Why Do Electromagnetic Waves Propagate Independently of Their Source?

[BobaJ]

Homework Statement

Use Maxwell's equations to elaborate an coherent explication of why electromagnetic waves propagate independently of the source that produces them.

Homework Equations

Maxwell's equations in vacuum:

• $\nabla * E=0$
• $\nabla * B=0$
• $\nabla \times E = -\frac{\partial B}{\partial t}$
• $\nabla \times B=\mu _0 \epsilon _0 \frac{\partial E}{\partial B}$

The Attempt at a Solution

Honestly I don't know how to begin or proof it.

 

[nrqed]

Homework Statement

Use Maxwell's equations to elaborate an coherent explication of why electromagnetic waves propagate independently of the source that produces them.

Homework Equations

Maxwell's equations in vacuum:

• $\nabla * E=0$
• $\nabla * B=0$
• $\nabla \times E = -\frac{\partial B}{\partial t}$
• $\nabla \times B=\mu _0 \epsilon _0 \frac{\partial E}{\partial B}$

The Attempt at a Solution

Honestly I don't know how to begin or proof it.

This is actually done in most textbooks. Hint: have you seen the wave equation? Do you know any identities containing curl and divergence of a vector field?

 

[BobaJ]

This is actually done in most textbooks. Hint: have you seen the wave equation? Do you know any identities containing curl and divergence of a vector field?

Maybe you can recommend me a textbook where it is done?
Yes, I have seen the wave equation (and how to derive it from Maxwell´s equations).

 

[nrqed]

Maybe you can recommend me a textbook where it is done?
Yes, I have seen the wave equation (and how to derive it from Maxwell´s equations).

AH ok. Then this is what shows that the waves propagate independently of the sources. When an em wave is produced, it propagates because the fields obey the wave equation.

 

[BobaJ]

Ahhh, so when they refer to different sources, they are just referring to either electric or magnetic ones?

So just by getting, the equations $\nabla^2 E=\mu_0 \epsilon_0 \frac{\partial^2 E}{\partial t^2}$ and $\nabla^2 B=\mu_0 \epsilon_0 \frac{\partial^2 B}{\partial t^2}$ I'm showing that the EM waves are independent of the source that produces them.

And they are propagating because they obey the wave equation.

So, when they are asking me to elaborate an coherent explanation to proof it, they are in reality asking me to derive and demonstrate the two equations previously mentioned? Is that right?

 

[nrqed]

Ahhh, so when they refer to different sources, they are just referring to either electric or magnetic ones?

So just by getting, the equations $\nabla^2 E=\mu_0 \epsilon_0 \frac{\partial^2 E}{\partial t^2}$ and $\nabla^2 B=\mu_0 \epsilon_0 \frac{\partial^2 B}{\partial t^2}$ I'm showing that the EM waves are independent of the source that produces them.

And they are propagating because they obey the wave equation.

So, when they are asking me to elaborate an coherent explanation to proof it, they are in reality asking me to derive and demonstrate the two equations previously mentioned? Is that right?

Well, you are showing that the propagation of the em waves is independent of the sources. As for what they want as a "coherent explanation", I cannot say. It depends on what your instructor has in mind so I would not dare try to guess in case I would induce you in error.