What do you mean Curl of vector. Write its significance.

Curl: The curl of a vector field \(\vec A\) is a vector whose magnitude is the maximum circulation of \(\vec A\) per unit area, as the area shrinks to zero, and whose direction is normal to that area (right-hand rule).

\[ \nabla\times\vec A=\begin{vmatrix}\hat i&\hat j&\hat k\\ \dfrac{\partial}{\partial x}&\dfrac{\partial}{\partial y}&\dfrac{\partial}{\partial z}\\ A_x&A_y&A_z\end{vmatrix},\qquad (\nabla\times\vec A)\cdot\hat n=\lim_{\Delta S\to0}\frac{\oint\vec A\cdot d\vec l}{\Delta S} \]

Significance:

• It measures the rotation (circulation) of a field about a point, like a paddle wheel spinning in flowing water.
• If \(\nabla\times\vec A=0\), the field is irrotational (conservative), for example an electrostatic field, so \(\vec A=-\nabla V\).
• If \(\nabla\times\vec A\neq0\), the field is rotational (a solenoidal or vortex field), for example the magnetic field around a current (\(\nabla\times\vec H=\vec J\)).
• For a rotating body, \(\nabla\times\vec v=2\vec\omega\). It appears in Maxwell's equations (Faraday and Ampere laws).