To solve this problem, we need to analyze the motion of both the positively charged particle and the negatively charged particle in a uniform electric field. Let's break it down step by step.
Understanding the Motion of Charged Particles
When a charged particle moves through an electric field, it experiences a force due to the field. The force acting on a charged particle is given by the equation:
F = qE
where F is the force, q is the charge of the particle, and E is the electric field strength. This force causes the particle to accelerate, changing its velocity over time.
Analyzing the Positive Charged Particle
For the positively charged particle with charge q, when it is launched with an initial velocity v1 perpendicular to the electric field, it will experience a force that causes it to deviate from its original path. The deviation angle given is 60°, and we know that its speed is halved after a certain time. Therefore, its final speed can be expressed as:
v_final = v1 / 2
During its motion, the particle undergoes both horizontal and vertical motion. The horizontal component of velocity remains constant (since there is no force acting in that direction), while the vertical component changes due to the electric force. The relationship between the initial and final velocities can be analyzed using the equations of motion.
Considering the Negative Charged Particle
Now, let's consider the negatively charged particle with charge -2q. It is also launched with the same initial velocity v1 in the same electric field. The force acting on this particle will be:
F = -2qE
This means the force will act in the opposite direction compared to the positively charged particle. However, since the mass of both particles is the same, we can analyze the acceleration experienced by the negatively charged particle:
a = F/m = -2qE/m
Calculating the Final Velocity of the Negative Charged Particle
To find the final speed v2 of the negatively charged particle after the same time interval, we can use the kinematic equations. The change in velocity due to the electric force will be similar in magnitude but opposite in direction compared to the positive particle. Thus, we can express the final velocity as:
v2 = v1 + a * t
Since we know the acceleration is negative for the negatively charged particle, we can substitute:
v2 = v1 - (2qE/m) * t
Finding the Ratio of the Squares of the Velocities
Now, to find the ratio of the squares of the final and initial velocities:
(v2)²/(v1)² = [(v1 - (2qE/m) * t)²] / (v1)²
Expanding this expression will give us:
(v2)²/(v1)² = [v1² - 2v1(2qE/m)t + (2qE/m)²t²] / (v1)²
Now, we can simplify this ratio. The first term will be 1, and the second term will depend on the values of q, E, m, and t. However, since we are interested in the relationship, we can focus on the significant terms that arise from the motion of both particles.
Final Thoughts
In conclusion, the ratio of the squares of the velocities of the negatively charged particle to the positively charged particle can be derived from their respective equations of motion. The key takeaway is that while both particles experience forces due to the electric field, their charges determine the direction of their acceleration, leading to different final velocities. The exact numerical ratio will depend on the specific values of the electric field and the time interval considered.
