# Increasing distance between pt-charges increases voltage diff between pts near them?

1. Sep 26, 2011

### DocZaius

Today I was reflecting about the statement that increasing the distance between two large charged plates increases voltage, when I started to wonder what the behavior was near point charges (more voltage? as much? less?).

Q and -Q are charges. P1 and P2 are locations.

Q @ x=0m
P1 @ x=1m
P2 @ x=2m
-Q @ x=3m

$V=\frac{q}{4\pi\epsilon_{0}r}$
V=0 at r=infinity.

Let |Q| = $4\pi\epsilon_{0}$ so that being 1 meter from +Q (and only +Q) gives 1 V.

It's clear that voltage at $V_{P1}=\frac{1}{2}$ and $V_{P2}=-\frac{1}{2}$ and that $\Delta V=1$

Now increase the distance between the two points and charges. There are now 2 meters between the points, but each is just as close to their respective charges.

Q @ x=0m
P1 @ x=1m
P2 @ x=3m
-Q @ x=4m

Now: $V_{P1}=\frac{2}{3}$ and $V_{P2}=-\frac{2}{3}$ and $\Delta V=\frac{4}{3}$

As P2 and -Q get moved far away (but with P2 staying 1 meter from -Q), it's clear that $\Delta V$ will approach 2.

Is it fair for me to deduce from this that increasing distance between point charges also increases voltage between points near them? If so, this would be surprising as I thought that only increasing distance between infinitely large plates increased voltage, because of the constant field intensity near infinitely large plates. I did not know this effect could also work for the simple case where field intensity actually drops with distance (as it does with point charges).

The implication is that if I wanted to give a charged particle maximum kinetic energy as it accelerates between these two charges, I would prefer these two charges to first be as distant as possible. This seems somehow counter-intuitive. Also, every time I see someone talk about increased distance resulting in increased voltage, it is for the case of charged plates. The more fundamental case of point charges also obeying this rule would seem to me to warrant priority as an example of this behavior.

Last edited: Sep 26, 2011
2. Sep 26, 2011

### Curl

Re: Increasing distance between pt-charges increases voltage diff between pts near th

If you have q and -q and you separate them further, then yes the voltage increases,. It is not linear though like for charged planes.

3. Sep 27, 2011

### fizzle

Re: Increasing distance between pt-charges increases voltage diff between pts near th

Remember, voltage is the line integral of the electric field. Even though the electric field between the two points is less due to the increased separation (1.33X) of the charges, the increased distance in the integral (2X) more than makes up for it.

Here's a nifty calculator to determine the voltage and field at a point given one or two charges:

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/e2p.html