The ans given is four times, but if we increase the number of turns by two times without change in length of the solenoid, shouldn't the radius also decrease by 2 times? Hence, won't the self inductance become eight times?:
26. When the number of turns in a solenoid are doubled without any change in the length of the solenoid, it self inductance becomes.
(1) Half (2) Double (3) Four Times (4) Eight times.
Here it is mentioned that only the turns are increased while keeping other variables as constant.So there will be no difference in the radius as well.
The self inductance of a solenoid is given as
L = μ0N2A / l = L1
N is the number of turns of the solenoid
A is the area of each turn of coil
l is the length of the solenoid
and μ0 is the permeability constant
so, if the number of turns was to be doubled the self-inductance would be
L2 = μ0 (2N)2A / l
L2 = 4L1
it would be quadrupled or increase fourfold.