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 need to calculate the time it takes for one charge to reach the position of the other charge.
First, let's assume that the positive charge (+q) is at the origin (0,0) and the negative charge (-q) is at the position (+r,0).
Since the charges are released from rest, we can use the equations of motion to determine the position of the negative charge as a function of time.
Let's use the equation: x = x0 + v0*t + (1/2)*a*t^2
In this case, since the negative charge is moving in the x-direction only, we can ignore the y-component of motion. Therefore, the above equation simplifies to: x = x0 + v0*t
Since the negative charge is at position (+r,0) initially, its initial position (x0) is +r. The initial velocity (v0) is 0.
For the negative charge, the acceleration (a) is given by: F = ma, where F is the electric force.
The electric force between two point charges is given by: F = k*q1*q2 / r^2, where k is the electrostatic constant = 8.99 x 10^9 N m^2/C^2, q1 and q2 are the magnitudes of the charges, and r is the distance between the charges.
Plugging in the values, we get: F = (8.99 x 10^9 N m^2/C^2) * (1 μC) * (1 μC) / (r^2)
Now, since the negative charge is attracted towards the positive charge, the direction of the force is towards the positive charge. So, we'll take the direction of the electric force as positive.
Therefore, a = F / m = [(8.99 x 10^9 N m^2/C^2) * (1 μC) * (1 μC) / (r^2)] / 0.001 kg = (8.99 x 10^15 N m^2/C^2 / (r^2)) m/s^2
Plugging in the value of r as 1 m, we get a = 8.99 x 10^15 N m^2/C^2 m/s^2 = 8.99 x 10^15 N/C.
Now, we can substitute the values into the equation x = x0 + v0*t:
r = r + 0*t + (1/2)*a*t^2
Since the positive charge is fixed at the origin, x0 = 0.
Therefore, r = 0 + 0 + (1/2)*(8.99 x 10^15 N/C)*(t^2)
Simplifying, we get: r = (4.495 x 10^15 N/C)*(t^2)
Solving for t, we get: t^2 = r / (4.495 x 10^15 N/C)
Taking the square root of both sides, we get: t = sqrt(r / (4.495 x 10^15 N/C))
Finally, plugging in the value of r as 1 m, we get:
t = sqrt(1 m / (4.495 x 10^15 N/C))
t = sqrt(2.227 x 10^-16) m
t ≈ 4.72 x 10^-9 seconds
Therefore, the time at which the charges collide is approximately 4.72 x 10^-9 seconds.