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Homework Statement
A fluxmeter is a device sometimes used for measuring magnetic fields. It is a suspended coil instrument in which the suspension has virtually no restoring twist. The instrument is used in series with a search coil which is removed from or rotated in the field to be measured. The fluxmeter coil and the search coil form a contiuous Circuit of total resistance ##R##. If the instrumental constant for the fluxmeter is ##k## (That is for small ##\theta## you can approximate the flux as ##\Phi = k \theta + \Phi_0##), determine the deflection ##\Delta \theta ## of the fluxmeter for a change of flux ##\Delta \Phi## through the search coil.
Homework Equations
[tex]\frac{d \Phi}{dt}=-V[/tex]
[tex]I=V/R[/tex]
[tex]T=I\frac{\partial \Phi}{\partial \theta}[/tex]
The Attempt at a Solution
I don't know what the relation is between the search coil and fluxmeter coil, but i am first examinining the search coil. I use the relation ##\Phi = k \theta + \Phi_0## to give
[tex]\frac{d \Phi}{dt}=k \frac{d \theta}{dt}=-V[/tex]
which leads to the current in the curcuit
[tex]I=V/R=-\frac{k d \theta / dt}{R}[/tex]
Now there is a torque on the coil given by
[tex]T=I\frac{\partial \Phi}{\partial \theta}=Ik=-\frac{k^2 d \theta / dt}{R}[/tex]
And from the laws of mechanics it is given by
[tex]T= I_m \frac{d^2\theta}{dt^2}[/tex]
Where ##I_m## is the moment of inertia for the Circuit. This leads to the differential equation
[tex]\frac{d^2 \theta}{dt^2}+a\frac{d \theta}{dt}=0[/tex]
where ##a=\frac{k^2}{RI_m}##. The solution is
[tex]\frac{d \theta}{dt}=Ce^{-at}=\omega (t)[/tex]
But since ## \omega (0) = 0## then ##C=0## so ##\theta = constant## and there is no deflection at all in the search coil. So (if this is right) why should there be a deflection in the fluxmeter?