A long solid cylindrical straight conductor carries a current `i_(0)` . Find the magnetic energy stored per meter inside it.

Explanation

The current enclosed within the shaded circle
`I = (i_(0))/(pi R^(2)) pir^(2)=(i_(0)r^(2))/(R^(2))`
Applying Ampere s law for the dotted loop we obtain `oint B .d l= mu_(0)i`
`implies B (2pir) = mu_(0)(u_(0)r^(2))/(R^(2))`
`implies B = (mu_(0)i_(0)r)/(2piR^(2))`
Energy stored in the elementary volume dV (`=2pi tau l dr)` is given as
`dU =(B^(2))/(2mu_(0)) dV`
`implies ` The total energy stored in the conductor
`U = int ""(B^(2))/(2 mu_(0)) dV= (1)/(2mu_(0)) int_(0)^(R) (mu_(0)i_(0)r)/(2piR^(2))^(2) (2pir ldr)`
`= (mu_(0)i_(0)^(2)l)/(4piR^(4)) int_(0)^(R) r^(3)dr = (mu_(0)i_(0)^(2)l)/(4pi)xx(1)/(4) = (mu_(0)i_(0)^(2))/(16pi)" " ( because l=1)`