How Do Eo and Bo Relate in Maxwell's Equations for a Vacuum?

In summary, the given electric and magnetic fields, with k, w, Eo, and Bo as constants, satisfy the Maxwell equations in a vacuum with zero charge and current densities. In order for this solution to hold with non-zero Bo and Eo, the relation k = w/x must hold, and Bo and Eo are related by Bo = kEo/w. The partial derivative of Bz with respect to time is uncertain.
  • #1
SaveFerris
4
0
1. Show that:

electric field E(x,t) = [0, Eo, 0] * f(kx-wt)

magnetic field B(x,t) = [0, 0, Bo] * f(kx-wt)

(where k, w, Eo, Bo are constants) satisfy the Maxwell equations in a vacuum where
charge and current densities are zero.

What relation between k and w must hold for a solution with Bo and Eo not equal to 0? How are Bo and Eo related in this case?


3. The Attempt at a Solution

I have finished the first part of this question proving that the Maxwell equations are solved, but am a little stuck on the second part!

i thought that if kx = wt then the equation would still equal zero, but as it's a function of kx-wt I am not sure this works? if i could do that i thought finding k in terms of w might be the relationship but I am not sure. this would give k = wt/x.

Thank you for any help!
 
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  • #2
What is, for example,

[tex]\frac{\partial B_z}{\partial t}?[/tex]
 

Related to How Do Eo and Bo Relate in Maxwell's Equations for a Vacuum?

1. What are Maxwell's equations in a vacuum?

Maxwell's equations in a vacuum are a set of four fundamental equations that describe the behavior of electric and magnetic fields in empty space. They were first formulated by James Clerk Maxwell in the 19th century and are essential for understanding electromagnetic phenomena.

2. What do the Maxwell equations in a vacuum represent?

The Maxwell equations in a vacuum represent the relationship between electric and magnetic fields and their sources, including electric charges and currents. They also describe how these fields propagate through space.

3. What are the four Maxwell equations in a vacuum?

The four Maxwell equations in a vacuum are Gauss's law, which relates electric fields to electric charges; Gauss's law for magnetism, which relates magnetic fields to magnetic charges; Faraday's law of induction, which describes how changing magnetic fields create electric fields; and Ampere's law, which relates magnetic fields to electric currents.

4. What is the importance of the Maxwell equations in a vacuum?

The Maxwell equations in a vacuum are crucial for understanding and predicting a wide range of electromagnetic phenomena, such as the behavior of light, electricity, and magnetism. They have also been used to develop technologies such as radio, television, and wireless communication.

5. Are the Maxwell equations in a vacuum always valid?

Yes, the Maxwell equations in a vacuum are always valid in empty space. However, they may need to be modified in certain situations, such as when dealing with materials with different electromagnetic properties or at very high energies.

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