Why does an accelerated charge radiate away energy?

Question

My textbook says:

"Neils Bohr objected to the idea of an electron orbiting a nucleus in a circular orbit. An electron experiences centripetal acceleration and an accelerated charge radiates away energy. So such an orbit would be unstable: the electron would spiral into the nucleus."

But why does a charge radiate away energy when accelerated? From my understanding of circular motion, if the electron is in circular motion, then the centripetal force that is acting on the electron only changes the electron's direction and not its linear velocity. And hence the electron's kinetic energy should remain constant.

Therefore, if the energy that the charge radiates shouldn't come from the kinetic energy then what type of energy would it radiate away when accelerated and why? Thanks.

Answer 1

It emits light, because it "stirs up" the electromagnetic field. To understand this, just dip your finger into a still pond and move it in a circle. Water waves will emanate from your finger. These waves have energy, which means energy is being taken away from you. Same goes for the charges.

In fact, this follows almost automatically from the finite propagation speed of light. The electric field of a stationary charge obeys Coulomb's law. If the charge suddenly starts moving, the field won't obey Coulomb's law anymore, but it can't instantly change everywhere because of the finite propagation speed. Instead a "shockwave" of information goes out from the charge at speed $c$. This shockwave contains electromagnetic energy and travels at the speed of light -- it is light.

Answer 2

An accelerating charge radiates energy because, according to Maxwell's equations, it produces an electromagnetic wave.

what type of energy would it radiate when accelerated and why?

In addition to kinetic energy, the electron-nucleus system also has energy stored in the electric field between the electron and the positively charged nucleus.

Answer 3

From my understanding of circular motion, if the electron is in circular motion, then the centripetal force that is acting on the electron only changes the electron's direction and not its linear velocity. And hence the electron's kinetic energy should remain constant.

We could speculate that the electron's kinetic energy would remain constant, if it did not radiate. But Bohr's whole argument was that the electron would lose its orbit, i.e., would lose its kinetic energy because it would radiate if it was orbiting the atom.

Therefore, if the energy that the charge radiates shouldn't come from the kinetic energy then what type of energy would it radiate away when accelerated and why?

Based on the argument above, Bohr, presumably, was thinking that the radiation energy would come from the electron's kinetic energy. This premise is supported by an observation on a macro level that an electron rotating in a uniform magnetic field is gradually losing its energy and spirals down.

The electron radiates electromagnetic energy and it radiates it because it accelerates. The type of radiation specifically associated with the circular motion, i.e., due to the centripetal acceleration, is known as a synchrotron radiation, because it occurs in synchrotrons and other circular particle accelerators and is considered a drawback due to the loss of energy associated with radiation.

The radiation power of a charged particle due to its acceleration is quantified by the Larmor formula:

$P=\frac {q^2a^2} {6\pi\epsilon_0 c^3},$

where $a$ is acceleration of a particle. More details related to the radiation due to the circular motion could be found in this Wikipedia article.

Answer 4

Why don't orbital electrons fall into the nucleus when they move with acceleration? Larmor's relation is usually used to calculate the relevant force, but it only makes sense for acceleration in the direction of movement. Orbits only have centrifugal acceleration, which is perpendicular to the motion, so the Larmor relation does not apply to them. The energy emitted by the orbitals is negligible, and is it replaced by the interaction with the electromagnetic field of the atom?

Bohr's model of the atom solve a non-existent problem. But this model opened the door to a land of no return; many scientists are fascinated by the "beauty of this land"?

Answer 5

From my understanding of circular motion, if the electron is in circular motion, then the centripetal force that is acting on the electron only changes the electron's direction and not its linear velocity. And hence the electron's kinetic energy should remain constant.

Being on the place of the moving around a nucleus electron, you will feel a force like in a carousel. The force drags you outwards, however, if the connection to the carousel is cut off, you fly tangentially away from the turning circle. Feeling a force, you are under acceleration and hence a circular motion (except the motion in free space around a massive body) is an acceleration.

To make it even more visible, if you are sitting in a car blindfolded and the driver accelerates or turns right or left you are able to decide between this acceleration and the turns. But, if during your ride your car seat will be turned by 90°, you will call the turns falsely acceleration and braking; and the accelerations you will feel as the turns. In reality, blindfolded you couldn’t anymore follow your experiences and the circular motion is not decidable from linear acceleration.

But why does a charge radiate away energy when accelerated?

What you mention was found out when electrons are moving in a magnetic field. In the case, the trajectory of the electron is perpendicular to the magnetic field, the electron undergoes a turn perpendicular to both the magnetic field and the direction of it’s movement. It was observed that this time the electron emits photons and its kinetic energy decreases. The trajectory of the electron becomes a spiral path until the kinetic energy gets converted fully to radiation and the electron stops in the center of the spiral path.

No rope nor magnetic field in the case of the nucleus and the electron; the electron simply couldn’t circle around the nucleus and Bohr’s model was discarded.