Simple Derivation Maxwell Equations

In summary, the divergence of a vector field is zero at all points in space, so you can conclude that the function is constant.
  • #1
kiwi101
26
0

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
 
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  • #2
kiwi101 said:

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.
 
  • #3
You're on the right track, just remember that spatial derivitives (like the divergence) commute with the time derivitives.
 
  • #4
So
0 = -∂/∂t(∇°B)
Does the -∂/∂t and the divergence cancel? Or do they make a second order derivative?
 
Last edited:
  • #5
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.
 
  • #6
That means that the function is constant regardless of time at all points of space
 
  • #7
kiwi101 said:
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:
  • #8
oh okay
thank you so much guys!
 

Related to Simple Derivation Maxwell Equations

1. What are Maxwell's equations?

Maxwell's equations are a set of four partial differential equations that describe the fundamental laws of electromagnetism. They were developed by James Clerk Maxwell in the 19th century and are considered one of the most important discoveries in physics.

2. What is the purpose of deriving Maxwell's equations?

The purpose of deriving Maxwell's equations is to understand the underlying principles and mathematical relationships that govern electromagnetism. This allows for a deeper understanding of how electromagnetic fields behave and how they can be manipulated.

3. How are Maxwell's equations derived?

Maxwell's equations can be derived from a combination of experimental observations and mathematical principles. They are based on Gauss's law, Ampere's law, Faraday's law, and the absence of magnetic monopoles.

4. What are the implications of Maxwell's equations?

Maxwell's equations have many important implications in physics and engineering. They explain the behavior of electromagnetic waves, which are essential for understanding light, radio waves, and other forms of radiation. They also provide the foundation for technologies such as electric motors, generators, and communication systems.

5. How are Maxwell's equations used in real-world applications?

Maxwell's equations are used in a wide range of real-world applications, from designing electronic circuits to understanding the behavior of light. They are also used in fields such as telecommunications, satellite communications, and radar technology. Additionally, they have played a crucial role in the development of modern technology and continue to be a fundamental part of many scientific and technological advancements.

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