The potentials across the \(x\) and \(y\) plates of a cathode-ray oscilloscope produce deflections given, respectively, by \(A\) sin \(\omega t\) and \(A \cos \omega t\). Show that the trace is a circle of radius \(A\). What is the effect on the trace of applying at the same time a modulation to the accelerating potential so that it varies by \((1+\sin \omega t)\) times its original constant value?

Electrical potential, read as the 'electrical potential energy per unit charge,' is a fundamental concept when discussing the behavior of electrons in circuits and devices such as a cathode-ray oscilloscope (CRO). In our example, the potentials applied across the CRO's plates are responsible for deflecting the electron beam. The potential difference creates an electric field that pushes the electrons, causing them to move.

Imagine this effect as rolling a ball across a surface with hills and valleys; where the ball goes depends on the shape of the terrain (potential). In the CRO, the electron beam's path is controlled by varying the potential across the x and y plates.

Specifically, modulation of the accelerating potential of a CRO, as introduced in the exercise, changes the energy of the electrons. As a result, the deflection becomes a combination of the original movement and the added 'push' from the modulation, altering the trajectory of the electrons and, consequently, the trace on the screen.

Harmonic motion refers to a type of periodic oscillation where the restoring force is proportional to the displacement. It's like a pendulum swinging back and forth or a spring bobbing up and down. This motion is mathematically characterized by sine and cosine functions which depict the projection of uniform circular motion onto an axis.

In our CRO example, we adopt the sine and cosine functions to describe the movement of the electron beam because these functions inherently capture the essence of harmonic motion. The sine function ( \(A \textrm{sin} \textrm{omega} t \$) represents the deflection on one plate, and the cosine function ( \)A \textrm{cos} \textrm{omega} t \$) on the other. Together, they form a circle, which is the fundamental shape of uniform harmonic motion in two dimensions.

When the beam follows a sinusoidal pathway across the plates, the display shows a classic example of harmonic motion—a circular trace—reflecting the periodic and uniform characteristics of the sine and cosine functions.

The trigonometric identity \(\textrm{sin}^2 \textrm{omega} t + \textrm{cos}^2 \textrm{omega} t = 1 \$ plays a critical role in showing why the display of a cathode-ray oscilloscope can be a circle as in the given exercise. This identity is fundamental to trigonometry, stating that the sum of the squares of sine and cosine functions (with the same argument) is always one, regardless of the value of \textrm{omega} t.

In the context of the oscilloscope, this identity confirms that the combined squared deflections of the x and y plates, at any point in time, equal the square of the amplitude ( \)A^2 \$). It beautifully simplifies the complex motion into a single, constant value, confirming the circular path of the electron beam.

The modulation effect that was applied to the potential alters the straightforward circle into a more complex shape. The trig identity has thus underscored a key pivot around which the beam's motion—and hence the trace—can rapidly transition from simple to complex, emphasizing the power of both the mathematical identity and the control the potential has on the oscilloscope's output.