The curl of certain vector fields

[hholzer]

Given the two vector fields:

<br /> \vec E and \vec B<br />

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

<br /> curl(\vec E) = -\frac{\partial \vec B } { \partial t }<br />

and further that:

<br /> curl(curl(\vec E)) = curl(-\frac{\partial \vec B } { \partial t }) <br /> = -\frac{\partial curl(\vec B) } {\partial t }<br />

I tried to prove these by defining the vector fields:

<br /> \vec E = C\frac{ \mathbf e_r } {p^2}<br />

and

<br /> \vec B = &lt;0, 0, B&gt;<br />

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.

 

[Matthollyw00d]

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.

 

[hholzer]

Hm, then in general how was the result reached that they are equal?