In this problem, you derive the expression for the self-inductance of a long sol

Question

In this problem, you derive the expression for the self-inductance of a long solenoid
[L = µ0n2pr2script i].
The solenoid has n turns per unit length, length script i, and radius r. Assume that the current flowing in the solenoid is I.

(a) Write an expression for the magnetic field inside the solenoid in terms of n, script i, r, I, and universal constants.
B = _________________

(b) Assume that all of the field lines cut through each turn of the solenoid. In other words, assume the field is uniform right out to the ends of the solenoid—a good approximation if the solenoid is tightly wound and sufficiently long. Write an expression for the magnetic flux through one turn.
F = ______________

(c) What is the total flux linkage through all turns of the solenoid? (Give your answer in terms of N, the total number of turns, instead of n.)
NF = ______________

(d) Use the definition of self-inductance [NF = LI] to find the self-inductance of the solenoid.
L = ___________

Explanation / Answer

a ) magnetic field inside a solenid is         B = _0 n * I         ( n = number of turns per unit length n = N / i ) b ) magnetic flux through one turn is            F = r^2 * B                = r^2 ( _0 n * I ) c ) The total flux linked through all the turns is           N F = L * I                = _0 N^2 * r^2 / i * ( I )            ( where i = length )     d ) Using the definition self inductance of a solenoid is              L = N F / I                  = _0 N^2 * r^2 / i * ( I )   / I                  = _0 N^2 * r^2 / i             

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