# Charge attached to a spring

Guys check this, its really interesting. Lets imaging a charge attached to a spring of charge q and mass m, and imagine strecht the spring a distance a, from its equilibrium position. Now the charge its being accelerating by the spring and therefore by maxwell electromagnetic theory it will radiate, the energy that the charge will radiate , will come out of its mechanical energy, and because of this the charge should stop, after radiated all the initial mechanical energy. But if i write the equation of motion, i only have the force of the spring accting on the charge, so if i solve the equation of motion, this equation will predditc that charge will oscillate for ever. So from the fact that the charge is radiating, i should consider some dissipated force, in order to reproduce in the equation of motion, what energy conservation says. But if the charge its alone in the universe, its just being influence by the spring, and nothing else, so perhpas its an auto force??? can anyone tell me, How should i write my equation of motion, and what its this dissipative force im not considering?

Thx for the help and sorry if my english its not the best

## Answers and Replies

olgranpappy
Homework Helper
It's called the "reaction force" and is proportional to the "jerk" (the first time derivative of the velocity). For more info see textbooks. For example, Heitler's book: "The Quantum Theory of Radiation" the chapter on Classical Theory of Radiation section 4.

It's called the "reaction force" and is proportional to the "jerk" (the first time derivative of the velocity).

"acceleration" is the first derivative of velocity w.r.t. time. "jerk" is the time derivative of acceleration or the 2nd derivative of velocity.

olgranpappy
Homework Helper
"acceleration" is the first derivative of velocity w.r.t. time. "jerk" is the time derivative of acceleration or the 2nd derivative of velocity.

righto, that's what i meant to write--slip of the pen.

So from the fact that the charge is radiating, i should consider some dissipated force, in order to reproduce in the equation of motion, what energy conservation says.
I am very attracted to springs. They are made of spiralling thread resistive of torsion, beacause bending of the thread itself is easy. This torsion has as counterpart: warming up. The dissipation must be in the warming up of the thread that is my answer.
greetings Janm

The concept of a charge on a spring is a good one, and the spring does not have to be dissipative for the system to lose energy. In Panofsky and Phillips "Classical Electricity and Magnetism" chapter 19 "Radiation from an Accelerated Charge" the authors go through the Lienard-Wiechert potential and vector triple products to show that there is a classical (non-quantum) field that drops off as 1/r and not 1/r^2, meaning power is being radiated. Panofsky and Phillips does have an equation giving the total radiated power of an accelerating electron (it has two terms proportional to acceleration squared) that can be put into the equation for an oscillating charge (with mass) on a spring.

In Panofsky and Phillips "Classical Electricity and Magnetism" chapter 19 "Radiation from an Accelerated Charge".
Hello Bob
I have the second edition of the book and try to find out which equation you mean. Is it par. 19-4 "The "convection potential",
which starts with the Lorentz expression: F=e(E+u x B) and then equation (19-25)?
Years ago I studied the Lienard-Wiechert potentials a little, attracted to the concept that field-changes propagate with the velocity of light.
greetings Janm

I have the second printing (first edition) of Panofsky and Phillips dated August 1856. I was referring to equation 19-37 in paragraph 19-5 "Radiation with no restrictions on acceleration or velocity" on page 308. In my book, the paragraph on convection potential is # 18-4 on page 293.

I have the second printing (first edition) of Panofsky and Phillips dated August 1856. I was referring to equation 19-37 in paragraph 19-5 "Radiation with no restrictions on acceleration or velocity" on page 308. In my book, the paragraph on convection potential is # 18-4 on page 293.

So your book comes from 1856. 13 years after W.R.Hamilton found his quaternions and 17 years before maxwell wrote his "treatise on electricity and magnetism". I come back on the equations you mentioned and by the way my research of the Lienard Wiechert potentials is based on Laundau L.D. and Lifschitz : the classical theory of fields (translated from the Russian) A. Wesley 1960 par 62.