Question

F'(r) = (2pxy)i+(3qyz^2-py^2)j-(qz^3)k
Where p and q are constants. (i,j,k are the vectors, couldn't seem to do ˆ above them)

Show this is a solenoidal field.

Answer 1

Answer on Question #77462, Physics / Electromagnetism | for completion

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$$F'(r) = (2pxy)i + (3qyz^2 - py^2)j - (qz^3)k$$

Where $p$ and $q$ are constants. (i,j,k are the vectors, couldn't seem to do ~ above them)

Show this is a solenoidal field.

Solution:

To prove that $F$ is solenoidal field we need to show that $div F = 0$.

$$div F = \frac{\partial}{\partial x} (2pxy) + \frac{\partial}{\partial y} (3qyz^2 - py^2) + \frac{\partial}{\partial z} (-qz^3) = 2py + 3qz^2 - 2py - 3qz^2 \equiv 0.$$

We have proved that $F$ is solenoidal field.

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