Question

Show explicitly that ▽ . B-0 near a long straight wire carrying a current I. HINT: you may use Cartesian coordinates, but another choice might make it easier

Answer 1

Answer:

$\bigtriangledown.B=0$ is proved.

Explanation:

The magnetic field in the long current carrying wire is,

$B=\frac{\mu_{0}I }{2\pi r } \phi$

Here, I is the current, B is the magnetic field.

Now, by using cylindrical coordinates for the divergence of B.

$\bigtriangledown.B=\frac{1}{s} \frac{d}{d\phi} B$

Put the value of B in above equation.

$\bigtriangledown.B=\frac{1}{s} \frac{d}{d\phi}(\frac{\mu_{0}I }{2\pi r } \phi)\\\bigtriangledown.B=0$

Hence, it is prove that for a long current I carrying wire magnetic field divergence that is $\bigtriangledown.B=0$.