A long straight wire of circular cross-section of radius $a$ carrie...

Question

A long straight wire of circular cross-section of radius $a$ carries steady current $I$ uniformly distributed across its cross-section. Using Ampere's law, calculate the magnetic field for: (a) $r <a> a$ (outside).

Accepted Answer

Using Ampere's Circuital Law: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$

(a) Inside wire ($r <a> a$):
$I_{enc} = I$ (full current)
$$B \cdot 2\pi r = \mu_0 I$$
$$B = \frac{\mu_0 I}{2\pi r}$$
$B$ decreases as $1/r$ outside the wire.

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