Is Solenoid Inductance Proportional to Turns Squared?

[Alin S]

Homework Statement

In Solenoid, is emf E= N. d(phi)/dt, or is it simply d(phi)/dt? Some books contain the latter one. It was not a big deal until i gave a thought about it. Since increasing the N, we are actually increasing the value of n, thus increasing B= (mu) .n.I
So, by placing N in front of the differential, are not we considering the same thing twice?

Homework Equations

The Attempt at a Solution

 

[Merlin3189]

Have a look at, for example, Hyperphysics article on inductance.

Yes you are using the number of turns twice (I'm not sure what N and n are) for exactly the reason you say: the flux depends on the number of turns and the emf depends on the number of turns x rate of change of flux, so inductance is proportional to turns².