Question

A variable capacitance angular velocity pickup is made with two concentrically mounted parallel semicircular plates with a small separation between them as shown in figure. A dc voltage is connected across the terminals of the capacitance sensor as shown. For a constant angular velocity $\omega$, the waveform of the current i(t) will be

• A. dc
• B. sinusoidal
• C. triangular
• D. square

Answer 1

The correct option is A dc

$\because i\left(t\right)=\frac{dQ}{dt}$

$Q=CE$

$i\left(t\right)=\frac{d}{dt}(C\times E)$

$=E\frac{d}{dt}\left(\frac{ϵA}{d}\right)=\frac{Eϵ}{d}\frac{d}{dt}\left(\frac{\theta r^2}{2}\right)$

$=\frac{Eϵ}{d}\frac{r^2}{2}\frac{d\theta }{dt}$

$\because \theta =\omega t$

$\frac{d\theta }{dt}=w$

$i\left(t\right)=\frac{Eϵr^2}{d\times 2}\times \omega$

$\therefore i\left(t\right)\propto \omega$

$\because \omega \to constant$

$\therefore i\left(t\right)\to constant$