# Electron Under Gravity: Lose Potential Energy in First Centimeter

• Reshma
In summary, the electron loses potential energy as it falls under the influence of gravity. This energy is radiated away in the form of electromagnetic radiation.
Reshma

## Homework Statement

An electron at rest is released from rest and falls under the influence of gravity. In the first centimeter, what fraction of potential energy lost is radiated away?

## Homework Equations

Lienard-Wichert potential for the electron of charge e is given by:
$$\phi = \frac{e}{R(1 - \beta \cdot \hat R)}$$

In this case the charge is accelerated by a gravity (a = g).

## The Attempt at a Solution

Reference: Electrodynamic Radiation by Marion and Heald

The problem hasn't mentioned whether the speed of the electron is relativistic.

If the speed of the electron is less than c ($\beta << 1$) then $R(1 - \beta \cdot \hat R) \rightarrow 0$ and the potential can be written as:
$$\phi = \frac{e}{R}$$
The accelerated field can be given as:
$$\vec E = \frac{e}{c^2 R^3}\left((\vec R \cdot \vec g)\vec R - R^2\vec g\right)$$

I don't know how the potential energy can be calculated here. If the direction of g and R are the same, shouldn't E = 0?

I assumed a non-relativistic case here. Am I going wrong here?

Reshma said:
The problem hasn't mentioned whether the speed of the electron is relativistic.
Do you really think that something can reach relativistic speeds after falling for 1 centimeter?

Hint: Look up the Larmor formula for energy radiated by an accelerating charge.

Doc Al said:
Do you really think that something can reach relativistic speeds after falling for 1 centimeter?

Hint: Look up the Larmor formula for energy radiated by an accelerating charge.

Thanks, I got it! Here goes:
Considering the motion of the electron along the y-axis.
The dipole moment is $\vec p = -ey\hat y$ and $y^2 = {1\over 2}gt^2$
Hence,
$$\vec p = -{1\over p}get^2\hat y$$

$$\ddot{\vec p} = -{1\over p}ge\hat y$$

By Larmor's formula, the radiated power is given by,

$$P = {2\over 3c^3}(ge)^2$$

Now, the time t it takes to fall a distance 'h' is given by: $h = {1\over 2}gt^2$

$t = \sqrt{{2h\over g}}$

So the energy released in falling a distance 'h' is: E = Power x time

$$E_{rad.} = Pt = {2g^2e^2\over 3c^3}(\sqrt{{2h\over g}})$$

Meanwhile the potential energy lost is PE = mgh.

So the fraction radiated is:
$$f = {E_{rad}\over E_{pot}} = {2g^2e^2\over 3c^3}(\sqrt{{2h\over g}} \times {1\over mgh} = {2e^2\over 3mc^3}(\sqrt{{2g\over h}})$$

Just another question, why does the dipole moment appear when there is only a single charge involved?

Last edited:

## 1. What is an electron under gravity?

An electron under gravity refers to the motion of an electron in a gravitational field, which causes the electron to experience a force due to the Earth's gravitational pull. This force can affect the electron's potential energy and alter its movement.

## 2. How does an electron lose potential energy in the first centimeter?

An electron loses potential energy in the first centimeter as it moves closer to the Earth's surface. This is because the gravitational force between the electron and the Earth increases as the distance between them decreases. As a result, the electron's potential energy decreases and it gains kinetic energy, causing it to accelerate towards the Earth.

## 3. What factors affect the amount of potential energy an electron loses under gravity?

The amount of potential energy an electron loses under gravity depends on the mass of the electron, the mass of the Earth, and the distance between them. A heavier electron or a more massive Earth will result in a greater potential energy loss, while a larger distance between the electron and the Earth will result in a smaller potential energy loss.

## 4. Can an electron gain potential energy under gravity?

Yes, it is possible for an electron to gain potential energy under gravity. This can occur if the electron moves away from the Earth's surface, increasing the distance between them and decreasing the strength of the gravitational force. As a result, the electron's potential energy will increase while its kinetic energy decreases.

## 5. How does the potential energy of an electron under gravity relate to its motion?

The potential energy of an electron under gravity is directly related to its motion. As the electron loses potential energy, it gains kinetic energy and accelerates towards the Earth. Similarly, as it gains potential energy, it loses kinetic energy and decelerates away from the Earth. This relationship follows the principle of conservation of energy, where energy cannot be created or destroyed, only transferred from one form to another.

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