The curl of certain vector fields

In summary, the conversation discusses the identity between the electric vector field and the magnetic vector field, and how it is proven through the use of various vector fields. However, the individual's reasoning is found to be incorrect due to the nature of the fields. It is then concluded that the two fields are equal based on Maxwell's equation.
  • #1
hholzer
37
0
Given the two vector fields:

[tex]
\vec E and \vec B
[/tex]

Where the first is the electric vector field
and the second is the magnetic vector field, we
have the following identity:

[tex]
curl(\vec E) = -\frac{\partial \vec B } { \partial t }
[/tex]

and further that:

[tex]
curl(curl(\vec E)) = curl(-\frac{\partial \vec B } { \partial t })
= -\frac{\partial curl(\vec B) } {\partial t }
[/tex]

I tried to prove these by defining the vector fields:

[tex]
\vec E = C\frac{ \mathbf e_r } {p^2}
[/tex]

and

[tex]
\vec B = <0, 0, B>
[/tex]

where C and B are constants. But I ended up with
zero for curl(E), which cannot be right. So my reasoning
is in error somewhere.

Any insight appreciated.
 
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  • #2
If p is your radial variable, then the electric field varies only radially and hence has curl zero, so that makes sense. Further, since you have a static magnetic field, its time derivative should also be zero, putting the two equal and satisfying Maxwell's equation.
 
  • #3
Hm, then in general how was the result reached that they are equal?
 

Related to The curl of certain vector fields

1. What is the curl of a vector field?

The curl of a vector field is a mathematical operation that describes the rotation or circulation of a vector quantity in a given space.

2. How is the curl of a vector field calculated?

The curl of a vector field is calculated using the cross product of the gradient operator and the vector field itself. This results in a new vector field that represents the magnitude and direction of the curl at each point in the space.

3. What is the physical significance of the curl of a vector field?

The curl of a vector field is physically significant as it describes the flow of a vector quantity, such as fluid or electromagnetic fields. It can also indicate the presence of vortices or rotational motion in a system.

4. How does the curl of a vector field relate to divergence?

The curl and divergence of a vector field are related through the fundamental theorem of calculus. Divergence represents the net flow in and out of a region, while the curl represents the rotational flow within that region.

5. Can the curl of a vector field be zero?

Yes, the curl of a vector field can be zero if the field is conservative, meaning that its path is independent of its starting point. In this case, the vector field is said to be irrotational.

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