Question

According to Stokes's Theorem, what can you conclude about the circulation in a field whose curl is \(\mathbf{0}\) ? Explain your reasoning.

Short answer

If the curl of a field is \(\mathbf{0}\), according to Stokes's theorem, the circulation of the field around any closed loop in the field is also zero. This means there's no rotational force being applied.

Step-by-step solution

Understand the Concept of Curl

Curl measures the twisting force a vector field applies to a point, and is a vector that points in the direction of the maximum rate of rotation. If the curl is zero, it means there's no rotational force. It's always a good idea to start with the most basic level of understanding.

Application of Stokes’s Theorem

According to Stokes's theorem, the circulation of a vector field \( \mathbf{F} \) around a closed loop, which is the line integral of \( \mathbf{F} \) around this loop, is equal to the double integral of the curl of \( \mathbf{F} \) spread over any surface bounded by the loop. The theorem is given by \( \oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} \).

Evaluate for Curl is Zero

In our case, the curl of the vector field is zero (\(\nabla \times \mathbf{F} = 0\)). That means, the double integral on the right hand side of the equation becomes zero. Consequently, the circulation of the field, the left hand side of the equation i.e., \( \oint_C \mathbf{F} \cdot d\mathbf{r} \), should also be zero.