Two identical +9.5uF point charges are initially 3.5cm from each other.
If they are released at the same instant from rest, how fast will each be moving when they are very far away from each other? Assume they have identical masses of 2.0mg (P.s. I am not in calculus) so the algebraic way would be great!
As in the problem below if you do not know how to integrate you must look up the formula. It looks like:
U = potential energy = k Q1 Q2/R
To determine the final velocities of the charges when they are very far away from each other, we can use the principle of conservation of energy. The initial potential energy between the charges will be converted into kinetic energy as they move apart, causing them to accelerate.
First, we need to calculate the initial potential energy of the system. The potential energy between two point charges can be given by the equation:
PE = k * (Q1 * Q2) / r
Where:
- PE is the potential energy
- k is the electrostatic constant (8.99 x 10^9 Nm^2/C^2)
- Q1 and Q2 are the charges (which are identical and given as +9.5 μC)
- r is the separation between the charges (given as 3.5 cm, convert to meters by dividing by 100: 0.035 m)
PE = (8.99 x 10^9 Nm^2/C^2) * [(9.5 x 10^-6 C) * (9.5 x 10^-6 C)] / 0.035 m
Now, let's calculate the initial potential energy.
PE = 8.99 x 10^9 * [(9.5 x 10^-6) * (9.5 x 10^-6)] / 0.035
PE ≈ 2.43 J (rounded to two decimal places)
Since the system starts from rest, this potential energy will be converted entirely into kinetic energy when the charges are very far apart.
According to the principle of conservation of energy, the energy is given by:
KE = (1/2) * m * v^2
Where:
- KE is the kinetic energy
- m is the mass of each charge (given as 2.0 mg, convert to kilograms: 2.0 x 10^-6 kg)
- v is the final velocity of each charge
Since the charges are identical, the kinetic energy of each charge would be identical when they are very far apart. Therefore, we can write:
KE = 2 * [(1/2) * (2.0 x 10^-6 kg) * v^2] (using the multiplication factor of 2 for identical charges)
Now, equating the initial potential energy to the final kinetic energy:
PE = KE
2.43 J = 2 * [(1/2) * (2.0 x 10^-6 kg) * v^2]
2.43 J = (2.0 x 10^-6 kg) * v^2
Divide both sides by (2.0 x 10^-6 kg):
v^2 = (2.43 J) / (2.0 x 10^-6 kg)
v^2 ≈ 1.215 x 10^6 m^2/s^2
Now, take the square root of both sides:
v ≈ √(1.215 x 10^6 m^2/s^2)
v ≈ 1102.8 m/s
Therefore, each charge will be moving with a speed of approximately 1102.8 m/s when they are very far away from each other.
Note: This is an approximate calculation considering the given data.