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Simple Derivation Maxwell Equations

  • Thread starter kiwi101
  • Start date
26
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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
 

TSny

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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.
 
8
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You're on the right track, just remember that spacial derivitives (like the divergence) commute with the time derivitives.
 
26
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So
0 = -∂/∂t(∇°B)
Does the -∂/∂t and the divergence cancel? Or do they make a second order derivative?
 
Last edited:

TSny

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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.
 
26
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That means that the function is constant regardless of time at all points of space
 

TSny

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Gold Member
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2,659
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?
 
Last edited:
26
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oh okay
thank you so much guys!
 

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