Potential Difference at One Location
Written by Alex George
The Main Idea
Usually, we are interested in the potential difference between two different points. However, it is also useful to determine potential at a single location.
In this case, we must define the potential at a location (ex. Location A) as the potential difference between a point infinitely far from all other charged particles and the location we defined.
As potential has a value at all locations at space, it is a scalar field in it of itself.
A Mathematical Model
Potential at One Location
[math]\displaystyle{ V_{a} = V_{a} - V_{\infty } }[/math] where [math]\displaystyle{ V_{a} }[/math] is the potential location of interest and [math]\displaystyle{ V_{\infty } }[/math] is the potential at a location infinitely far away.
This equation is only valid when [math]\displaystyle{ V_{\infty } }[/math] is equal to zero.
Potential near a Point Charge
Parameters: A point charge with charge [math]\displaystyle{ q }[/math]. We are a distance [math]\displaystyle{ x }[/math] from this point charge and we want to find the potential difference between our point charge and the potential at infinity.
[math]\displaystyle{ V_{x} = V_{x} - V_{\infty } = -\int_{\infty }^{x} \frac{1}{4\pi\varepsilon _{0} } \frac{q}{x^2} dx = \frac{1}{4\pi\varepsilon _{0} } (\frac{q}{x}-\frac{q}{\infty}) = \frac{1}{4\pi\varepsilon _{0}} \frac{q}{x} }[/math]
The sign of the potential depends on the sign of the charge on the particle:
if [math]\displaystyle{ q \lt 0, V_{x} \lt 0 }[/math]
if [math]\displaystyle{ q \gt 0, V_{x} \gt 0 }[/math]
Potential Energy at a Single Location
Once the value of the potential at one location is known, the potential energy of a system can be calculated.
If a charged object was placed at the location where the potential is known, then the potential energy of the system would be given the the equation:
[math]\displaystyle{ U_{A} = qV_{A} }[/math]
Substituting in the derived equation from the previous section for [math]\displaystyle{ V_{A} }[/math], we obtain the following:
[math]\displaystyle{ U_{A} = q_{1} (\frac{1}{4\pi\varepsilon _{0}} \frac{q_{2}}{x}) }[/math] where [math]\displaystyle{ x }[/math] is the separation between the two particles, and [math]\displaystyle{ q_{1} }[/math] and [math]\displaystyle{ q_{2} }[/math] are the respective charges of the particles.
A Computational Model
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