Curl Operator/Examples

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Examples of Curl Operator

Rotation of Rigid Body

Consider a rigid body $B$ in rotary motion with angular velocity $\omega$ about an axis $OA$, where $O$ is some fixed point inside $B$.

Let $P$ be an arbitrary point inside $B$.

Let $B$ also be subject to a linear velocity $\mathbf v_0$ in an arbitrary direction.

Let the instantaneous velocity of $P$ be $\mathbf V$


Then:

$\bsomega = \dfrac 1 2 \curl \mathbf V$

where $\bsomega$ is the angular velocity (axial) vector along the axis $OA$ in the sense according to the right-hand rule.


Motion of Fluid

Consider an infinitesimal volume of fluid $F$.

It may have $3$ kinds of motion:

$(1): \quad$ Moving with a linear velocity as a whole
$(2): \quad$ Changing its shape
$(3): \quad$ In rotary motion.

At any instant, $F$ may be regarded as a rigid body.

Hence from Curl of Rotation of Rigid Body, the curl of the velocity of $F$ is twice its angular velocity where its axis of rotation at that instant is the same as that of the curl.


Rotational-and-irrotational-motion.png


Consider the diagram above.

On the left, the element $E_1$ has itself rotated in moving to ${E_1}'$.

If every element of the body of fluid has rotated the same amount, $\curl \mathbf V$ would be twice the angular velocity about $O$.


On the right, on the other hand, the element $E_2$ has not actually rotated in moving to ${E_2}'$.

Hence there is no $\curl \mathbf V$ and its angular velocity is zero.


Magnetic Field of Conductor

Consider a conductor of electricity $C$.

Let $C$ be carrying a steady current $I$.

Let $P$ be an arbitrary point in the magnetic field $\mathbf H$ induced by $I$.

Let a small plane surface be placed at $P$, turned into a position so that the contour integral of the magnetic force taken around its boundary is the greatest possible.

This value per unit area is the curl of $\mathbf H$.

This is the magneto-motive force per unit area at $P$.

If $P$ is within $C$ at the point where current density is $\mathbf J$, this will be the total current passing normally through the closed curve when the contour integral is greatest.

We have from the Ampère-Maxwell Law that:

$\curl \mathbf H = \mathbf J$

For a point in the magnetic field outside the conductor there is no current density and so:

$\curl \mathbf H = 0$