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libelec
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Homework Statement
A variable current I(t) = 2Acos(100Hz*t) is passed through a long, thin solenoid of R = 2,5 cm and 900 spirals per meter in length. Calculate the induced EMF inside the solenoid and the self-inductance coefficient L.
Homework Equations
Magnetic field inside a long, thin solenoid: [tex]\vec B = {\mu _0}\eta I{\rm{ \hat k}}[/tex], where [tex]\eta [/tex] is the spiral density (900 spirals per meter).
Induced EMF inside a solenoid: [tex]\varepsilon = \frac{{{\mu _0}{N^2}IS}}{L}[/tex], where S is the transversal suface.
The Attempt at a Solution
My problem is the lack of data. I don't have the length of the solenoid, so what I calculated remains a function of L. This is what I did:
1) I calculate the flux of B through one spiral: [tex]\Phi = {\mu _0}\eta I(t).{\pi ^2}R[/tex].
2) I multiply that by N (number of spirals), to get the total flux through the solenoid: [tex]{\Phi _T} = N{\mu _0}\eta I(t).{\pi ^2}R[/tex].
3) Since it changes with time, because I changes with time, I derive the total flux to get the induced EMF: [tex]\varepsilon = - \frac{{d{\Phi _T}}}{{dt}} = - N{\mu _0}\eta {\pi ^2}R\frac{{dI}}{{dt}} = N{\mu _0}\eta {\pi ^2}R.2A.100Hz\sin (100Hz*t)[/tex]
Then I can't calculate it. I'm missing the total number of spirals N, or the length L, such that [tex]\eta [/tex] = N/L.
What can I do to find the induced EMF. The problem asks for a numerical solution (in function of t).