# Simple Derivation Maxwell Equations

#### kiwi101

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

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

3. 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

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#### TSny

Homework Helper
Gold Member
3. 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 spacial 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?

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#### TSny

Homework Helper
Gold Member
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

Homework Helper
Gold Member
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?

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#### kiwi101

oh okay
thank you so much guys!