Calculate the time constant `tau` of a straight solenoid of length `l = 1.0 m` having a single-layer winding of copper wire whose total mass is equal to `m = 1.0 kg`. The cross-sectional diameter of the solenoid is assumed to be considerably less than its length.

Explanation

The time constant `tau` given by
`tau = (L)/(R) = (L)/(rho_(0) (l_(0))/(S))`
where, `rho_(0)` = resistivity, `l_(0)` = length of the winding wire, `S` = cross section of the wire.
But `m = l rho_(0) S`
So eliminating `S,tau = (L)/((rho_(0) l_(0))/(m//rho l_(0))) = (mL)/(rho rho_(0) l_(0)^(2))`
From problem `3.315 l_(0) = sqrt((4pi r l L)/(mu_(0)))`
(note the interchange of `l` and `l_(0)` because of difference in notation here.)
Thus, `tau = (mL)/(rho rho_(0) (4pi)/(mu_(0)) L l) = mu_(0) 4pi (m)/(rho rho_(0) l) = 0.7 ms`