Solution to Maxwells equations

In summary, the problem statement asks for a solution to Maxwell's equations in terms of the potentials, so by solving for E and H in terms of the potentials and substituting in, you show that Maxwell's equations are reduced to the final 3 equations.
  • #1
likephysics
636
2

Homework Statement


I am trying to solve prob 4.107 in Schaums' Vector analysis book.

Show that solution to Maxwells equations -

[tex]\Delta[/tex]xH=1/c dE/dt, [tex]\Delta[/tex]xE= -1/c dH/dt, [tex]\Delta[/tex].H=0, [tex]\Delta[/tex].E= 4pi[tex]\rho[/tex]
where [tex]\rho[/tex] is a function of x,y,z and c is the velocity of light, assumed constant, are given by

E = -[tex]\Delta[/tex][tex]\phi[/tex]-1/c dE/dt, H= [tex]\Delta[/tex]xA

where A and [tex]\phi[/tex], called the vector and scalar potentials, respectively satisfy the equations
[tex]\Delta[/tex].A + 1/c d[tex]\phi[/tex]/dt =0
[tex]\Delta[/tex]^2 [tex]\phi[/tex] - 1/c (d^2 [tex]\phi[/tex]/dt^2) = -4pi[tex]\rho[/tex]
[tex]\Delta[/tex]^2 A = 1/c^2 (d^2A/dt^2)

Homework Equations





The Attempt at a Solution



I don't understand the problem. Should I show that E = -[tex]\Delta[/tex][tex]\phi[/tex]-1/c dE/dt, H= [tex]\Delta[/tex]xA satisfies the vector and scalar potential equations?
 
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  • #2
Hi likephysics! :smile:

(have a delta: ∆ and a phi: φ and try using the X2 tag just above the Reply box :wink:)
likephysics said:
Should I show that E = -[tex]\Delta[/tex][tex]\phi[/tex]-1/c dE/dt, H= [tex]\Delta[/tex]xA satisfies the vector and scalar potential equations?

No, you should assume that the A,φ equations are satisfied, and then prove that E and H (derived from A and φ) satisfy Maxwell's equations. :smile:
 
  • #3
tiny-tim said:
No, you should assume that the A,φ equations are satisfied, and then prove that E and H (derived from A and φ) satisfy Maxwell's equations. :smile:

I disagree, that seems to be proving the reverse statement of what the problem statement asks for.

I would assume that E and H satisfy Maxwell's equations (so that they are solutions to said equations, as per the first premise of the problem statement), then substitute in your expressions for them in terms of the vector and scalar potentials (the second premise of the problem statement) and use appropriate vector product rules to show that Maxwell's equations, in terms of the potentials, reduce to the final 3 equations you are given (the intended conclusion).
 

1. What are Maxwell's equations?

Maxwell's equations are a set of four fundamental equations that describe the behavior of electric and magnetic fields. They were developed by James Clerk Maxwell in the 19th century and are essential to understanding the behavior of electromagnetic waves.

2. What is the solution to Maxwell's equations?

The solution to Maxwell's equations is a set of mathematical equations that describe the behavior of electric and magnetic fields in space and time. These equations can be used to predict the behavior of electromagnetic waves and the interactions between electric and magnetic fields.

3. How are Maxwell's equations used in science?

Maxwell's equations are used in many areas of science, including electromagnetism, optics, and telecommunications. They are fundamental to understanding the behavior of electric and magnetic fields and are essential for the development of technologies such as radio, television, and wireless communication.

4. What is the significance of Maxwell's equations?

Maxwell's equations are significant because they unified the previously separate fields of electricity and magnetism and provided a complete description of the behavior of electromagnetic waves. They also laid the foundation for the development of modern physics and technology.

5. Are there any limitations to Maxwell's equations?

While Maxwell's equations are incredibly powerful and accurate, they do have some limitations. They cannot fully explain the behavior of certain materials, such as superconductors, and they do not take into account the effects of quantum mechanics. However, they are still the basis for many scientific and technological advancements.

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