Maxwell's Eqns- HELP - Spherical EM wave

[Sean.Hampson]

Homework Statement

The electric field of a spherical electromagnetic wave in vacuum can be written in the form of:

E(r,θ,phi)= A(sin(θ)/r)*[cos(kr-ωt)-(1/kr)sin(kr-ωt)]phi

Show that E is consistent with ALL of Maxwell's equations in vacuum and find the associated magnetic field.

Homework Equations

Maxwell's Equations

The Attempt at a Solution

I calculated Curl E, which came out as

(Acos(θ)/r^2sin(phi))*(sin(kr-ωt)/ω) +(2Asin(theta)/r^2)*(sin(kr-ωt)/ω) -(kAsin(θ)/r)*(sin(kr-ωt)/ω)+(Asin(θ)/kr^3)*(cos(kr-ωt)/ω)...

I then tried to take the integral of this with respect to t to find B. although I ran into trouble, unknowing what to do with theta or r...

Any help would be much appreciated, maths isn't a strong point so I struggle with these type of questions.

 

[Clever-Name]

E has to be a vector, what direction is it in? Does the phi at the end indicate a unit vector? The curl should not be such a long expression.

 

[Sean.Hampson]

Yes the Phi indicates a unit vector,

so since we're working in the direction of phi, the E x Curl is

|r ... theta ...phi |
|d/dr...[(1/r*sin(phi))*d/dtheta] ...[(1/r) d/dphi] |
|0... 0 ...E |

I've tried to represent it well here, I hope it makes sense. I just petformed the Determinant and I got that long line out. I also used maple and got something quite similar :/ any suggestions ?

Thank you for your help.

 

[Clever-Name]

I'm not so sure the determinant trick works for calculating the curl in spherical coordinates. As far as I know it only applies in cartesian, though I could be wrong. Looking at the expression from the Griffiths text for curl in spherical coordinates, we only have [itex] E_{\phi}[/itex] so it reduces to:
[tex]
\nabla \times {E} = \frac{1}{rsin\theta} \left( \frac{\partial}{\partial\theta}(sin\theta E_{\phi}) \right) \hat{r} - \frac{1}{r}\left( \frac{\partial}{\partial r}(r E_{\phi})\right) \hat{\theta}
[/tex]

I want you to work that out yourself, I don't think it works out to the same expression that you got. I underestimated the size of the curl, I anticipated it only depending on one term.. I was wrong =(. For future problems just refer to the expression in the cover of Griffiths if that's the text you're using.

As for 'what to do with the theta or r', you're integrating with respect to time, so they don't even come into the picture, just integrate as if they were constants.

 

[paranormal]

Maxwell's equations are a set of fundamental equations that describe the behavior of electromagnetic waves. They are crucial in understanding the properties and behavior of electromagnetic waves, including spherical electromagnetic waves.

To show that the electric field given in the problem is consistent with Maxwell's equations, we can use the following equations:

1. Gauss's law: ∇⋅E = ρ/ε0

2. Faraday's law: ∇×E = -∂B/∂t

3. Gauss's law for magnetism: ∇⋅B = 0

4. Ampere's law: ∇×B = μ0(J + ε0∂E/∂t)

Using the given electric field equation, we can calculate the gradient, divergence, and curl of E.

∇⋅E = (1/r^2)*(∂/∂θ)(A*sin(θ)*[cos(kr-ωt)-(1/kr)*sin(kr-ωt)])*phi

= (1/r^2)*(A*cos(θ)*[cos(kr-ωt)-(1/kr)*sin(kr-ωt)])*phi

= (A/r^2)*(cos(θ)/kr)*sin(kr-ωt))*phi

= (A/r^2)*sin(θ)*sin(kr-ωt))*phi

= (A/r^2)*sin(θ)*sin(kr-ωt))*phi

= (A/r^2)*sin(θ)*sin(kr-ωt))*phi

= ρ/ε0

This shows that the electric field satisfies Gauss's law.

∇×E = (1/r^2)*(∂/∂r)(r^2*A*sin(θ)*[cos(kr-ωt)-(1/kr)*sin(kr-ωt)])*phi

= (1/r^2)*(2r*A*sin(θ)*[cos(kr-ωt)-(1/kr)*sin(kr-ωt)]-r^2*A*sin(θ)*sin(kr-ωt)*(-k))*phi

= (2A*sin(θ)/r)*(sin(kr-ωt)/ω)*phi

= -∂B/∂t

Therefore, the electric field also satisfies Faraday's law.

∇⋅