This video is about charges in orbit.
We learned in subtopic 7.3 that in the rather forward model of the atom, negatively charged
electrons orbit the positively charged nucleus in circular orbits.
Even though this model is not fully correct, which we will discuss in more detail in subtopic
12.1, for now we'll work with this model and assume that the electron remains in a
circular orbit due to the force between the positively charged nucleus and the negatively
charged electron.
Let's examine the simplest atom, which is the hydrogen atom.
In the hydrogen atom, a single electron orbits a proton.
Let's assume that the charge of the electron is negative lowercase q, the charge of the
proton is positive uppercase q, the distance between the electron and the proton, so the
orbital radius is r and that the mass of the electron is m.
In subtopic 5.1 we learned about Coulomb's law.
In this subtopic we returned to this law and once again saw the equation that expresses
this law.
In subtopic 6.1 we learned about circular motion.
We discovered that when an object moves in a circle at constant speed, a centripetal
force is present.
Here is the equation to calculate the centripetal force.
In our example here, the centripetal force is provided by the force that the proton exerts
on the electron.
Therefore we can write that mb squared over r is equal to k uppercase q times lowercase
q divided by r squared.
You might have noticed that I omitted the negative sign that would appear on the right
hand side of this equation when we substitute negative q for the charge of the electron.
The negative sign here shows that the force between the electron and the proton is attractive.
Since for this calculation we are interested in the magnitude of the force, I omitted the
negative sign from the equation.
Moving further we can simplify by r and after dividing by m and taking the square root we
get that v is equal to k times uppercase q times lowercase q divided by mr under the
square root.
So this is the expression for the orbital speed when a charge orbits another charge.
In our example here we can take this one step further because the magnitude of the electrons
and the protons charge is the same.
It is the elementary charge.
This means that we can replace both uppercase q and lowercase q by e, the symbol of the
elementary charge.
This gives us that v is equal to k times e squared over mr under the square root.
Let's move on to calculate the total energy of the orbiting electron.
Et, the total energy of the electron, has two components, potential energy and kinetic
energy.
Here is the equation that we learned for electrical potential energy earlier in this subtopic
and here is the kinetic energy equation from subtopic 2.3.
Substituting we get that Et is equal to k times positive q times negative q divided
by r.
Since here potential energy is negative, I will not omit the signs of the charges when
calculating the potential energy.
To this we will add the kinetic energy which is 1 half times m times v squared, where v
is the orbital speed of the electron that we just found.
So we get k times uppercase q times lowercase q divided by mr under the square root to the
power of 2.
Working further we have that Et is equal to negative k uppercase q lowercase q over r
plus 1 half and in the expression for the kinetic energy the square root and the square
cancel and we can also cancel m giving us 1 half times k times uppercase q times lowercase
q divided by r.
Combining the two terms we get that Et is equal to negative k times uppercase q times
lowercase q over 2r, which again considering that the charge of the electron and the proton
is equal to the elementary charge, we get that the total energy is equal to negative
k times e squared over 2r.
Note that both the orbital speed and the total energy are very similar to what we found
for the orbital speed and the total energy for a satellite that is in orbit around the
planet.
This once again shows that fields in physics have very similar characteristics.
This wraps up our discussion about charges in orbit and completes the final video of
Subtopic 10.2.
In the first video of Subtopic 11.1 we'll start learning about electromagnetic induction.