An electron and a proton are initially very far apart (effectively an infinite distance apart). They are then brought together to form a hydrogen atom, in which the electron orbits the proton at an average distance of 5.11 10-11m. What is EPEfinal - EPEinitial, which is the change in the electric potential energy?
They attract, so the potential energy will go down as they get close. (the result will be negative)
the force is a constant *1/r^2 = k/r^2
(You can do the charges and Coulomb's law)
So the real question is what is the integral from r = infinity to r = R (which is given as 5.11*10^-11) of 1/r^2 dr ?
int from infinity to R of dr/r^2 = -1/r at R - (-1/r) at infinity
= -1/R
now use that with k Q1 Q2 /r^2
to get
- k (charge of electron)^2 / R
for the change in potential energy coming in from infinity.
Of course that loss of potential went somewhere, so the thing must have that amount of kinetic energy, which is probably the next problem.
To calculate the change in electric potential energy (EPE) between two charged particles, you need to use the formula:
ΔEPE = EPEfinal - EPEinitial
In this case, the electron and the proton are initially very far apart, effectively at an infinite distance. At infinite separation, the electric potential energy between them is zero since they do not interact.
However, when they come together to form a hydrogen atom, the electron orbits the proton at an average distance of 5.11 x 10^(-11) m. To calculate the electric potential energy of the system at this distance, you need to use the equation for electric potential energy:
EPE = (k * q1 * q2) / r
Where:
- k is Coulomb's constant (approximately 9 x 10^9 N m^2 / C^2)
- q1 and q2 are the charges of the particles (in this case, the charge of an electron and a proton)
- r is the separation between the particles (5.11 x 10^(-11) m)
Since the electron and proton have equal magnitude but opposite charges, the product q1 * q2 will be the same for both particles, but with opposite signs.
Therefore, to calculate the change in electric potential energy, substitute the values into the formula:
ΔEPE = EPEfinal - EPEinitial
ΔEPE = [k * (-e) * (+e) / (5.11 x 10^(-11))] - 0
ΔEPE = -2 * (k * e^2) / (5.11 x 10^(-11))
Here, e represents the elementary charge, which is approximately 1.6 x 10^(-19) C.
Now, you can substitute the known values into the equation and calculate the change in electric potential energy.