Two electrons are fiXed 2 cm apart .another eleCtron is shot from infinity and stops midway between the two .what is the initial speed?
To solve this problem, we can use the principles of electrostatic forces and conservation of energy.
First, let's assume that the three electrons are aligned in a straight line. We are given that the two fixed electrons are 2 cm apart, so their separation distance (r) is 2 cm or 0.02 m.
When the third electron is shot from infinity and stops midway between the two fixed electrons, it means that the electrostatic force between the third electron and the two fixed electrons is exactly equal.
Using Coulomb's law, the electrostatic force between two electrons can be given by:
F = k * (q1 * q2) / r^2
Where:
- F is the electrostatic force
- k is the electrostatic constant (9 x 10^9 N m^2/C^2)
- q1 and q2 are the charges of the interacting particles
- r is the separation distance between the particles
In this case, both the two fixed electrons and the third electron have an equal magnitude of charge, which we can denote as e (the elementary charge). Therefore, q1 = q2 = e.
Since the electrostatic forces between the third electron and the two fixed electrons are equal, we can write:
F1 = F2
Using Coulomb's law and substituting the known values:
k * (e * e) / (r1^2) = k * (e * e) / (r2^2)
Since r1 = r2 = r (0.02 m), we have:
k * (e * e) / (r^2) = k * (e * e) / (r^2)
The charges and electrostatic constant cancel out, leaving us with:
1 / (r^2) = 1 / (r^2)
This equation tells us that the electrostatic forces are equal, regardless of the charges involved.
Now, let's consider the conservation of energy. At infinity, the initial kinetic energy (K1) of the third electron is zero. As it moves closer to the fixed electrons, it comes to a stop and converts all its initial kinetic energy into potential energy (U).
The potential energy can be calculated using the formula:
U = k * (q1 * q2) / r
Substituting the known values:
U = k * (e * e) / r
To find the initial speed of the third electron, we need to equate the initial kinetic energy (K1 = 0) to the potential energy (U):
0 = U
Since U = k * (e * e) / r, we have:
0 = k * (e * e) / r
Rearranging the equation:
e * e = 0
This means that the charge of the electron, e, must be zero. However, this contradicts our knowledge that electrons do have a charge. Therefore, this scenario is not physically possible, and we cannot determine the initial speed of the third electron.
find potential energy of stopped electron .01 meters from another one.
k e^2/.01
Double that.
That is the work done to bring the electron in from infinity.
That is the kinetic energy required at infinity (1/2) m v^2