Two electrons start at rest with a seperation of 5e-12 metres. Once released, the electrons accerelate away from each other. Calculate the speed of each electron when they are a very large distance apart.
To calculate the speed of each electron when they are at a very large distance apart, we can use the principles of conservation of energy.
Initially, the electrons are at rest, so the total initial kinetic energy is zero. As they move away from each other, the electrostatic force causes them to repel each other, doing work on the electrons and increasing their kinetic energy.
The work done by the electrostatic force is equal to the change in potential energy between the starting point and the final point when they are at a very large distance apart.
The potential energy of two charged particles separated by a distance can be calculated using the formula:
PE = k * (q1 * q2) / r
where
PE is the potential energy,
k is the electrostatic constant (8.99 × 10^9 N*m^2/C^2),
q1 and q2 are the charges of the two particles,
r is the separation distance between them.
In this case, since both electrons have the same charge (q1 = q2 = -1.6 × 10^-19 C) and the separation distance is 5 × 10^-12 m, we can calculate the change in potential energy (ΔPE).
ΔPE = PE_final - PE_initial
ΔPE = 0 - (k * (q1 * q2) / r)
ΔPE = - (8.99 × 10^9 N*m^2/C^2) * ((-1.6 × 10^-19 C)²) / (5 × 10^-12 m)
Now, since the total initial kinetic energy is zero, the final kinetic energy for each electron will be equal to the absolute value of the change in potential energy.
KE = ΔPE
KE = 8.99 × 10^9 N*m^2/C^2 * (1.6 × 10^-19 C)^2 / (5 × 10^-12 m)
KE ≈ 4.589 × 10^-18 J
The kinetic energy of an object can be calculated using the formula:
KE = (1/2) * m * v^2
where
KE is the kinetic energy,
m is the mass of the object,
v is the velocity/speed of the object.
In this case, the mass of each electron (m) is approximately 9.1 × 10^-31 kg.
Thus, we can rearrange the formula and solve for v (velocity/speed):
v = sqrt((2 * KE) / m)
v = sqrt((2 * 4.589 × 10^-18 J) / (9.1 × 10^-31 kg))
Calculating this equation, we find that the speed of each electron when they are a very large distance apart is approximately 6.66 × 10^6 m/s.
To calculate the speed of each electron when they are a very large distance apart, you can apply the principle of conservation of energy.
Initially, the electrons are at rest, and hence their initial kinetic energy is zero. As they accelerate away from each other, they gain kinetic energy, which is balanced by a decrease in their potential energy.
The potential energy between two point charges can be calculated using the formula:
U = (k * q1 * q2) / r
where U is the potential energy, k is the electrostatic constant (9 x 10^9 Nm²/C²), q1 and q2 are the charges (in this case, the charge of an electron, which is -1.6 x 10^-19 C), and r is the separation between the electrons.
As the electrons move apart, their potential energy decreases, and their total energy remains constant. So, the decrease in potential energy is converted into an increase in kinetic energy.
To find the speed when they are very far apart, we can assume that their final potential energy is negligible (close to zero). Therefore, we can equate the initial potential energy to the final kinetic energy:
(k * q1 * q2) / r = (1/2) * m * v^2
where m is the mass of an electron (9.1 x 10^-31 kg) and v is the final speed of each electron.
Rearranging the equation, we can solve for v:
v = √ [(2 * (k * q1 * q2)) / (m * r)]
Now let's substitute the values into the equation:
v = √ [(2 * (9 x 10^9 Nm²/C² * -1.6 x 10^-19 C * -1.6 x 10^-19 C)) / (9.1 x 10^-31 kg * 5 x 10^-12 m)]
v ≈ 5.16 x 10^6 m/s
Therefore, each electron will have a speed of approximately 5.16 x 10^6 m/s when they are a very large distance apart.