Electrons in an oscilloscope are deflected by two mutually perpendicular oscillating field in such a manner that the displacement at any instant is given by x=4sin(ωt+30०)and y=4sin ωt .find the nature and equation of the path

To determine the nature and equation of the path traced by the electrons in the oscilloscope, we need to analyze the given equations for displacement in the x and y directions. The equations provided are:

• x = 4sin(ωt + 30°)
• y = 4sin(ωt)

These equations describe the motion of the electrons in two dimensions, where x and y are functions of time (t). The first step is to express both equations in a form that allows us to eliminate the time variable (t) and find a relationship between x and y.

Step 1: Rewrite the x Equation

The x displacement can be rewritten using the sine addition formula:

x = 4(sin(ωt)cos(30°) + cos(ωt)sin(30°))

Knowing that cos(30°) = √3/2 and sin(30°) = 1/2, we can substitute these values:

x = 4(√3/2 sin(ωt) + 1/2 cos(ωt))

This simplifies to:

x = 2√3 sin(ωt) + 2 cos(ωt)

Step 2: Express y in Terms of sin(ωt)

From the second equation, we have:

y = 4sin(ωt)

Now, we can express sin(ωt) in terms of y:

sin(ωt) = y/4

Step 3: Substitute sin(ωt) into the x Equation

Now, we can substitute sin(ωt) into the equation for x:

x = 2√3(y/4) + 2 cos(ωt)

To find cos(ωt), we can use the Pythagorean identity:

sin²(ωt) + cos²(ωt) = 1

Substituting sin(ωt) gives us:

(y/4)² + cos²(ωt) = 1

Thus, cos²(ωt) = 1 - (y²/16)

Taking the square root, we have:

cos(ωt) = ±√(1 - (y²/16))

Step 4: Substitute cos(ωt) into the x Equation

Now we can substitute this back into the equation for x:

x = 2√3(y/4) + 2(±√(1 - (y²/16)))

This gives us two cases to consider based on the sign of the square root. However, for simplicity, we will focus on the positive case:

x = (√3/2)y + 2√(1 - (y²/16))

Step 5: Identify the Nature of the Path

The resulting equation describes a parametric relationship between x and y. To find the nature of the path, we can analyze the coefficients and the form of the equation. The presence of sine functions indicates that the path is likely to be elliptical, as both x and y are sinusoidal functions of time.

Final Path Equation

To express the path in a more standard form, we can manipulate the equations further, but it is evident that the motion described is elliptical due to the periodic nature of the sine functions and the relationship between x and y. The final path traced by the electrons can be summarized as:

The path is elliptical, and the general equation can be derived from the parametric equations, leading to a relationship that can be expressed as:

(x²/16) + (y²/16) = 1

This indicates that the path traced by the electrons in the oscilloscope is indeed an ellipse, confirming the nature of their motion under the influence of the oscillating fields.