How to Identify divergence and curl graphically

Question

I came across this solution to a problem in Griffith's Introduction to Electrodynamics where we had to construct a non uniform field whose curl and divergence are zero. The picture is the equation of vector field $y\,\hat x + x\,\hat y + 0\,\hat z$ Even though mathematically the formulas for divergence and curl gives zero, I am unable to understand this graphically,as a line lying on the positive x axis will feel a net torque on it, I feel this goes against the basic example (a stick floating in the river) given to visualise a curl. I want to know how to identify if there is a curl or divergence graphically and whether a field can produce torque without having any curl.

Answer 1

As jbag has stated. The torque analogy is only valid for closed loops.

It comes from stokes theorem.

$\iint \nabla × \vec{E} \cdot da =\oint \vec{E} \cdot \vec{dl}$

For a circular loop: In the context of electric fields

Torque = $\oint \vec{r} × \rho \vec{E} dl$

|Torque| = $\oint |r| \rho |\vec{E}|sin(\theta)dl $

$|\vec{E}|sin(\theta)$ is the component of E perpendicular to $\vec{r}$, which also happens to be $(\vec{E} \cdot \vec{dl}) * \frac{1}{dl}$

Meaning

|Torque| = $|r| \rho\oint \vec{E} \cdot \vec{dl}$

Using stokes theorem:

|Torque| = $|r| \rho\iint \nabla × \vec{E} \cdot \vec{da}$

For a circular (closed) loop of a small size, the torque is proportional to the curl.

There are probably easier ways to show this but it's just the first I thought of.

Answer 2

A stick on the x-axis will feel a net torque, however a stick on the y axis will fill a torque in the opposite direction. Curl is really a measure of the total torque a “box” would feel.

Answer 3

You can see the divergence as the flow through a closed surface, that's the divergence theorem : \begin{equation} \oint_{\partial \tau}\vec{A}\cdot\hat{n}dS=\int_{\tau}\vec{\nabla}\cdot\vec{A}d\tau \end{equation}

Think about the electric field produced by an elementary charge, you can draw lines through a (virtual) sphere enclosing the charge.(If I remember well, Griffiths write about this in his book).

And in the case of this charge, the divergence is not zero, it gives you the Gauss' theorem that you maybe already know.

Hope this helps ;)

Answer 4

The usual thing I advise is to imagine a very small water wheel with many fins. If the water wheel would turn, then the field has curl.

In this case, there would indeed be a net anti-clockwise torque for fins parallel to the z-axis. But there would be a clockwise torque on the fins aligned with the y-axis.

Unfortunately it is not always obvious whether the wheel would turn and what is important is whether it turns relative to the flow itself.

Divergence is more straightforward. Construct a small, closed box and ask yourself whether there is a net flow out of (positive divergence) or into (negative divergence) the box.