Question

Tíwo simple harmonic waves of the same amplitude and frequency differ by a phase $\pi / 2$ When they are fed simultaneously to the $X$ and $Y$-plates of a CRO, the screen would display the trace of :

• A. a circle
• B. an ellipse
• C. a straight line
• D. a square

Answer 1

Answer: (A)

Two simple harmonic waves of same amplitude and frequency with phase difference $ \frac{\pi }{2} $ in $ x $ and $ y- $ directions respectively are written as:

$ x=a\sin \omega t $ ...(i)
$ y=a\sin \left( \omega t+\frac{\pi }{2} \right) $ ... (ii)
From Eqs.(i) and (ii),
$ \sin \omega t=\frac{x}{a} $ ... (iii)
$ \sin \left( \omega t+\frac{\pi }{2} \right)=\cos \omega t=\frac{y}{a} $ .... (iv)
Squaring and adding Eqs. (iii) and (iv), we have
$ {{x}^{2}}+{{y}^{2}}={{a}^{2}} $ This is an equation of a circle.