Question

Draw a vector field whose curl vanishes.

Please explain in detail how the vector field should look like.

Answer 1

Ans.) Curl of a vector field is its rotational tendency. Let a vector field F be defined at every point in space. Consider a closed curve in that space. Consider the component of F in the direction of tangent at all points of the closed curve. The curve may be traversed in any direction, clockwise or anticlockwise. Sum all these tangential components. The limit of this sum divided by the surface area enclosed by this curve as this area goes to zero defines the curl of the vector F. Curl F = (∂Fz/∂y - ∂Fy/∂z)i + (∂Fx/∂z) - ∂Fz/∂x)j + (∂Fy/∂x - ∂Fx/∂y)k. and it is evaluated at a point in space. Curl of a vector is also a vector and its direction is normal to the plane of the closed curve. If the vector does not show any rotational tendency, the curl is zero. For example, the electric filed diverges from a charge. It does not show any rotational tendency so curl of an electric field produced from a charge is zero. On the other hand, a current flowing in a straight wire produces a magnetic field whose lines of force form closed curves about the wire. Thus in this case, magnetic field vector possesses a curl.