B Understanding the Concept of Infinity as a Reference in Physics: Explained

[Nader AbdlGhani]

I'm facing a problem in my physics course which is accepting that infinity can be a reference point in both Electrostatics (calculating the voltage of a point) and Matter Properties (calculating the gravitational potential energy), how come we use a reference point which we don't know where it is, keep in mind that I don't have any problems dealing with infinity when we plug it in a mathematical relation, what I want is to understand the physical concept of choosing infinity as a reference.

 

[A.T.]

I want is to understand the physical concept of choosing infinity as a reference.

There is no general physical concept. It's often just a convenient convention

 

[PeroK]

I'm facing a problem in my physics course which is accepting that infinity can be a reference point in both Electrostatics (calculating the voltage of a point) and Matter Properties (calculating the gravitational potential energy), how come we use a reference point which we don't know where it is, keep in mind that I don't have any problems dealing with infinity when we plug it in a mathematical relation, what I want is to understand the physical concept of choosing infinity as a reference.

"A point at infinity" is simply a point far enough away that going any further would make a negligible difference to the system. E.g. the point at infinity is far enough away that the potential energy is a maximum (for GPE or attractive charges) or a minimum (for repulsive charges).

 

[Nader AbdlGhani]

There is no general physical concept. It's often just a convenient convention

Ok then, I can't get over that "convenient convention", and please tell me, what makes it legit ?

 

[Nader AbdlGhani]

"A point at infinity" is simply a point far enough away that going any further would make a negligible difference to the system. E.g. the point at infinity is far enough away that the potential energy is a maximum (for GPE or attractive charges) or a minimum (for repulsive charges).

Thanks for your reply, but can you tell me how we are able to calculate for instance the voltage of a point charge having a charge Q, it's coordinates are (X,Y) while setting our reference point as infinity ?

 

[PeroK]

Thanks for your reply, but can you tell me how we are able to calculate for instance the voltage of a point charge having a charge Q, it's coordinates are (X,Y) while setting our reference point as infinity ?

Voltage is the difference in electric potential, so the question is how to define electric potential. One definition of electric potential for a point charge $Q$ is:

$U = \frac{Q}{4\pi \epsilon_0 r} \$, where $r$ is the distance from the charge.

This gives a function of $r$ that tends to $0$ as $r \rightarrow \infty$. And, in many ways, this is the most natural and useful definition, given the relationship between $U$ and $r$. I'm not sure I would say this uses $\infty$ as a reference point, though.

You could equally well define:

$U = U_0 + \frac{Q}{4\pi \epsilon_0 r} \$, where $U_0$ is some constant.

If $Q$ is negative (or if $Q$ is positive and $U_0$ is negative), you will have some radius $r_0$ where $U(r_0) = 0$. But, it's not really making $r_0$ special.

 

[PeroK]

PS How you define $U(r)$ doesn't change the critical fact that the function $U(r)$ never attains its max or min, but tends to one of these as $r \rightarrow 0$ and the other as $r \rightarrow \infty$. In a sense, $r \rightarrow \infty$ has a physical meaning whether you like it or not!

 

[A.T.]

how come we use a reference point which we don't know where it is

This might be the core of your confusion: We aren't using the position as reference, just the finite value at which some function converges.

 

[ayans2495]

Voltage is the difference in electric potential, so the question is how to define electric potential. One definition of electric potential for a point charge $Q$ is:

$U = \frac{Q}{4\pi \epsilon_0 r} \$, where $r$ is the distance from the charge.

This gives a function of $r$ that tends to $0$ as $r \rightarrow \infty$. And, in many ways, this is the most natural and useful definition, given the relationship between $U$ and $r$. I'm not sure I would say this uses $\infty$ as a reference point, though.

You could equally well define:

$U = U_0 + \frac{Q}{4\pi \epsilon_0 r} \$, where $U_0$ is some constant.

If $Q$ is negative (or if $Q$ is positive and $U_0$ is negative), you will have some radius $r_0$ where $U(r_0) = 0$. But, it's not really making $r_0$ special.

Hello. Why are we giving it an arbitrary constant in the first place?

 

[ayans2495]

Voltage is the difference in electric potential, so the question is how to define electric potential. One definition of electric potential for a point charge $Q$ is:

$U = \frac{Q}{4\pi \epsilon_0 r} \$, where $r$ is the distance from the charge.

This gives a function of $r$ that tends to $0$ as $r \rightarrow \infty$. And, in many ways, this is the most natural and useful definition, given the relationship between $U$ and $r$. I'm not sure I would say this uses $\infty$ as a reference point, though.

You could equally well define:

$U = U_0 + \frac{Q}{4\pi \epsilon_0 r} \$, where $U_0$ is some constant.

If $Q$ is negative (or if $Q$ is positive and $U_0$ is negative), you will have some radius $r_0$ where $U(r_0) = 0$. But, it's not really making $r_0$ special.

Hello. Why do we use the arbitrary constant may I ask? What function does it serve? Is it the initial electric potential at that reference point?

 

[PeroK]

Hello. Why do we use the arbitrary constant may I ask? What function does it serve? Is it the initial electric potential at that reference point?

Mathematically, the potential when differentiated gives the electric field. And an anti-derivative has an arbitrary constant.

Physically, only the difference in potential is important, so you can add a constant without changing the physics. And, yes, this is equivalent to choosing a certain reference point as having zero potential.

 

[vanhees71]

Hello. Why do we use the arbitrary constant may I ask? What function does it serve? Is it the initial electric potential at that reference point?

None. That's an important point. The electrostatic potential is only determined up to an arbitrary additive constant without any physical significance. Only potential differences are related to physical observables. That's why you can choose the additive constant arbitrarily, and it is usually convenient to choose it such that the potential goes to zero at infinity.

 

[Demystifier]

I'm facing a problem in my physics course which is accepting that infinity can be a reference point in both Electrostatics (calculating the voltage of a point) and Matter Properties (calculating the gravitational potential energy), how come we use a reference point which we don't know where it is, keep in mind that I don't have any problems dealing with infinity when we plug it in a mathematical relation, what I want is to understand the physical concept of choosing infinity as a reference.

It's a reference for theorists, not for experimentalists. We cannot measure at infinity, but we can put $r\to \infty$ in equations.

 

[vanhees71]

We cannot measure the absolute electric potential anyway, as stressed above, but only potential differences ("voltages") between two points separated by a finite distance in the lab. So theory and experiment are in no contradiction here but perfectly match (as it should be)!