Question

A straight thick long wire of a uniform cross section of radius 'a' is carrying current I .use ampere circuital law to obtain the relation showing variation of magnetic field ( Br) inside and outside the wire with distance r(ra) of the field section. plot a graph showing variation of field B with distance r.

Answer 1

To Derive:

Variationof magnetic field intensity inside and outside a straight thick long wire of uniform cross section on radius "a".

Derivation:

For r < R, Applying Ampere-Circutal Law:

$\displaystyle \therefore \: \int B.dl = \mu_{0} i$

$\displaystyle= > \: B \int dl = \mu_{0} \times \bigg \{\frac{I}{\pi {R}^{2} } \times \pi {r}^{2} \bigg \}$

$\displaystyle= > \: B \times 2\pi r= \mu_{0} \times \bigg \{\frac{I}{\pi {R}^{2} } \times \pi {r}^{2} \bigg \}$

$\boxed{ \displaystyle= > \: B = \frac{ \mu_{0}Ir }{2\pi {R}^{2} } }$

For r = R:

$\displaystyle= > \: B = \frac{ \mu_{0}IR }{2\pi {R}^{2} }$

$\boxed{ \displaystyle= > \: B = \frac{ \mu_{0}I }{2\pi R} }$

For r > R:

$\displaystyle \therefore \: \int B.dl = \mu_{0} i$

$\displaystyle= > \: B \times 2\pi r= \mu_{0} \times I$

$\boxed{ \displaystyle= > \: B = \dfrac{\mu_{0} I }{2\pi r}}$

Refer to the attached graph.