Self-inductance

Self-inductance measures how effectively a coil opposes changes in its own current by linking flux with itself, and for a long solenoid it depends only on geometry through (L = \mu_0 n^2 A \ell), where (n) is the turns per unit length. This formula arises because the field inside a long solenoid is (B = \mu_0 n I), the flux through one turn is (BA), and the total flux linkage over all (N = n\ell) turns is (N B A), which divided by the current yields (L). Here (n = N/\ell = 1000/0.5 = 2000) turns per metre, so (n^2 = 4\times 10^{6}). Substituting, (L = (4\pi\times10^{-7})(4\times10^{6})(4\times10^{-4})(0.5)). Grouping the numbers gives (L = 4\pi\times10^{-7}\times 800 \approx 1.0\times10^{-3}) H, that is about 1.0 mH. The option 2.0 mH wrongly drops the squaring of turns density at one step. The option 0.5 mH omits the length factor entirely. The option 4.0 mH uses the total turns (N) where the turns density (n) belongs, inflating the result. This matches the NCERT solenoid model in which end effects are neglected for a long coil and the field is treated as uniform inside. A dimensional check shows (T·m/A)(m^{-2})(m^2)(m) reduces to T·m^2/A per ampere, i.e. the henry, and a value of around a millihenry is entirely realistic for an air-cored laboratory solenoid of this size.