Electrostatic Potential Concept

formal

This is the maths formulation alright, and so I suppose it applies also to gravitation.
and I suppose it applies also in the case we do not start from infinity. (point b)


Imagine we have in vacuum 55200 esu,0.0000148 C ,positive charge ( 4x 10^14 at 1m).
An electron is at rest at r0= 10^7m/ (^9 cm.) (acc= 4 m/sec^2, 400cm/s^2).
If we move it to r = 6.4x 10^6m/ (^8cm) (acc= 9.8 m/s^2 ,980cm/s^2)
is work done on the charge the same if v changes? or is it useless and we need only Maths and
KE= Δ PE = 2.25 x 10^ 7 ?

That is to say the same KE an electron would get anyway in a free fall from r0 to r?

formal

thank you for the link.
but the point is: is the definition given in the thread ,(you said it is correct), just a formal definition, meaning nothing in concrete.? or we can really use a test charge to measure Δ PE? or if we moved the two conductors of a capacitor?

In a cyclotron you have a set difference in E-PE, and every time a charge makes a jump from a Dee it gets the same amount of KE, independently of its velocity. Is that correct?

Dickfore

It is a formal definition. Potential energy is not operationally defined physical quantity, but a derived quantity.

formal

Thank you!
That is just what worried me.It really did not make much sense.
Now could you comment the two examples I made.
Is KE = 2.25x 10 ^7 J, final v= 1.8x 10^ 4 m/s, 18 Km/ sec if electron is at rest ? and much more if it is not?

And in a cyclotron why the increase in KE is the same at every passage even if speed is always different?
(P.S. should we say derived or derivative quantity?)

Dickfore

I can, but I won't.

Because it always passes through the same potential difference between the duants.

formal

That is why I asked for your comment on the other case.
In that case, in vacuum, an electron at rest gets less KE than an electron with v>0

Where is the difference?

Dickfore

The difference is that moving an electron in the field of another point charge has nothing to do with the concept of a cyclotron.

formal

Thank you,
could you, please , explain how in the example at post #18, a rapidly moving charge's magnetic field would influence work done on it?

Naty1

Going back to the OP:

This is the answer you are looking for...

It's just an operational definition for convenience. The reference could have been chosen as -100 volts at infinity; very often in circuit problems earth is assigned "zero" volts as the reference potential...

formal

Thank you,
the value at infinity is really irrelevant.
The point is the remainder of the definition:

is velocity important or not? if it is not , why bother?
if it is (as it figures), in what way is it relevant? is it a matter of MF?

please see previous post #25, and also 18,19

Naty1

It is important NOT to change the velocity of the test particle much. That's because in the definition of potential energy you want to measure the work done against the static potential...NOT the work done to accelerate the particle related to kinetic energy...which must be kept small by comparison.

The work done in the static case is not dependent on the path taken.

If you move a test particle with a positive charge to closer proximity with a positvely charged particle it takes positive work....If of opposite charge, negative work....negative potential results...analogous to a (attractive) gravitational field where particles come closer together.

formal

I think there is a misunderstanding, probably I haven't made it clear enough
I am not talking of (1) changing v or (2) changing path.
I am talking of absolute value of v, I stated it repeatedly

if a charge as v= 1, 10, 100 , does it make any difference?
is post #6 correct or false?

3) if a body in a gravitational field is at rest or has v = 1 or = 10 it matters! and how!
the greater is v, the greater is KE acquired (work done)
a charge in vacuum behaves differently from the same charge between two Dee's ? why?

Delta²

No this is not true either. As long as the field is not time- varying the force that does the work will not be time-varying either so the work done by the force as defined by

[tex] \int_{C} F(r)dr[/tex] will not depend on time, it will depend only on the path C (and not how long it takes to travel through that path). Moreover since the electrostatic field is conservative it will depend only on the start and end points of the path C.

Dadface

There is a simple high school analogy.If you lift a mass M through a height h against a uniform gravitational field of field strength g then the gain of gravitational potential energy is Mgh.This change is dependant on M and g and the initial and final positions only and is independant of the method of lifting.If the mass is lifted infinitely slowly at an angle the change is Mgh.If it is fired vertically up at a billion metres per second then as the mass pases through a height h the gain of PE is again Mgh.Whatever method is used energy is conserved.

thebiggerbang

OP here. I guess that the velocity of the charge that is being moved should tend to zero (vanishingly small) as if it is not so, then it implies that at some point of time the applied force was more than the force due to the electric field, there was no equilibrium.

Also, as we can't calculate the absolute energy of a system, we always consider the change in the energy of the system. For all practical purposes, we always use potential difference, and not the absolute potential. This difference will be always independent of the initial position of the charge (infinity on this case).

Mathematically, Potential difference to move a charge from A to B is given by

[itex]\Delta[/itex]P=P[itex]_{AB}[/itex]=P[itex]_{\infty B}[/itex]-P[itex]_{\infty A}[/itex]
=P[itex]_{B}[/itex]-P[itex]_{\infty}[/itex]-P[itex]_{A}[/itex]+P[itex]_{\infty}[/itex]
=P[itex]_{B}[/itex]-P[itex]_{A}[/itex]

I hope the formulation is correct

Studiot

Not exactly but almost.

Yes, as I said before, it is way of stating that none of the electric potential energy is converted to kinetic energy or that the electric force does not speed up the test charge.

However some other agent may be causing motion,independent of the electric effect.

Consider.

I float up to 10000ft in a balloon and stay there in equilibrium.

The balloon and contents have a certain PE due to the altitude..

If I now start the motor and propel the balloon horizontally at 1 mile/hr does the PE change if the altitude remains constant?

If I accelerate the balloon to 10 miles/hr does this change the PE?

formal

1) (EB) the explanation one reliable text gives for the motion being slow is:
"...this is the only case in which motion does not, of itself, cause work to be done elsewhere in the universe" ...." the vector curl E (del x E) must be zero"


2) , 3) If we move (a balloon or) a charge (horizontally or) in a normal direction to force E, PE , of course, does not change because work is not done against the force (as r0 -r = 0)(4*)

( 4*) text says work: W = q(o) * q/ 4π ε0 (1/ r0 -1/ r ) )

now if definition EB is correct, could you or someone help to interpret it?
in your previous post (#17) you interpret slowly as steadily, but if they wanted to mean steady they would just say steady.
If it is so, (and slow does not mean 'not accelerated') the absolute value of v is relevant:
OP pinpoints his previous 'without acceleration' to 'vanishingly small'. That is correct!That is what the definition is all about!

But the best way to explain it is to say what happens if v is greater, I suppose!
everyone has his own view: is post #6 correct? is post #24 correct? is post #20 correct?
Now consider this:

2,3) if you deflate your (balloon A) charge q(a) (if q(o)= 0.0000184 C)
will drop vertically and after 1 second it will gain
acc= v = 9.8 m/sec and KE = W(A)
if another (balloon B) charge q(b) is already (falling) moving alongside it at v 9.8 m/ sec it will change
its v from 9.8 to 19.6 m/ sec gaining KE W(B) > W(A) (= 4 W(A)) , while
in a cyclotron W(A)= W(B)=..W(C)...

In conclusion we have a (formal?, hypothetical?, meaningless?) definition of Electrostatic PE which states
that absolute value of v is relevant, whereas (as stated correctly in post #31)
it is not at all relevant in a gravitational field and
it is not at all relevant in an Electric field between two Dee's