Question 705031

SCQMEDIUM

Vorticity of a vector field $\vec{B}$ is defined as $\vec{\omega} = \nabla \times \vec{B}$ . Given $\vec{B} = kxyz\hat{r}$ , which one of the following is correct?

(A)

Vorticity is a null vector for all finite $x, y, z$

(B)

Vorticity is parallel to the vector field everywhere

(C)

The angle between vorticity and vector field depends on $x, y, z$

(D)

Vorticity is perpendicular to the vector field everywhere

Detailed Solution

Calculating the curl of $\vec{B} = kxyz\hat{r}$ in spherical coordinates leads to a vector with components only in the $\hat{\theta}$ and $\hat{\phi}$ directions. Since the original field $\vec{B}$ only has a component in the radial $\hat{r}$ direction, the dot product $\vec{\omega} \cdot \vec{B} = 0$ , meaning the vorticity is perpendicular to the vector field.

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