Relation between angular momentum and total engery

This gives us KE = .5m(L/(ma))^2 = .5L^2/(m^2a^2). Substituting this into our expression for total energy, we get E = -e^2/(4πε0r) + .5L^2/(m^2a^2). Finally, we can rewrite this in terms of L and a, which gives us the desired expression for the total energy of an electron in the Coulomb potential of a proton: E = -.5L^2/(m^2a^2) - e^2/(4πε0r).In summary, the total energy of an electron in
  • #1
steely
1
0

Homework Statement


a) Express the total energy of an electron in the Coulomb potential of proton through the electron's angular momentum L and the shortest distance a between the proton and the electron's orbit. Hint: The electron's velocity is perpendicular to it's position vector whenever it is distance a away from the proton.

b) For a fixed L, minimize the expression found in (a) with respect to a. Show that the minimum corresponds to the case of a circular orbit. State the minimum value of the total energy for fixed L.

Homework Equations


L=sqrt(l2+l)ћ
Lz=mlћ
L=r x mv=Iω
En=-mee4/[2(4πε0)2ћ2n2]
n > l > |ml| > 0
KE=.5Iω2

The Attempt at a Solution


E=U+KE => E= .5L*v/a - e^2/[4πε0a]
I have little confidence in that as a solution. Either I've stopped short or I'm going in the wrong direction, I'm not sure. I really only want help with part (a)... Once I get that far I should be handle (b) somewhat easily, I just wanted to provide context for the problem.
 
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  • #2


a) To express the total energy of an electron in the Coulomb potential of a proton, we need to consider the potential energy and kinetic energy of the electron. The potential energy is given by the Coulomb potential, which is proportional to the inverse of the distance between the electron and the proton, and the kinetic energy is given by the electron's angular momentum.

Let's start by looking at the potential energy. The Coulomb potential is given by V = -kq1q2/r, where k is the Coulomb constant, q1 and q2 are the charges of the two particles, and r is the distance between them. In this case, q1 is the charge of the electron and q2 is the charge of the proton. We can rewrite this as V = -e^2/(4πε0r), where e is the charge of the electron and ε0 is the permittivity of free space.

Next, let's consider the kinetic energy. We know that the electron's velocity is perpendicular to its position vector when it is a distance a away from the proton. This means that the electron's angular momentum, L, is given by L = mvr, where m is the mass of the electron and v is its speed. We can rewrite this as L = mva, since the velocity is perpendicular to the position vector, which has a magnitude of a.

Now, we can combine the potential and kinetic energies to get the total energy, E. This gives us E = V + KE = -e^2/(4πε0r) + .5L^2/mva^2. However, we want to express this in terms of the electron's angular momentum, L, and the shortest distance, a, between the proton and the electron's orbit. We can do this by substituting in our expression for L, which gives us E = -e^2/(4πε0r) + .5(mva)^2/mva^2. Simplifying this, we get E = -e^2/(4πε0r) + .5mv^2, since the a's cancel out.

But we still have r in our expression, and we want to express everything in terms of a. To do this, we can use the fact that the angular momentum is constant for a given orbit, which means that L = mva = constant. This means that v = L/(ma), and
 

Related to Relation between angular momentum and total engery

What is the relation between angular momentum and total energy?

The relation between angular momentum and total energy is described by the conservation of energy and angular momentum principles. In a closed system, the total energy, which includes kinetic energy and potential energy, remains constant while the angular momentum remains constant in both magnitude and direction.

How is angular momentum related to rotational motion?

Angular momentum is a measure of an object's tendency to continue rotating in its current direction. It is directly proportional to the object's moment of inertia and angular velocity. As the moment of inertia increases, the angular velocity decreases, resulting in a constant angular momentum.

What happens to the angular momentum if an external torque is applied?

If an external torque is applied to a rotating object, the angular momentum will change in magnitude or direction or both. This change in angular momentum is equal to the external torque applied multiplied by the time during which it acts.

How does the conservation of energy apply to rotational motion?

The conservation of energy states that the total energy in a closed system remains constant. In rotational motion, this means that the sum of kinetic energy and potential energy remains constant as long as there is no external torque acting on the system.

What is the significance of the relation between angular momentum and total energy?

The relation between angular momentum and total energy is significant because it allows us to predict the motion of objects in rotational systems. By understanding how changes in angular momentum affect the total energy of a system, we can make predictions and calculations for a variety of scenarios in physics and engineering.

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