A long, cylindrical tube of inner and outer radii a and b carries a current i distributed uniformly over its cross section. Find the magnitude of the magnetic field at a point (a) just inside the tube (b) just outside the tube.

Explanation

`r lt R`
`r lt R` (inside)
`B=(mu_(0)ir)/(2piR^(2))`
`r=R`(on the surface)
`B=(mu_(0)i)/(2piR)`
`r gt R` (outside)
`B=(mu_(0)i)/(2pir)`
(b)
(i) Just inside the tube `(r lt a)`
Enclosed current in circle of radius `r=0, B=0`
Assume this current along the axis of cylinder and apply formula of long wire
(ii) Just outside the tube `(r le b)`
Enclosed current = i
`B=(mu_(0)i)/(2pir)=(mu_(0)i0)/(2pib)`
(c)
where `r`. distance from axis
(i) `r lt R`
`B=(mu_(0)ir)/(2piR^(2))`
(ii) `R lt r lt 2R`
`B=(mu_(0)i)/(2pir)`
(iii) `2R lt r lt 3R`
Let current in a circle of inner radius `2R` and outer radius `r` be `i_(1)`,
`i_(1)=(ixxpi[r^(2)-(2R)^(2)])/(pi[(3R)^(2)-(2R)^(2)])`
`=(i(r^(2)-4R^(2)))/(5R^(2))`
Net current `i-i_(1)=i[(5R^(2)-r^(2)+4R^(2))/(5R^(2))]`
`=(i(9R^(2)-r^(2)))/(5R^(2))`
`B2pir=(mu_(0)i(9R^(2)-r^(2)))/(5R^(3))`
`B=(mu_(0)i(9R^(2)-r^(2)))/(10 piR^(2)r)`
(iv) `r gt 3R`
Net current `= 0`
`B = 0`