Question

If \( i, j, k \) are the orthogonal unit vectors in Cartesian \( x \)-\( y \)-\( z \) coordinate system, the curl of the vector \( -2y\hat{i} + x\hat{j} \) is:

• A. \( 3\hat{k} \)
• B. \( -3\hat{k} \)
• C. \( -\hat{k} \)
• D. \( \hat{k} \)

Hint:

The curl of a vector field measures the rotation or "twist" of the field at a point. It is particularly useful in fluid dynamics and electromagnetism.

Answer 1

The Correct Option is A

We are asked to find the curl of the vector field \( \mathbf{F} = -2y\hat{i} + x\hat{j} \). The curl of a vector field \( \mathbf{F} = P\hat{i} + Q\hat{j} + R\hat{k} \) is given by:
\[ \nabla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \hat{i} - \left( \frac{\partial R}{\partial x} - \frac{\partial P}{\partial z} \right) \hat{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \hat{k} \]
In our case, \( P = -2y \), \( Q = x \), and \( R = 0 \). Now, calculating the partial derivatives:
- \( \frac{\partial R}{\partial y} = 0 \) and \( \frac{\partial Q}{\partial z} = 0 \),
- \( \frac{\partial R}{\partial x} = 0 \) and \( \frac{\partial P}{\partial z} = 0 \),
- \( \frac{\partial Q}{\partial x} = 1 \) and \( \frac{\partial P}{\partial y} = -2 \).
Thus, the curl is:
\[ \nabla \times \mathbf{F} = \left( 0 - 0 \right) \hat{i} - \left( 0 - 0 \right) \hat{j} + \left( 1 - (-2) \right) \hat{k} = 3\hat{k} \]
Therefore, the curl of the vector is \( 3\hat{k} \), which corresponds to option (A).