# The curl of certain vector fields

1. Jun 27, 2010

### hholzer

Given the two vector fields:

$$\vec E and \vec B$$

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

$$curl(\vec E) = -\frac{\partial \vec B } { \partial t }$$

and further that:

$$curl(curl(\vec E)) = curl(-\frac{\partial \vec B } { \partial t }) = -\frac{\partial curl(\vec B) } {\partial t }$$

I tried to prove these by defining the vector fields:

$$\vec E = C\frac{ \mathbf e_r } {p^2}$$

and

$$\vec B = <0, 0, B>$$

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.

2. Jun 27, 2010

### 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.

3. Jun 27, 2010

### hholzer

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