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Obtain an expression for the mutual indu...

Obtain an expression for the mutual inductance of two long co-axial solenoids.

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Consider two solenoids as shown in figure. The length of primary coil be l and are a of cross section A. Let `N_1 and N_2` are the total number of turns in the primary and secondary solenoids. Let n, and n, be the number of turns per unit length `(n_(1)=(N_(1))/(l) and n_2 = (N_2)/(l))`. Current in the primary coil is i.
`therefore` .. Magnetic field inside the primary (B) `=mu_(0)n_(1)I=mu_(0) (N_(1))/(l) I......(1)`
Magnetic flux through each turn of primary ` phi_B=vecBa,vecA=mu_(0)(N_(1))/(l)IxxA.......(2)`
The same magnetic flux is linked with the secondary coil.
`therefore` Total magnetic flux linked with secondary `=mu_(0)=(cancel(N_(1)i))/(l)xxAxxN_(2) "--"(3)`
If M be mutual inductance of the two coils, the total flux linked with the secondary is Mi
`therefore Mi=(mu_(0)N_(1)N_(2)iA)/(l)...(4)`
`M=(mu_(0)N_(1)N_(2)A)/(l)...(5) (therefore A=pir^(2))`
`(or )M=(mu_(0)N_1N_2(pir^(2)))/(l)....(6) (therefore mu_(r)=mu/(mu_0))`
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