Two point charges +q=1 μC and
−q=−1 μC 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 τ in seconds at which they collide.
To find the time τ at which the charges collide, we can use the equations of motion for both charges.
Let's consider the positive charge, denoted as q1, located at position r1 = r0, and the negative charge, denoted as q2, located at position r2 = -r0.
The electric force between the two charges is attractive and given by Coulomb's Law:
F = k * |q1 * q2| / r^2
where k is the Coulomb's constant (8.99 * 10^9 N m^2/C^2), q1 = +1 μC = 1 * 10^-6 C, q2 = -1 μC = -1 * 10^-6 C, and r is the distance between the two charges. In this case, r is equal to 2 times the magnitude of r0.
The gravitational force acting on each charge can be neglected since its magnitude is significantly smaller than the electric force.
Using Newton's second law, F = ma, where m is the mass of each charge and a is their common acceleration, we can write the equations of motion for the charges:
For q1:
k * |q1 * q2| / (2 * r0)^2 = m * a
For q2:
k * |q1 * q2| / (2 * r0)^2 = -m * a
Since the charges have opposite signs, they accelerate towards each other with the same magnitude but in opposite directions.
Plugging in the given values, we have:
k * |(1 * 10^-6 C) * (-1 * 10^-6 C)| / (2 * (1 m))^2 = (1 g) * a
Simplifying the equation:
8.99 * 10^9 N m^2/C^2 * 10^-12 C^2 / 4 m^2 = (0.001 kg) * a
8.99 * 10^9 N / 4 = a
2.2475 * 10^9 N = a
Now we can find the time it takes for the charges to collide.
From the equations of motion, we know that the distance traveled by each charge is given by:
d = 1/2 * a * t^2
where d is the distance traveled, a is the acceleration, and t is the time.
For q1 and q2, the distance traveled will be equal to the magnitude of r0, i.e., |r0| = 1 m.
Substituting the values:
1 = 1/2 * (2.2475 * 10^9 N) * t^2
Simplifying:
2 = (4.495 * 10^9 N) * t^2
t^2 = 2 / (4.495 * 10^9 N)
t^2 ≈ 0.44487 * 10^-9 s^2
Taking the square root of both sides:
t ≈ 0.66690 * 10^-5 s
Therefore, the time τ at which the two charges collide is approximately 0.66690 * 10^-5 seconds.
To find the time τ at which the charges collide, we can use principles of electrostatics and classical mechanics.
First, let's consider the forces acting on each charge. Since the charges have opposite signs, they will experience an electrostatic attraction towards each other. The force between two point charges q1 and q2 separated by a distance r is given by Coulomb's Law:
F = k * |q1 * q2| / r^2
where k is the electrostatic constant.
In this case, the force between the charges is:
F = k * |q * (-q)| / (2r)^2
= k * q^2 / (4r^2)
Now, let's consider the forces acting on each charge individually. The force acting on each charge can be related to its acceleration using Newton's second law:
F = m * a
where m is the mass of the charge and a is its acceleration.
Since the charges have the same mass and they are experiencing the same magnitude of attraction force, their accelerations will be equal in magnitude. Let's denote this common acceleration as a.
Using Newton's second law, we have:
m * a = k * q^2 / (4r^2)
Plugging in the given values: m = 1g = 0.001kg, q = 1μC = 1e-6C, and r = 1m, and the value of electrostatic constant k = 9 * 10^9 N m^2 / C^2, we can solve for the acceleration a:
a = k * q^2 / (4r^2 * m)
= (9 * 10^9 N m^2 / C^2) * (1e-6C)^2 / (4 * (1m)^2 * 0.001kg)
Now, let's find the time τ at which the charges collide. Since both charges are released from rest at t = 0, their initial velocities are zero. We can use the equations of motion:
v = u + at
s = ut + (1/2)at^2
where v is the final velocity, u is the initial velocity, a is the acceleration, s is the displacement, and t is the time.
In this case, the initial displacement between the charges is 2r, and at the time of collision, the displacement will be 0. Therefore, we can write the equation:
0 = 2r + (1/2)at^2
Simplifying this equation, we have:
(1/2)at^2 = 2r
at^2 = 4r
t^2 = 4r / a
Now, we can substitute the values of r and a into this equation to find the time τ:
t^2 = 4 * (1m) / ((9 * 10^9 N m^2 / C^2) * (1e-6C)^2 / (4 * (1m)^2 * 0.001kg))
Simplifying this expression and taking the square root, we can find the value of τ.