Electric Potential of a Point Charge Calculator
Calculate the electric potential V at any distance from a point charge. Enter the source charge Q, the distance r, and the medium's relative permittivity to get the result in volts and kilovolts.
Electric Potential from a Point Charge
Point Charge Electric Potential
Where:
- \(V\) — electric potential at the point, in volts (V)
- \(k = 8.988 \times 10^9\ \text{N·m}^2/\text{C}^2\) — the Coulomb constant
- \(Q\) — source charge, in coulombs (C)
- \(\varepsilon_r\) — relative permittivity of the medium (1 for vacuum or air)
- \(r\) — distance from the charge to the measurement point, in meters (m)
The formula gives the potential relative to infinity (\(V(\infty) = 0\)): the work per unit charge needed to bring a small positive test charge from infinitely far away to the distance \(r\).
Point Charge Electric Potential – Calculation Example
Example: a charge of \(Q = 1\ \mu\text{C}\) sits in air, and you want the potential at a distance of \(r = 10\ \text{cm}\).
\[ V = \frac{8.988 \times 10^9 \times 1 \times 10^{-6}}{1 \times 0.1} = \frac{8.988 \times 10^3}{0.1} \]
Result: \(V \approx 89\,876\ \text{V} \approx 89.9\ \text{kV}\).
How to Use the Calculator
- Enter the magnitude of the Source Charge (Q) and pick its unit (nC, µC, mC or C). The sign of the potential follows the sign of the charge.
- Enter the Distance (r) from the charge to the measurement point and select the unit (mm, cm, m, in or ft).
- Optionally set the Relative Permittivity (εr) of the surrounding medium — leave it at 1 for vacuum or air, or use ≈2.2 for oil, ≈4.7 for glass, ≈80 for water.
- Click Calculate. The result appears as the potential in volts (V) and kilovolts (kV), along with a step-by-step calculation breakdown.
What Is the Electric Potential of a Point Charge?
The electric potential \(V\) at a point in space describes how much electric potential energy a charged object would have there, per unit of charge. It is a scalar quantity: unlike the electric field, which points in a specific direction, the potential is just a single number — positive near positive charges, negative near negative charges. Potential is always measured relative to a chosen reference level, and for an isolated point charge the conventional choice is \(V = 0\) at infinity.
For a point charge, the potential falls off as \(1/r\): doubling the distance halves the potential. This is a slower falloff than the electric field, which drops as \(1/r^2\). Because potential is a scalar, the total potential from several charges at a point is simply the algebraic sum of the individual potentials — a property that makes it much easier to work with than vector fields.
A useful way to interpret the result: \(V\) equals the work per coulomb required to move a small positive test charge from infinitely far away to that point. If the calculator returns \(V = 90\ \text{kV}\), a charge of \(+1\ \mu\text{C}\) brought to that location would have \(90 \times 10^3 \times 10^{-6} = 0.09\ \text{J}\) of electric potential energy. The potential also determines the electric field through \(E = -dV/dr\), so a steep potential gradient means a strong field.
Assumptions, Limitations and Practical Tips
- Point charge assumption: the formula is exact for a point charge, and equally valid outside a uniformly charged sphere (a charged conducting sphere behaves as if all its charge were concentrated at its center).
- Zero reference at infinity: results assume \(V(\infty) = 0\). In circuits or near large grounded objects, measured potentials may differ because a different reference (e.g., earth ground) is used.
- Uniform medium: the calculation assumes a single, uniform, linear dielectric filling all the space around the charge. If the charge sits at a boundary between two media, the simple formula no longer applies.
- Distance must be positive: the potential is undefined exactly at the location of a point charge (it diverges as \(r \to 0\)), so the distance must be greater than zero.
- Sign matters: enter a negative charge to get a negative (attractive) potential; the magnitude of the result is unchanged.
- Dielectric media reduce the potential by the factor \(\varepsilon_r\) — water, for instance, cuts the potential by roughly a factor of 80 compared with air.
To explore the vector counterpart of this quantity, try the Point Charge Electric Field Calculator. If you are interested in other inverse-square and energy-related physics, the Gravitational Force Calculator and the Rotational Kinetic Energy Calculator cover related concepts.
Frequently Asked Questions
How do I calculate the electric potential of a point charge by hand?
Multiply the charge by the Coulomb constant \(k = 8.988 \times 10^9\) and divide by the distance. For example, with \(Q = 2\ \mu\text{C}\) at \(r = 0.5\ \text{m}\) in air: \( V = (8.988 \times 10^9 \times 2 \times 10^{-6}) / 0.5 \approx 35\,950\ \text{V}\). Always convert the charge to coulombs and the distance to meters first, and divide the result by the relative permittivity if the medium is not vacuum or air.
What is a typical or 'normal' value for electric potential?
There is no single normal value — the potential depends entirely on the charge size and distance. Everyday static electricity can reach thousands of volts (a 1 µC charge at 10 cm gives about 90 kV), while a 1 C charge at 1 m produces about 9 GV. High voltage does not necessarily mean danger: the potential energy involved, \(U = qV\), matters too, and small charges store very little energy.
Can the potential be negative, and what does a negative result mean?
Yes. The sign of the potential always matches the sign of the source charge: a negative charge creates a negative potential. A negative value means a positive test charge would have negative potential energy there — it is attracted to the location rather than repelled. This calculator takes the charge magnitude, so if you need the signed potential, apply the charge's sign to the result manually.
Why does potential fall off as 1/r while the electric field falls off as 1/r²?
The electric field is the negative gradient of the potential, \( E = -dV/dr \). Taking the derivative of \( V = kQ/r \) gives \( E = kQ/r^2 \), which is the inverse-square law. Practically, this means potential decreases more slowly with distance: at three times the distance, the potential is one third of its original value, while the field is only one ninth.
How do I find the total potential from several point charges?
Because potential is a scalar, you simply add the individual potentials algebraically — no vector components needed. Compute \( V = kQ_i/\varepsilon_r r_i \) for each charge at the point of interest and sum the results, keeping the signs. For instance, a +2 µC charge and a −1 µC charge at the same 10 cm distance give a combined potential of \( 179\,752 - 89\,876 \approx 89\,876\ \text{V}\).
What happens if I enter a distance of zero or very close to zero?
The formula diverges as \( r \to 0 \): the potential grows without bound, and exactly at a point charge it is undefined. Physically, a true point charge has zero size, so no finite potential exists at its location. The calculator requires a positive distance and will reject zero. For real charged objects with a finite radius, use a distance from the center that is at least equal to the object's radius.
How do I convert charge or distance units before calculating?
Use \(1\ \text{nC} = 10^{-9}\ \text{C}\), \(1\ \mu\text{C} = 10^{-6}\ \text{C}\), \(1\ \text{mC} = 10^{-3}\ \text{C}\), and for distance \(1\ \text{cm} = 0.01\ \text{m}\), \(1\ \text{in} = 0.0254\ \text{m}\), \(1\ \text{ft} = 0.3048\ \text{m}\). The most common mistake is forgetting to convert, which changes the answer by a factor of 10 to 1000 — a quick sanity check is that 1 µC at 1 m always gives about 8988 V in air.