Simple Derivation Maxwell Equations

[kiwi101]

Homework Statement

Derive the 2 divergence equations from the 2 curl equations and the equation of continuity.

Homework Equations

∇°D=ρ
∇°B = 0
∇xE = -∂B/∂t
∇xH = J + ∂D/∂t
∇°J = -∂ρ/∂t (equation of continuity)

The Attempt at a Solution

1)∇xE = -∂B/∂t
∇°(∇xE) = ∇°(-∂B/∂t)
0 =∇°(-∂B/∂t) (divergence of curl of vector field is 0)
I'm stuck now I don't know what to do


2)∇xH = J + ∂D/∂t
∇°(∇xH) = ∇°(J + ∂D/∂t)
0 =-∂ρ/∂t + ∇°∂D/∂t
Once again I am stuck here too


Please guide me guys

 

[TSny]

The Attempt at a Solution

1)∇xE = -∂B/∂t
∇°(∇xE) = ∇°(-∂B/∂t)
0 =∇°(-∂B/∂t) (divergence of curl of vector field is 0)
I'm stuck now I don't know what to do

See this and apply it to your situation where one of the variables is time.

 

[smithhs]

You're on the right track, just remember that spatial derivitives (like the divergence) commute with the time derivitives.

 

[kiwi101]

So
0 = -∂/∂t(∇°B)
Does the -∂/∂t and the divergence cancel? Or do they make a second order derivative?

 

[TSny]

If the partial derivative with respect to time of a function of space and time is zero at all points of space, then what can you conclude about the function?

Think of $\small \nabla \cdot \bf{B}$ as some function of space and time.

 

[kiwi101]

That means that the function is constant regardless of time at all points of space

 

[TSny]

That means that the function is constant regardless of time at all points of space

Right, so $\small \nabla \cdot \bf{B}$ does not depend on time. It can be a function of space only.

Strictly speaking, I think this is as far as you can go mathematically. But can you add a reasonable physical argument that will allow you to go further?

 

[kiwi101]

oh okay
thank you so much guys!