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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.

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  • $\begingroup$ I think your question begs for a logical correction. Your core argument is that since the centripetal force in a circular motion acts at right angles to the velocity, it cannot change the kinetic energy and thus, the particle shouldn't be able to radiate away any energy because if the kinetic energy is not changing then where can the energy to be radiated away come from? But this argument is valid only for the case in which the particle is in a circular motion and not for a generic accelerated motion. So, that is the logical correction that I think must be pointed out in your question. $\endgroup$ – Dvij Mankad Sep 4 '18 at 4:10
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    $\begingroup$ See en.wikipedia.org/wiki/Larmor_formula $\endgroup$ – J.G. Sep 4 '18 at 5:46
  • $\begingroup$ Consider you hold a charge and another charge is attracted to it, but is held by a spring. Nothing is moving, no energy is emitted. Now, if you move you charge a bit closer to the other charge, that charge would attract to your charge stronger and stretch the spring a bit more. It takes energy to stretch the spring, where did this energy come from? Clearly from your hands, but how? Through the electromagnetic interaction between the charges. In other words, this energy was emitted by your charge and absorbed by the other charge. This would happen even if you move your charge along a circle. $\endgroup$ – safesphere Sep 4 '18 at 6:41
  • $\begingroup$ A charge in circular motion would be essentially an alternating electric current. $\endgroup$ – Dmitry Grigoryev Sep 4 '18 at 8:09
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    $\begingroup$ Iti s basic that change in direction also means change in velocity, altough it doesn't change $\\vec{v}|$ (modulus), but direction changes, so there is an acceleration. $\endgroup$ – FGSUZ Sep 5 '18 at 10:36
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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.

enter image description here

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.

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  • $\begingroup$ thanks for this Knzhou, and for the very useful drawing. I have seen it a couple of times before and always wondered what the magnetic field looks like in the vicinity of the kink. Can you sketch it out for me? -Niels $\endgroup$ – niels nielsen Sep 4 '18 at 4:25
  • $\begingroup$ @nielsnielsen For a plane wave $\mathbf{E} = \mathbf{v} \times \mathbf{B}$, so similarly here I think the magnetic field is pointing in/out of the page, with magnitude proportional to $|\mathbf{E}|$. $\endgroup$ – knzhou Sep 4 '18 at 4:36
  • $\begingroup$ This Phet simulation might be of interest? phet.colorado.edu/sims/radiating-charge/… $\endgroup$ – Farcher Sep 4 '18 at 6:32
  • $\begingroup$ Thank you very much for the answer. You mentioned "because it 'stirs up' the electromagnetic field," do you mean that an accelerated charge will 'stir up' its own electric and magnetic field? Therefore in the case of an electron in orbital, is it that the electron stirs up its own electric and magnetic field? $\endgroup$ – Bøbby Leung Sep 4 '18 at 9:42
  • $\begingroup$ @BøbbyLeung I wouldn't say that. There is just one electromagnetic field, not one electromagnetic field for every charge. If a flashlight emits some light and it travels millions of miles into space, "whose" field is it? $\endgroup$ – knzhou Sep 4 '18 at 19:34
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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.

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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.

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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.

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