In one line
Curl of a vector field is a vector that measures how much the field circulates or rotates around a point, with direction along the axis of rotation given by the right-hand rule.
Explained simply
Drop a tiny paddle wheel into a flowing river. If the water on one side flows faster than on the other, the wheel spins. The curl tells how fast it spins and about which axis. If the wheel does not spin anywhere, the flow has zero curl and is called irrotational.
How it works
- Curl is the cross product of \(\nabla\) with \(\vec{A}\), written as a determinant: \(\nabla\times\vec{A} = \begin{vmatrix}\hat{i} & \hat{j} & \hat{k}\\ \partial_x & \partial_y & \partial_z\\ A_x & A_y & A_z\end{vmatrix}\).
- Expanded: \(\hat{i}\left(\dfrac{\partial A_z}{\partial y} - \dfrac{\partial A_y}{\partial z}\right) - \hat{j}\left(\dfrac{\partial A_z}{\partial x} - \dfrac{\partial A_x}{\partial z}\right) + \hat{k}\left(\dfrac{\partial A_y}{\partial x} - \dfrac{\partial A_x}{\partial y}\right)\).
- Meaning: circulation \(\oint\vec{A}\cdot d\vec{l}\) per unit area of a tiny loop, in the limit of zero area, gives the component of curl normal to that loop.
- If \(\nabla\times\vec{A} = 0\) the field is irrotational (conservative) and can be written as the gradient of a scalar.
- Physics link: \(\nabla\times\vec{E} = -\partial\vec{B}/\partial t\) (Faraday) and \(\nabla\times\vec{H} = \vec{J} + \partial\vec{D}/\partial t\) (Ampere-Maxwell). Electrostatic fields have zero curl.
Key formula
\(\nabla\times\vec{A}\) as the determinant above. Its unit is that of \(A\) per metre. Useful identities: \(\nabla\times(\nabla\phi) = 0\) and \(\nabla\cdot(\nabla\times\vec{A}) = 0\).
Quick example
For \(\vec{A} = -y\hat{i} + x\hat{j}\) (a field that swirls around the z-axis): \(\hat{k}\) component \(= \partial(x)/\partial x - \partial(-y)/\partial y = 1 + 1 = 2\), the other components are 0. So \(\nabla\times\vec{A} = 2\hat{k}\): uniform rotation about the z-axis. For \(\vec{A} = x\hat{i} + y\hat{j} + z\hat{k}\), the curl is 0, so it is irrotational.
Common mistakes
- Getting the sign of the \(\hat{j}\) term wrong while expanding the determinant.
- Saying a field with curved field lines must have non-zero curl; curl depends on local rotation, not on line shape.
- Writing curl as a scalar.
How MU asks it
Asked as "Explain the physical significance of curl" (3 marks), "Show that a field is irrotational" (3 marks), or as part of Maxwell's equations. Write the determinant, expand it correctly, explain with the paddle-wheel idea, and state the irrotational condition.