# Where does the energy to accelerate particle in an electric field come from?

Consider a hollow conducting tube with a highly positively charged object close to one of its ends. The medium between the conducting tube and the charged object is strong enough to prevent an electric breakdown. Now, if I pass protons through the tube, they don't feel any force from the charged object as they are inside a conductor. (Since there is a hole, the charges feel a negligible force on them. The hole is small enough so that only a negligible or tiny part of the electric field enters the hollow region inside of it.) When the particles exit the tube, they now feel a large force on them as they are near the charged object.

The induced surface charge will alter the electric field. Now, all the electric field lines are perpendicular to the conductors surface and the side of the tube closer to the charged body has a negative charge.

Can the particles leave the conducting tube, or will they be attracted to the conductors negative charge that is induced. If they do leave, they will be accelerated to relativistic speeds, assuming the charged body is charged enough.). So, where is this energy coming from? Since the larger the number of protons that pass through the tube, the more energy is required to accelerate them. I am assuming that the protons would actually collide with the surface of the conductor just after they leave it, unless they have sufficient kinetic energy to start with. Even then, I don't understand where that potential energy between the charged body and the protons is coming from or how it is created all of a sudden.

• It is not clear why you think protons will be accelerated to relativistic speeds. If they left the tube and are approaching positively charged object, they will be slowing down at first. A diagram of the system could help to make your idea clear. In any case, in general, kinetic energy gained by charged particles in electric field is accompanied by loss of equivalent EM energy in their vicinity. – Ján Lalinský Jan 18 '17 at 22:29
• When the protons exit the tube, the are beside the charged body. Hence, the particles accelerate(which were previously in a really small electric field in the conductor). They only accelerate when they leave the tube. I know they slow down as the conductor is not Ideal and it is not a closed surface. But it reduces the energy needed to bring the charge near the charged body a lot. I dont understand where the energy to accelerate the protons is coming from as we do not add enough potential energy by performing work on the charge to bring it close to the charged object. – Chandrahas Jan 19 '17 at 12:57
• If protons accelerate due to external electric field, they gain their energy at the expense of the EM field energy. There need not be any external work done. The energy is already in the system, it just changes form - from EM energy to kinetic energy of particles. – Ján Lalinský Jan 19 '17 at 17:43
• How does the EM energy show up if there is no work done? – Chandrahas Jan 20 '17 at 14:38
• The EM energy is already there, because there are many interacting charged particles in the system. The extractability of energy is not always apparent when the system does not change - when the stored energy stays the same we tend to forget that it can be extracted when the conditions are right. It's like with one-time battery you can buy in a supermarket. Nobody stored the energy into it by doing work. The energy was already there in the chemicals when the battery was assembled. These chemicals by themselves store energy. No work in the physics sense was necessary to get them. – Ján Lalinský Jan 20 '17 at 22:57