An electron and a proton are a distance r = 8.5 × 10-9 m apart. How much energy is required to increase their separation by a factor of 6?
k Q1*Q2*[1/r - 1/(6r)] = ke^2*[5/(6r)]
k is the Coulomb's Law constant and e is the electron charge
To find the energy required to increase the separation between the electron and proton by a factor of 6, we can use the formula for the electrostatic potential energy between two charged particles:
U = k * (q1 * q2) / r
Where U is the potential energy, k is the Coulomb's constant (8.99 × 10^9 N m^2/C^2), q1 and q2 are the charges of the particles, and r is the separation distance.
In this case, the charges of an electron and a proton are opposite in sign, with the same magnitude, q1 = e and q2 = -e, where e is the elementary charge (1.6 × 10^-19 C).
To increase their separation by a factor of 6, the new separation distance is r_new = 6 * r.
Substituting the values into the formula, we have:
U_new = k * (q1 * q2) / r_new
U_new = k * (e * (-e)) / (6 * r)
U_new = k * (e^2) / (6 * r)
Now, let's calculate the energy required:
U_new = (8.99 × 10^9 N m^2/C^2) * ((1.6 × 10^-19 C)^2) / (6 * (8.5 × 10^-9 m))
U_new = 0.3192 J
Therefore, the energy required to increase the separation between the electron and proton by a factor of 6 is approximately 0.3192 Joules.
To find the energy required to increase the separation between the electron and proton by a factor of 6, we can use Coulomb's law and the formula for electrical potential energy.
Coulomb's law states that the force between two charged particles is given by the equation:
F = k * (q1 * q2) / r^2
where F is the force, q1 and q2 are the charges of the particles, r is the separation distance, and k is the electrostatic constant (k ≈ 9 * 10^9 Nm^2/C^2).
From Coulomb's law, we know that the force between the electron and proton is attractive (since they have opposite charges), and it decreases as the separation distance increases.
The electrical potential energy (E) between two charged particles is given by the equation:
E = k * (q1 * q2) / r
To find the energy required to increase the separation by a factor of 6, we need to calculate the initial energy (E_initial) and the final energy (E_final) and then find the difference between them.
1. Initial separation (r_initial) = 8.5 × 10^-9 m
2. Final separation (r_final) = 6 * r_initial
Now let's calculate the initial energy (E_initial) and final energy (E_final):
1. E_initial = k * (q_electron * q_proton) / r_initial
2. E_final = k * (q_electron * q_proton) / r_final
Note: In this case, the charges of the electron and proton are q_electron = -1.6 * 10^-19 C and q_proton = 1.6 * 10^-19 C, respectively.
Substituting the values into the equations, we can calculate the initial and final energies:
1. E_initial = (9 * 10^9 Nm^2/C^2) * ((-1.6 * 10^-19 C) * (1.6 * 10^-19 C)) / (8.5 × 10^-9 m)
2. E_final = (9 * 10^9 Nm^2/C^2) * ((-1.6 * 10^-19 C) * (1.6 * 10^-19 C)) / (6 * (8.5 × 10^-9 m))
By evaluating these formulas, we can find the values for E_initial and E_final. Finally, subtracting E_initial from E_final will provide the amount of energy required to increase the separation by a factor of 6.