Electric and magnetic field problems (curl/divergence)

In summary, the conversation discusses finding a magnetic field that satisfies certain conditions and finding a relationship between k and w. The main issue is understanding the curl operator and how to use it in this context. The del operator is used to represent the curl and divergence of a vector function. The correct matrix for this problem is shown, and once the operators are understood, the rest of the problem should be straightforward.
  • #1
ParoxysmX
21
0

Homework Statement



Consider the electric field E(t,x,y,z) = Acos(ky-wt)k

1. Find a magnetic field such that [itex]\partial_t[/itex]B + [itex]\nabla[/itex] X E = 0
2. Show that [itex]\nabla[/itex] . E = 0 and [itex]\nabla[/itex]. B = 0
3. Find a relationship between k and w that enables these fields to satisfy

[itex]\nabla[/itex] X B = [itex]\mu_{0}[/itex][itex]\epsilon_{0}[/itex][itex]\frac{\partial E}{\partial t}[/itex]

The Attempt at a Solution



Really the problem here is the first one. I understand (sort of) the curl operator, but how do you find [itex]\nabla[/itex] X E? Would you start with a matrix of

[i j k
0 kAcos(ky-wt) 0
0 0 Acos(ky-wt)]

Then find the determinant, which is Acos(ky-wt)(1+k)i - 0j + 0k?
 
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  • #2
No, that matrix is not correct. The cross product of the del operator [itex] \nabla [/itex] and a vector function is just an alternate convention for denoting curl. You can why here: http://en.wikipedia.org/wiki/Del#Curl. This should also explain why divergence can be denoted: [itex]\nabla \cdot [/itex]

So your matrix should be: [itex] \left[ \begin{array}{ccc}
\hat{i} & \hat{j} & \hat{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\
0 & 0 & {\scriptsize A\cos(ky-wt)} \end{array} \right][/itex]

The rest of the problem should be relatively trivial once you know what the operators do.
 
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  • #3
Ah I see. Thanks for your help.
 

1. What is the difference between electric and magnetic fields?

Electric fields are created by stationary or moving electric charges, while magnetic fields are created by moving electric charges or by magnetic materials.

2. How are electric and magnetic fields related?

Electric and magnetic fields are related through Maxwell's equations, which show that a changing electric field can create a magnetic field and a changing magnetic field can create an electric field.

3. What is curl in the context of electric and magnetic fields?

Curl is a mathematical operation that describes the circulation or rotation of a vector field. In the context of electric and magnetic fields, curl represents the strength and direction of the magnetic field created by a changing electric field.

4. What is divergence in the context of electric and magnetic fields?

Divergence is a mathematical operation that describes the flow or spread of a vector field. In the context of electric and magnetic fields, divergence represents the strength and direction of the electric field created by a changing magnetic field.

5. How are curl and divergence used in solving electric and magnetic field problems?

Curl and divergence are used in Maxwell's equations to mathematically describe the behavior of electric and magnetic fields. They are essential tools for solving complex problems involving these fields and are used extensively in fields such as electromagnetics, optics, and electrical engineering.

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