}{\partial t^2}$

Again, the wave equation for $\mathbf{B}$ follows:

$\nabla^2 \mathbf{B} = \mu \epsilon \frac{\partial^2 \mathbf{B}}{\partial t^2}$

Therefore, the Helmholtz equation for the magnetic field is:

$\nabla^2 \mathbf{B} + \beta^2 \mathbf{B} = 0$

Step 3: Conclusion

We have demonstrated that both the electric and magnetic fields for a linearly polarized plane wave propagating in the $z$-direction satisfy the Helmholtz equation. Specifically:

• The electric field $\mathbf{E}$ satisfies $\nabla^2 \mathbf{E} + \beta^2 \mathbf{E} = 0$,
• The magnetic field $\mathbf{B}$ satisfies $\nabla^2 \mathbf{B} + \beta^2 \mathbf{B} = 0$.

Thus, both fields satisfy the Helmholtz equation in free space, as required.

Would you like more details or clarifications?

Here are five related questions you could explore:

1. How does the Helmholtz equation relate to the wave equation in electromagnetism?
2. Can the Helmholtz equation be used in non-vacuum media? If so, how would it change?
3. How do the solutions of the Helmholtz equation for different wave polarizations differ?
4. How does the phase constant $\beta$ affect wave propagation in free space?
5. How would the Helmholtz equation change if there were sources of charges or currents?

Tip: When working with electromagnetic waves, always keep in mind the relationship between the electric and magnetic fields and their perpendicularity in space, as this is key to understanding their propagation.