To tackle this problem, we need to analyze the motion of the electron under the influence of a uniform electric field. The key here is to understand how the electric field affects the electron's motion in both the x and y directions. Let's break this down step by step.
Understanding the Forces at Play
When the electron is shot along the positive x-axis with an initial speed \( u \), it experiences no forces in the x-direction, so it continues moving with constant velocity. However, the uniform electric field \( E \) directed along the positive y-axis exerts a force on the electron. This force can be calculated using Coulomb's law:
- The force \( F \) on the electron is given by \( F = eE \), where \( e \) is the charge of the electron.
- Since the electron has a negative charge, the force will actually act in the negative y-direction, resulting in \( F = -eE \).
Calculating the Y-Coordinate
Next, we can use Newton's second law to find the acceleration of the electron in the y-direction:
Acceleration: The acceleration \( a \) can be expressed as:
\( a = \frac{F}{m} = \frac{-eE}{m} \), where \( m \) is the mass of the electron.
Now, since the electron starts from rest in the y-direction (its initial velocity \( v_{y0} = 0 \)), we can use the kinematic equation for displacement:
\( y = v_{y0}t + \frac{1}{2}at^2 \)
Substituting the values we have:
\( y = 0 + \frac{1}{2}\left(-\frac{eE}{m}\right)t^2 \)
This simplifies to:
\( y = -\frac{eEt^2}{2m} \)
Thus, we have shown that the y-coordinate of the electron after a time \( t \) is indeed:
y = -\frac{eEt^2}{2m}
Describing the Trajectory in the XY Plane
Now, let’s analyze the trajectory of the electron in the xy-plane. We know that the x-coordinate changes uniformly with time since there is no force acting in that direction:
\( x = ut \)
We can express time \( t \) in terms of \( x \):
\( t = \frac{x}{u} \)
Now, substituting this expression for \( t \) into the equation for \( y \):
\( y = -\frac{eE}{2m}\left(\frac{x}{u}\right)^2 \)
This can be rewritten as:
\( y = -\frac{eE}{2mu^2}x^2 \)
This equation is in the form of \( y = kx^2 \), where \( k = -\frac{eE}{2mu^2} \). Since \( k \) is negative, this indicates that the trajectory is a downward-opening parabola.
Final Thoughts
In summary, we have derived the y-coordinate of the electron's motion under the influence of an electric field and shown that its trajectory in the xy-plane is a parabola. This illustrates the fascinating interplay between electric forces and motion in physics, highlighting how charged particles behave in electric fields.
