Two point charges +q=1 μC and −q=−1 μC with mass m=1 g are fixed at the positions ±r⃗ 0 with |r0|=1 m. The charges are released from rest at t=0. Find the time τ in seconds at which they collide.
Hint: Can you do it without integrating by using Kepler's laws?
Details and assumptions
k=1/4πϵ0=9×10^9 m/F
To find the time τ at which the charges collide, we can use Kepler's laws which describe the motion of objects under the influence of gravity or electrostatic forces. In this case, we have two charged particles, so we can consider it as an electrostatic problem.
Kepler's laws apply to two-body systems, where the forces involved follow an inverse square law. In our case, the force between the charges follows Coulomb's law, which is an inverse square law, just like gravity.
Let's proceed with finding the time of collision using Kepler's laws:
1. First, let's determine the force between the charges using Coulomb's law:
Coulomb's law states that the force between two point charges is given by:
F = k * |q1 * q2| / r^2
Given:
Charge 1 (q1) = +1 μC = 1 * 10^(-6) C
Charge 2 (q2) = -1 μC = -1 * 10^(-6) C
Distance between the charges (r) = 2 * r0 (since the charges are placed at positions ± r0)
Plugging these values into Coulomb's law, we get:
F = (9 * 10^9 N m^2/C^2) * |(1 * 10^(-6) C) * (-1 * 10^(-6) C)| / (2 * r0)^2
Simplifying further, we get:
F = -9 N
The negative sign indicates that the force is attractive.
2. Now, let's use the second law of motion to determine the acceleration:
F = m * a
Given:
Mass (m) = 1 g = 1 * 10^(-3) kg
Plugging in the values, we get:
-9 N = (1 * 10^(-3) kg) * a
Solving for acceleration (a), we find:
a = -9 * 10^3 m/s^2
The negative sign indicates that the acceleration is in the opposite direction of the force, i.e., towards each other.
3. Finally, let's use the third law of motion to find the time of collision:
Using the equations of motion, we have:
v = u + at
x = ut + (1/2)at^2
Since the charges start from rest, the initial velocity (u) is zero, i.e., v = 0.
Plugging in the values, we get:
0 = (1/2) * (-9 * 10^3 m/s^2) * t^2
Solving for time (t), we find:
t^2 = 0
This implies that the time of collision (τ) is zero, i.e., the two charges collide instantly at t = 0.
Therefore, the time τ at which the charges collide is zero seconds.
Note: It's important to consider that Kepler's laws are primarily used to describe the motion of celestial bodies under the influence of gravity. While we can apply similar principles to electrostatics, there may be some differences and limitations. In this case, we were able to find the time of collision without resorting to integration by utilizing basic equations of motion and Coulomb's law.