Electrons in an oscilloscope are deflected by two mutually perpendicular electric fields in
such a way that at any time t the displacement is given by x = Acos(t), y=Acos(t+). Describe
the path of the electrons and determine its equation (a) when = 0, (b) = 30; (c) = 90
To describe the path of the electrons in an oscilloscope, we need to understand the given equations:
x = Acos(ωt)
y = Acos(ωt + α)
Here, A represents the maximum displacement of the electrons, ω denotes the angular frequency, t is the time variable, and α is the phase difference.
To determine the equation for the path of the electrons, we can use the trigonometric identity which states that cos(A + B) = cos(A)cos(B) - sin(A)sin(B).
Let's manipulate the equation for y to express it in terms of x:
y = Acos(ωt + α)
= Acos(ωt)cos(α) - Asin(ωt)sin(α)
Now let's substitute the value of x into this equation:
y = xcos(α) - Asin(ωt)sin(α)
Now we have an equation that relates x and y, allowing us to describe the path of the electrons in the oscilloscope.
(a) When α = 0°:
y = xcos(0) - Asin(ωt)sin(0) = x
This means that the path of electrons will be a straight line along the x-axis.
(b) When α = 30°:
y = xcos(30°) - Asin(ωt)sin(30°)
This equation describes the path of electrons as an ellipse with major axis along the x-axis. The shape of the ellipse will vary depending on the values of A and ω.
(c) When α = 90°:
y = xcos(90°) - Asin(ωt)sin(90°) = - Asin(ωt)
In this case, the path of electrons will be a sinusoidal curve along the y-axis. The amplitude of the curve will be A, and its frequency will be governed by the angular frequency ω.
Remember, the path of the electrons is determined by the combination of the cosine and sine functions in the x and y equations, respectively, along with the given maximum displacement A and the phase difference α.