Self-inductance of a coil is the property of the coil in which it opposes the change of current flowing through it.Self induction of long solenoid of inductance L, A long solenoid is one which length is very large as compared to its cross section area. the magnetic field inside such a solenoid is constant at any point and given by B=\\frac { { \\mu }_{ 0 }NI }{ l } { \\mu }_{ 0 }=\\quad absolute\\quad magnetic\\quad permeability\\\\ N=\\quad total\\quad number\\quad of\\quad turns Magnetic flux through each turn of solenoid \\phi =B\ imes area\\quad of\\quad each\\quad turn\\\\ \\phi =\\dfrac { { \\mu }_{ 0 }NI }{ l } \ imes A\\\\ total\\quad flux=flux\ imes total\\quad number\\quad of\\quad turns\\\\ N\\phi =N(\\frac { { \\mu }_{ 0 }NI }{ l } \ imes A)\\quad \\quad ..........(1) If L is the coefficient of inductance of solenoid N\\phi =LI\\quad \\quad \\quad \\quad \\quad \\quad ...............(2)\\\\ from equation 1 and 2 LI=N(\\dfrac { { \\mu }_{ 0 }NI }{ l } \ imes A)\\\\ L=\\dfrac { { \\mu }_{ 0 }{ N }^{ 2 }A }{ l }\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad(3) The magnitude of emf is given by \\quad |e|\\quad or\\quad e\\quad =L\\dfrac { dI }{ dt } \\quad \\quad \\quad \\quad ...............(4)\\\\ multiplying\\quad I\\quad to\\quad both\\quad sides\\\\ eIdt=LIdI\\\\ but\\quad I=\\dfrac { dq }{ dt } \\\\ Idt=dq Also work done (dW)= voltage X Charge(dq)or dW = eXdq = eIdtsubstituting the values in equation 4 dW = LIdtBy integrating both sides \\int _{ 0 }^{ w }{ dW } =\\int _{ 0 }^{ { I }_{ 0 } }{ LIdt } \\\\ W=\\dfrac { 1 }{ 2 } L{ { I }_{ 0 } }^{ 2 } this work done is in increasing the current flow through inductor is stored as potential energy (U) in the magnetic field of inductor U=\\frac { 1 }{ 2 } L{ { I }_{ 0 } }^{ 2 }