# Reference Electrostatic Potential

What is the criteria in choosing infinity as zero potential ?

e.g. an infinite plate with uniform charge density.

What is the physical meaning of not be able to choose a position as reference potential ??

My understanding of it is that, mathematically, because the functional form of the Coulombic potential is $$V = \frac{kq}{r}$$ no real number results in a value of zero for the potential. The only "value" of r that satisfies this is infinity. Physically, as I understand it, it means that the electromagnetic force has an infinite range, because the electric potential has a value all the way out to infinity.

You can chose whatever you like for zero potential, the only way to connect electrostatic potential to something physical is to take its gradient, so it is underconstrained up to a constant term.

V or V + 301.20 will both lead to the same electric field, E = -grad(V) == -grad(V + 301.20). So whether you define the potential to be zero or 301.20 at infinity doesn't matter. Likewise, you can put the zero potential wherever you want. Put it in Norway for all the physics cares, you'll get the same electric field as a result.

You might just spend more time doing unnecessary mathematics.

If I have an infinitely large conductor plate with uniform charge density.

E = σ/2ε (suppose it is in x direction)

V = -∫E dx (from x0 to x)

V = -(σ/2ε)(x - x0 )

From this expression, I can't choose reference at infinity (i.e. x0 --> infinity) because the whole expression V would be infinity.

Is it because there is infinitely many charges ??

Essentially, any time you actually have CHARGES out at infinity, you won't be able to take infinity as your reference point, just like you can't take the location of a point charge as a reference point. The potential will blow up in either case.

Nugatory
Mentor
If I have an infinitely large conductor plate with uniform charge density.

As mikeph said, you can choose the zero point of potential to be anywhere you want, so we generally choose whatever makes the problem easiest to solve.

For the infinite plate with uniform charge density, it's most convenient to take the surface of the plate as the point of zero potential.

rude man
Homework Helper
Gold Member
What is the criteria in choosing infinity as zero potential ?

e.g. an infinite plate with uniform charge density.

What is the physical meaning of not be able to choose a position as reference potential ??

The reason infinity is chosen as zero potential is to be able to say that "the potential of an object is the work necessary to bring that object from infinity to the position of that potential". This holds for any conservative field like gravity or electrostatics.

You can't always have zero potential at infinity, for example a charged wire of infinite length cannot have zero potential at infinity. Same for your infinite charged plate.

From this expression, I can't choose reference at infinity (i.e. x0 --> infinity) because the whole expression V would be infinity.

How?

rude man
Homework Helper
Gold Member
If I have an infinitely large conductor plate with uniform charge density.

E = σ/2ε (suppose it is in x direction)

V = -∫E dx (from x0 to x)

V = -(σ/2ε)(x - x0 )

From this expression, I can't choose reference at infinity (i.e. x0 --> infinity) because the whole expression V would be infinity.

Is it because there is infinitely many charges ??

Yes. When dealing with infinities, life can get complicated! As you found out, you cannot assign zero potential to your plate at infinity.

You can assign it as zero at the plate as another poster has suggested, or you can assign zero potential to any point a finite distance from the plate, but not an infinite distance.

BTW the E field for a conductor is E = σ/ε. It's σ/2ε for a charged dielectric sheet.