Obtain an expression for e.m.f. induced in a coil rotating with uniform angular velocity in a uniform magnetic field. Show graphically the variation of e.m.f. with time (t).
Rewsistance of a potentiometer witre is `0*1" "Omega//cm`. A cell of e.m.f. `1*5` V is balanced at 300 cm on this potentiometer wire, Calculate teh current and balancing length for another cell of e.m.f. `1*4` V on the same potentiometer wire.

Explanation

Consider a colil of N turns with effective area NA placed in uniform magnetic field as shown in the figure below

the coil is rotated continusly with constant angular velocity `omega,` At `t=0`e.i., initially, the plane of the coil is perpendicular to teh magnetic induction B. The magnetic flux `phi` throught the coil at time t is given by,
` phi = " NBA cos " theta" ...(1)"`
The plane of the coil is perpendicualr to the magnetic field then after time t, is retated by angle `theta`,
Hence `omega=theta/t"or "theta=omegat`
By equation (i),
`phi ="NBA cos "omegat`
By Faraday's law and Lenz's law,
`E = - (d phi)/("dt")=- d/("dt")("NBA cos" omegat)`
E = NBA `omega sin omegat`
This is an expression for inducred e.m.f.
The graph showing the variation with time t or angle of rotation ot the e.m.f. induced in the coil is given below :

Given : `" R/L"=Sigma=0*1" "Sigma//cm, V_(2)=1*4, V, L_(1)=300 cm, V_(1)=1*5 V`
We know that,
`V_(1)= 1 Sigma L_(1)`
`:." Current, "I = V_(1)/(SigmaL_(1))`
`(1.5)/(0*1xx300)`
`=0*05A`
`"Now, "K=V_(1)/_(L_(1))`
`=(1*5)/(300)`
`=5xx10^(-3)V//cm`
Balancing length, `L_(2)=V_(2)/K=(1*4)/(5xx10^(-3))=280 cm.`