Proving EM Waves Equations: E = Emsin(kx-ωt) and B = Bmsin(kx-ωt)

[noppawit]

Show that the electric field: E = E_msin(kx-ωt) and magnetic field: B=B_msin(kx-ωt) satisfy the following equations:

$$-\frac{\partial B}{\partial x} = \mu_{0}\epsilon_{0}\frac{\partial E}{\partial t}$$

and

$$\frac{\partial E}{\partial x} = -\frac{\partial B}{\partial t}$$

I have no idea about this. Would you please guide me for solving this.

Thanks.

 

[AEM]

Your equations didn't come out properly so we can't help you.

 

[noppawit]

Sorry for my bad Latex typing.

I've edited.

 

[AEM]

Show that the electric field: E = E_msin(kx-ωt) and magnetic field: B=B_msin(kx-ωt) satisfy the following equations:

$$-\frac{\partial B}{\partial x} = \mu_{0}\epsilon_{0}\frac{\partial E}{\partial t}$$

and

$$\frac{\partial E}{\partial x} = -\frac{\partial B}{\partial t}$$

I have no idea about this. Would you please guide me for solving this.

Thanks.

In a situation like this, the usual procedure is to substitute the two given expressions into the differential equation, take the derivatives and see what happens. As I look at what you've been given (without working it out myself) I think you may have to figure out how to relate $$\mu_o \epsilon_o$$ to k and $$\omega$$.
That should get you started. Let us know if you run into trouble.

 

[AEM]

In a situation like this, the usual procedure is to substitute the two given expressions into the differential equation, take the derivatives and see what happens. As I look at what you've been given (without working it out myself) I think you may have to figure out how to relate $$\mu_o \epsilon_o$$ to k and $$\omega$$.
That should get you started. Let us know if you run into trouble.

Upon reading my previous post, I see a small problem with what I wrote. When you substitute the expressions for E an B into each equation, you'll end up with a $$B_m$$ on one side and a $$E_m$$ on the other. That's a nuisance. However, I'll bet you can figure out a way to combine those two equations into one equation and then do the substitution.

 

[berkeman]

Show that the electric field: E = E_msin(kx-ωt) and magnetic field: B=B_msin(kx-ωt) satisfy the following equations:

$$-\frac{\partial B}{\partial x} = \mu_{0}\epsilon_{0}\frac{\partial E}{\partial t}$$

and

$$\frac{\partial E}{\partial x} = -\frac{\partial B}{\partial t}$$

I have no idea about this. Would you please guide me for solving this.

Thanks.

IIRC, you would start with Maxwell's equations...