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
Here is one way to do this.
I assume you are in calculus. Lets pretend we can do this charge by charge.
First charge:
work done=INT force dx from x=.03 to inf
where force=kq2q1/x^2
integrate it over that domain of x.
So that is the energy this partical has.
SEcond charge.
Now, with the first charge is gone, the second charge doesn't move.
So the total work in the system is what we figured on the first charge. So if both particles were released, this energy would have been divided evenly, so now we know the KE of each particle
KE=1/2 work done above
1/2 m v^2=1/2 work done above
solve for v
I am not in calculus. Is there any other way to to this through algebra?!
calculate potential energy when close to each other.
You probably have the formula which Bob Pursley just told you how to derive with calculus by integrating the force
it is something like
U = k Q1 Q2 / R
The (1/2)m v^2 of each particle is half of that
To determine the speed of each charge when they are very far away from each other, we can consider the conservation of energy.
Initially, the charges are at rest, and they have electrostatic potential energy given by:
U = k * q₁ * q₂ / r
Where:
- U is the electrostatic potential energy,
- k is the electrostatic constant (approximated to 9 * 10^9 N * m² / C²),
- q₁ and q₂ are the charges,
- r is the distance between the charges.
Since the charges are identical, we can substitute q₁ = q₂ = q and rewrite the potential energy as:
U = (k * q²) / r
When the charges are very far away from each other, the distance between them can be considered infinite (r → ∞). At this point, the electrostatic potential energy at infinity is zero.
Therefore, the total energy of the system is conserved:
E = U + K
Where:
- E is the total energy of the system,
- U is the electrostatic potential energy,
- K is the kinetic energy.
Since the charges start from rest, their initial kinetic energy is zero. Therefore, the total energy of the system is equal to the initial electrostatic potential energy:
E = U = (k * q²) / r
At a very far distance, when the charges are no longer interacting, the potential energy is zero. At this point, all the initial potential energy is converted into kinetic energy:
E = K
Therefore, we have:
(k * q²) / r = (1/2) * m * v²
Where:
- m is the mass of each charge,
- v is the velocity of each charge.
We can now solve for v:
v² = [(2 * k * q²) / (m * r)]
Taking the square root yields:
v = √[(2 * k * q²) / (m * r)]
Now we can substitute the given values:
- k = 9 * 10^9 N * m² / C²,
- q = 9.5 * 10^(-6) C,
- m = 2.0 * 10^(-6) kg,
- r = 0.035 m (since the charges were initially 3.5 cm apart).
Plugging in these values, we get:
v = √[(2 * (9 * 10^9) * (9.5 * 10^(-6))²) / ((2.0 * 10^(-6)) * (0.035))]
Calculating this expression will give us the speed at which each charge will be moving when they are very far away from each other.