Potential Difference at One Location
Written by Kayden Less SPRING 26'
The Main Idea
Electric potential is a fundamental concept in electrostatics that describes the electric potential energy per unit charge at a point in space. It provides a way to analyze electric interactions using energy methods rather than forces, which often makes many problems easier to solve. Instead of tracking vector forces from multiple charges, electric potential allows those contributions to be added as scalars, making superposition problems much more manageable.
For a point charge, the electric potential at a distance r is given by
V= r kq
where k is Coulomb’s constant, q is the source charge, and r is the distance from the charge. This shows that potential decreases with distance and depends on both the magnitude and sign of the source charge.
Electric potential is closely related to work and energy. The change in electric potential between two points (voltage) tells how much work is done per unit charge moving between those locations. This relationship is central to understanding charged particle motion, circuits, capacitors, and energy storage.
Electric potential also connects directly to electric fields through the relationship
E =−∇V
showing that electric fields point in the direction of decreasing potential. This connection links energy-based reasoning and field-based reasoning into one framework.
The concept becomes especially powerful when dealing with systems of multiple charges, continuous charge distributions, and conductors, where direct force calculations may be difficult. Because of this, electric potential is one of the most important tools in electromagnetism and serves as a foundation for many later topics in physics and engineering.
A Mathematical Model
The 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 = q_{1} (\frac{1}{4\pi\varepsilon _{0}} \frac{q_{2}}{x}) = (\frac{1}{4\pi\varepsilon _{0}} \frac{q_{1}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 Visual Model
A visualization of Distance vs. Potential from a point charge is given by Figure 1:
Figure 1. Potential vs. Distance for a Point charge with Charge [math]\displaystyle{ e }[/math]
The electric field of the point charge can be obtained by differentiation. The negative gradient of the potential with respect to distance would yield the electric field:
[math]\displaystyle{ E_{x} = -\frac{\partial V}{\partial x} }[/math]
Here is a video explaining the above concepts with more examples [1].
Examples
Here are a couple of problems that illustrate the concepts presented above.
Simple Examples
Example 1 - Potential at a Location Near a Sphere
What is the electric potential 5 cm away from a 1 cm diameter metal sphere that has a -3.00 nC static charge?
Solution:
As the distance from the center of the sphere is greater than the radius, we can treat the metal sphere as a point charge.
From the above derivation of the potential for a single point charge, we have:
[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} } \bigg(\frac{q}{x}-\frac{q}{\infty} \bigg) }[/math]
[math]\displaystyle{ V_{x} = 9\times 10^{9} \frac{N\cdot m^{2}}{C^{2} } \bigg(\frac{-3.00\times 10^{-9}\ C}{5.00\times 10^{-2}\ m}-\frac{-3.00\times 10^{-9}\ C}{\infty}\bigg) = 9\times 10^{9} \frac{N\cdot m^{2}}{C^{2} } \frac{-3.00\times 10^{-9}\ C}{5.00\times 10^{-2}\ m} = -539 V }[/math]
Example 2 - Charge Required to Produce a Specific Potential
What is the excess charge on a Van de Graff generator that has a 25.0 cm diameter metal sphere that produces a voltage of 50 kV near the surface?
Solution
The potential of the surface will be the same as that of a point charge from the center of the sphere, 12.5 cm away. The excess charge can be derived from the equation for potential a distance away from a point charge:
[math]\displaystyle{ V_{x} = \frac{1}{4\pi\varepsilon _{0} } \bigg(\frac{q}{x}\bigg) }[/math]
[math]\displaystyle{ q = {4\pi\varepsilon _{0}} x V_{x} }[/math]
[math]\displaystyle{ q = \frac{1}{9\times 10^{9}} \frac {N\cdot m^{2}}{C^{2}} (12.5 \times 10^{-2} \ m) (50 \times 10^{3} \ V) = 6.94 \times 10^{-7} \ C }[/math]
Difficult Examples
Example 1 - Potential Along the Axis of a Ring
What is the potential a distance [math]\displaystyle{ z }[/math] from the center of the ring?
Solution:
The electric field of a ring is given by the following equation:
[math]\displaystyle{ E_{z,ring} = \frac{1}{4\pi\varepsilon _{0} } \frac {Qz}{(R^{2}+z^{2})^{\frac {3}{2}}} }[/math] where [math]\displaystyle{ z }[/math] is the distance from the center of the ring, [math]\displaystyle{ R }[/math] is radius of the ring, and [math]\displaystyle{ Q }[/math] is the charge on the ring.
This equation can be integrated with respect to [math]\displaystyle{ z }[/math] and the appropriate bounds to obtain the potential at a location [math]\displaystyle{ z }[/math] from the center of the ring:
[math]\displaystyle{ V_{z,ring} = -\int_{\infty}^{z} \frac{1}{4\pi\varepsilon _{0} } \frac {Qz}{(R^{2}+z^{2})^{\frac {3}{2}}} dz }[/math]
[math]\displaystyle{ V_{z,ring} = \frac{1}{4\pi\varepsilon _{0} } \frac {Q}{\sqrt{R^{2}+z^{2}}} }[/math]
Connectedness
- How is this topic connected to something that you are interested in? This topic is related to the concept of Potential Energy, or the capacity to do work in a system. Electric Potential is a derivation of potential energy and is a convenient way to express potential in terms of electric potential energy per unit charge. I am interested in bio instrumentation, so this concept has direct relevance in terms of circuit analysis and design.
- How is it connected to your major? My major is biochemistry and electric potential has several applications in biological systems. For instance, in mechanisms such as the firing of a neuron's action potential, the electric potential across the neuronal membrane must reach certain thresholds in order to propagate the neural signal.
- Is there an interesting industrial application? Electric potential has various applications in industry, particularly in the aspects of circuit design and analysis. A particular implementation in medicine would be a defibrillator.
Real-World Applications
Electric potential appears in many technologies and physical systems.
Capacitors store electrical energy through potential differences and are used in electronics, camera flashes, and energy storage devices.
Particle accelerators use voltage differences to accelerate charged particles for research and medical applications.
Electrostatic precipitators use electric potential to remove pollutants from industrial exhaust.
Power transmission systems rely on high voltages to reduce resistive losses during long-distance energy transfer.
Electric potential is therefore not only a theoretical concept, but also a practical engineering tool.
Common Exam Mistakes
Students often confuse electric potential with electric field.
Common mistakes include:
Using V = kq/r instead of distinguishing it from E = kq/r². Forgetting electric potential is a scalar quantity and adds algebraically. Ignoring the sign of charge when determining whether potential is positive or negative. Mixing up electric potential V and electric potential energy U. Forgetting potential is path independent and depends only on endpoints.
Exam Tip:
For multiple charges,
V_net = Σ(kq_i/r_i)
Use superposition and add potentials algebraically.
Difficult Example 2 — Superposition of Potential
Suppose two charges contribute to the electric potential at point P.
q₁ = +5 μC at r₁ = 0.20 m
q₂ = −3 μC at r₂ = 0.10 m
Using superposition:
V = k(q₁/r₁ + q₂/r₂)
Substituting:
V = (9 × 10⁹)[(5 × 10⁻⁶ / 0.20) − (3 × 10⁻⁶ / 0.10)]
V = -4
History
The SI unit for electric potential is the volt, which is named in honor of the Italian physicist Count Alessandro Volta (1745-1827), who invented the voltaic pile, the first chemical battery.
Volta had developed the voltaic pile using an effective pair of metals (zinc and silver) to create a steady electric current in the 1880s.
Today, the "conventional" volt is used and was defined in 1988 by the 18th general conference on Weights and Measures and adopted in 1990. The Josephson effect was employed to produce an exact frequency to voltage conversion and a cesium frequency standard is used to define the modern volt.
More information on Alessandro Volta can be found at Wikipedia [2] .
See also
Potential Difference in a Uniform Field
Potential Difference Path Independence
Potential Difference of point charge in a non-Uniform Field
Potential Difference in an Insulator
Further reading
Chabay, Ruth W. Matter and Interactions, Ch. 16.7. 4th ed. New York: Wiley, 2015. Print.
Urone, Paul Peter., et al. College Physics, Ch. 19.3. Houston, Texas: OpenStax College, Rice University, 2013. Print.
External links
Youtube Video on Electric Potential of a Point Charge
References
Chabay, Ruth W. Matter and Interactions. 4th ed. New York: Wiley, 2015. Print.
Urone, Paul Peter., et al. College Physics. Houston, Texas: OpenStax College, Rice University, 2013. Print.
Wikipedia.org