Two point charges +q=1 ìC and −q=−1 ìC with mass m=1 g are fixed at the positions ±r ⃗ 0 with |r 0 |=1 m . The charges are released from rest at t=0 . Find the time ô in seconds at which they collide.
To find the time at which the charges collide, we can use the concepts of electrostatic forces and Newton's laws of motion. Here's how you can solve this problem step by step:
1. Determine the force between the two charges:
The force between two point charges can be calculated using Coulomb's law, which states that the force (F) between two charges (q1 and q2) separated by a distance (r) is given by the equation:
F = k * (q1 * q2) / r^2,
where k is the electrostatic constant (approximately 9 × 10^9 N m^2/C^2).
In this case, the charges are equal in magnitude but opposite in sign, so the electrostatic force between them will attract them towards each other. Therefore, the force between the charges is:
F = k * (q^2) / r^2.
2. Apply Newton's second law of motion:
According to Newton's second law of motion, the force acting on an object is equal to the mass of the object multiplied by its acceleration (F = m * a). Since the charges are of negligible mass compared to the charges themselves, the only force acting on them is the electrostatic force.
So, we can equate the electrostatic force to the mass of either charge multiplied by its acceleration:
F = m * a.
3. Solve for acceleration:
Substituting the value of the force from step 1, we can find the acceleration of the charges:
k * (q^2) / r^2 = m * a.
Rearranging the equation, we get:
a = (k * q^2) / (m * r^2).
4. Determine the distance between the charges at any time (d(t)):
Since the charges are released from rest and are attracted towards each other, they will accelerate towards the center of mass position until they collide. At any time t, the distance between the charges can be given by:
d(t) = 2 * r - (a * t^2) / 2.
5. Find the time of collision:
The time of collision occurs when the distance between the charges becomes zero (d(t) = 0). Therefore, we can solve the equation:
2 * r - (a * t^2) / 2 = 0,
for t.
6. Plug in the values and solve:
Substitute the given values into the equations:
- q = -1 μC = -1 * 10^-6 C,
q = 1 μC = 1 * 10^-6 C,
m = 1 g = 1 * 10^-3 kg,
r = 1 m,
k = 9 * 10^9 N m^2/C^2.
Calculate the acceleration (a) using the equation from step 3.
Solve the equation in step 5 for t.
7. Calculate the time (ô) in seconds:
Once you have obtained the value of t, you will have the time in seconds (ô) at which the charges collide.
By following these steps, you should be able to find the time (ô) at which the charges collide.