Maxwell's Equations in Vacuum: Constraints on Wave

In summary, the problem is to show that gy = fz and gz = -fy by finding the appropriate relation between the f and g functions, taking into account that they go to zero as their parameter goes to +/- infinity and that B and E do not have the same units. The solution should involve integrating the functions without concern for a constant added on.
  • #1
MisterX
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Homework Statement



Condensed/simplified problem statement

[itex]\vec{E} = f_{y}(x-ct)\hat{y} + f_{z}(x-ct)\hat{z} \\
\vec{B} = g_{y}(x-ct)\hat{y} + g_{z}(x-ct)\hat{z} \\[/itex]

All the f and g functions go to zero as their parameters go to ±∞.

Show that gy = fz and gz = -fy

Homework Equations


[itex]\nabla \cdot \vec{E} = 0 \\
\nabla \cdot \vec{B} = 0 [/itex]
[itex]\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} [/itex]
[itex]\nabla \times \vec{B} = \frac{1}{c^{2}}\frac{\partial \vec{E}}{\partial t}
[/itex]

The Attempt at a Solution


[itex]\nabla \times \vec{E} = -\frac{\partial f_{z}(x-ct)}{\partial x}\hat{y} + \frac{\partial f_{y}(x-ct)}{\partial x}\hat{z} = -\frac{\partial \vec{B}}{\partial t} = \\-(g^{'}_{y}(x-ct)(-c)\hat{y} + g^{'}_{z}(x-ct)(-c)\hat{z}) = c(g^{'}_{y}(x-ct)\hat{y} + g^{'}_{z}(x-ct)\hat{z}) [/itex]

The -c factors come from the chain rule when differentiating B with respect to t. So then I have the following.

[itex]-\frac{\partial f_{z}(x-ct)}{\partial x} = -f^{'}_{z}(x-ct) = cg^{'}_{y}(x-ct) \\
\frac{\partial f_{y}(x-ct)}{\partial x} = f^{'}_{y}(x-ct) = cg^{'}_{z}(x-ct)[/itex]

The problem is to relate the functions directly and not by their derivatives. We are given that the functions all go to zero as the parameter goes to +/- infinity. So maybe it would be okay to just integrate without concern for a constant added on. But supposedly the c factor isn't there. The answer is supposed to have[itex]g_{y} = -f_{z}[/itex]. How do I get there?
 
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  • #2
B and E do not have the same units, so f and g must be related with an appropriate factor of c. There's probably a typo in the problem. Your argument about the integration constants makes sense.
 

FAQ: Maxwell's Equations in Vacuum: Constraints on Wave

1. What are Maxwell's Equations in Vacuum?

Maxwell's Equations in Vacuum are a set of four equations that describe the behavior of electric and magnetic fields in a vacuum. They were first developed by James Clerk Maxwell in the 1860s and are considered one of the cornerstones of classical electromagnetism.

2. What do Maxwell's Equations tell us about waves?

Maxwell's Equations describe how electric and magnetic fields interact with each other and with charged particles. This interaction produces electromagnetic waves, which can travel through a vacuum at the speed of light.

3. How do Maxwell's Equations constrain the behavior of waves?

Maxwell's Equations impose certain constraints on the behavior of electromagnetic waves. For example, they dictate that the speed of light is constant in a vacuum, and that the electric and magnetic fields must be perpendicular to each other and to the direction of wave propagation.

4. Can Maxwell's Equations be applied to other materials besides vacuum?

Yes, Maxwell's Equations can be applied to other materials besides vacuum. However, the behavior of electromagnetic waves in different materials may be affected by factors such as the material's electrical conductivity and magnetic permeability.

5. How are Maxwell's Equations used in modern science and technology?

Maxwell's Equations are used in a wide range of fields, from telecommunications and electronics to optics and materials science. They are essential for understanding and developing technologies such as radio, television, and wireless communication. They also play a key role in our understanding of light and its interactions with matter.

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