# Energy due to a charge is infinite?

1. ### A Dhingra

207
energy due to a charge is infinite??

hello..

My professor said "Law of conservation of energy is not applicable to waves." This puzzled me a lot and trying to think otherwise i have a doubt regarding energy due to electric field. It seems that if energy due to electric field and magnetic combined, this might turn out to be true for light.
Let us have a charge Q at rest, whose electric field exists in the whole space. If a test charge,q i.e. Q>>q, is placed in this region, then due to repulsion it will move away from the charge Q. It will continue this motion till it is at infinite distance. This suggests that the total energy gained by the charge q from Q is infinite, further suggesting that the energy due to the electric field is not getting used up.... i.e. i can bring as many test charges as i want but this repulsion will exist, no matter how much energy the previous one has taken. So how is that so?

What kind of energy is possessed by the electric field created by a charge? Is it never ending, and suggesting that it never got created (may be when the charge or the electric field was formed)? in that sense this need not obey the laws of conservation of energy!

may be what i have asked is very silly, but any help is appreciated..

### Staff: Mentor

Re: energy due to a charge is infinite??

This is incorrect. Simply because the distance is infinite does not imply that the energy is infinite.

What is the general expression for work? What does the energy depend on besides the distance? What happens if this other quantity goes to 0 and is that the situation here?

3. ### Simon Bridge

15,261
Re: energy due to a charge is infinite??

Welcome to PF;

What you want to investigate is how much energy it takes to accelerate a charged particle compared with a neutral particle.
[Dalespam has shown you the way to challenge the argument you were provided with.]

4. ### A Dhingra

207
Re: energy due to a charge is infinite??

Work is W=∫F.dx .
So in it if i apply the limits some distance r to ∞, i would get,
W = ∫(kQq/x2) dx = -kQq/x, x from r to ∞,
= -kqQ(1/∞-1/r) = -kQq/r..
this is definitely not ∞.

But I was thinking that the test charge is moving continuously (with non uniform, decreasing acceleration) almost for infinite time. In that case the sum of all the small patches under the curve should have given infinite energy(I was considering kinetic energy, on v-t graph)....but that doesn't seem to be happening...

5. ### facenian

216
Re: energy due to a charge is infinite??

As was already pointed out to you(by Dalespam) this process does not render an infinite ammount of energy but besides that this energy is not produced by the rest charge Q alone, this energy is assigned to the system of two charges and it is yust the energy recovered from the one used to assemble the two charges.
However there is a problem if you think of the energy of the charge Q alone since this energy is indeed infinite. This is a known problem of electrodynamics and is an indication that classical electrodynamics is not logically consistent unlike newtonian mechanics which does not suffer from consistency problems.

### Staff: Mentor

Re: energy due to a charge is infinite??

Exactly! Well done.

Again, you need to carry out the appropriate integral. However, you seem to think that the area under a v-t graph represents energy. It does not.

Think about what the appropriate integrand is, consider impulse, momentum, and the relationship between momentum and KE. Look at that integrand, is there some part of it that makes you think it might not be infinite? If you can, perform the integral.

Last edited: Feb 6, 2013
7. ### Khashishi

Re: energy due to a charge is infinite??

You must have misquoted your professor or taken eir words out of context. Law of conservation of energy applies everywhere, except possibly on a cosmological scale. (The latter uncertainty is due to our lack of understanding of gravity, ad hoc inflaton fields, and dark energy.)

Last edited: Feb 6, 2013
8. ### Simon Bridge

15,261
Re: energy due to a charge is infinite??

Or the prof is using a pedagogical technique hoping to stimulate inquiry, a bit like the famous goldfish problem (why does a goldfish not displace any water?) ... the statement itself looks like an oblique reference to what facenian was talking about.