A very long straight solenoid has a cross section radis `R` and `n` turns per unit lenghth. A direct current `I `flows throguh the solenoid. Suppose that `x` is the distance from the end of the the solenoid, measured along its axis. Find:
(a) the magnetic induction `B` on the axis as a funciton of `x`, draw an approximate plot of `B` vs ratio `x//R`,
(b) the distance `x_(0)` to the point on the axis at which the value of `B` differs by `eta = 1%` from that in the middle section of the solnoid.

Explanation

We proceed exactly as in the previious problem. Then (a) the magnitude induction on the axis at a distance `x` from one end is clearly.
`B = (mu_(0) nI)/(4pi) xx 2pi R^(2) int_(0)^(oo) (dz)/([R^(2) + (z -x)^(2)]^(3//2)) = (1)/(2) mu_(0) n I R^(2) int_(x)^(oo) (dz)/((z^(2) + R^(2))^(3//2))`
`= (1)/(2) mu_(0) n I int_("tan"^(-1) (x)/(R))^(pi//2) cos theta d theta = (1)/(2) mu_(0) n I (1 - (x)/(sqrt(x^(2) + R^(2))))`
`x gt 0` means that the field point is outside the solenoid. B then falls with `x. x lt 0` means that the field point gets more and more inside the solenoid . `B` then increases wtih`(x)` and eventually becomes constant equal to `mu_(0) n I`. The `B - x` graph is as given iin the answer script.
(b) We have, `(B_(0) - del B)/(B_(0)) = (1)/(2) [1 - (x_(0))/(sqrt(R^(2) + x_(0)^(2)))] = 1 - eta`
or, `- (x_(0))/(sqrt(R^(2) + x_(0)^(2))) = 1 - 2eta`
Since `eta` is small `(oo 1%), x_(0)` must be negative. Thus `x_(0) = -|x_(0)|`
and `(|x_(0)|)/(sqrt(R^(2) + |x_(0)|^(2))) = 1 - 2eta`
`|x_(0)|^(2) = (1 - 4eta + 4eta^(2)) (R^(2) + |x_(0)|^(2))`
`0 = (1 - 2 eta)^(2) R^(2) - 4eta (1 - eta) |x_(0)|^(2)`
or, `|x_(0)| = ((1 -2 eta) R)/(2sqrt(eta (1 - eta)))`