How this instrument works
Electric potential is bookkeeping for work. Carry a test charge from infinitely far away to a point sitting a distance r from a source charge q, and the energy spent per coulomb carried is kq/r. Because that figure is a plain scalar, contributions from a dozen scattered charges add by ordinary arithmetic with no components to resolve — which is why most field calculations build V first and differentiate afterwards rather than summing force vectors one by one.
The idea arrived with George Green, a Nottingham miller who had roughly one year of formal schooling. His 1828 essay on mathematical analysis applied to electricity and magnetism introduced the phrase 'potential function' along with the theorem now carrying his name; printed by subscription for barely fifty local backers, it sank without trace until William Thomson tracked down a copy in 1845 and arranged its reprinting in Berlin. Siméon Denis Poisson had reached adjacent ground in 1811, and Lagrange and Laplace had used such functions for gravitation decades earlier. Volta, whose name the unit borrows, wrote none of it.
Four conditions sit underneath that expression. The source has to be point-like, or a sphere you are standing outside of — step within a charged conducting shell and potential stops climbing, holding flat at its surface value. Zero has to be at infinity, a convention baked into the algebra, so an infinite line or sheet of charge cannot be handled here: those integrals diverge and demand some other reference. Surroundings have to be vacuum or near enough, since a dielectric divides everything by relative permittivity. And the charge has to sit still, because an accelerating one requires retarded Liénard–Wiechert potentials — news of its position travels at c, not instantly.
- Put your figure into Source charge and pick its unit — nC, µC or mC — rather than keying exponents. Keep any minus sign; this sheet preserves it.
- Set Distance from the charge as centre-to-point separation, from your charge's middle out to whatever spot you care about. Units run mm, cm or m; zero gets refused, since that expression has no value at r = 0.
- Read Electric potential in V, mV or kV. It is the value at that one spot measured against infinity, not a difference between two places.
- For several charges, run the sheet once per charge and add the answers with their signs. Scalars need no vector arithmetic.
Worked example — one coulomb, one metre
Set Source charge to 1 C and Distance from the charge to 1 m. Every other factor collapses to unity: V = k × 1 ⁄ 1 = 8,987,551,787.37 V. What returns is Coulomb's constant itself wearing volts, which makes this pairing a standard smoke test for any implementation — see 8.98755 × 10⁹ come back and both that constant and your division are intact.
Nine gigavolts is no laboratory quantity. The largest electrostatic accelerators ever built hold roughly 25 million volts on their terminals, and a lightning leader works with perhaps 100 million, so a coulomb parked on a single point would tower some ninety times above the storm. Bench figures land far lower: a microcoulomb at 10 cm reads 89.9 kV, while rubbing a balloon against wool shifts maybe 10 nC, worth about 1.8 kV five centimetres away.
Push Distance from the charge out to 2 m and the reading halves to 4,493,775,893.68 V. Not a quarter — potential thins as 1/r, whereas force between two charges thins as 1/r². That missing power of r reappears when you differentiate to recover the field, and it is precisely where people lose the thread.
Questions
What is the difference between electric potential and voltage?
Voltage nearly always means a potential difference — the gap between two places, which is what a voltmeter reads. Electric potential is the value at one place, referenced to a zero sitting infinitely far away, and that is what this sheet returns. Subtract two of its answers and you recover voltage in the everyday sense: around a 1 µC charge, the potential at 10 cm minus the potential at 20 cm gives 89.88 kV − 44.94 kV = 44.94 kV.
Why does potential fall as 1/r when force falls as 1/r²?
Because potential is force integrated along an inward path. Integrating 1/r² with respect to r yields 1/r, and that surviving factor of r converts newtons into joules per coulomb. Reverse that operation and field strength returns: E = −dV/dr = kq/r². In practice, doubling your distance halves potential but quarters field, and swapping those two rules is this quantity's most frequent error.
Can electric potential come out negative?
Yes, and the sign carries physics rather than bookkeeping. A negative source charge produces negative potential everywhere around itself, meaning a positive test charge brought in from far away releases energy instead of costing it. Whatever sign you type into Source charge is preserved here, unlike a Coulomb's law sheet reporting force as a bare magnitude. One nanometre from a single electron the potential runs about −1.44 V.
What does this give inside a charged metal sphere?
Nothing usable, because this formula describes space outside a source rather than a conductor's interior. Charge on an isolated metal sphere rides its surface, and throughout that interior potential holds constant at kq/R while field strength is zero. Enter a distance below its radius and you get an inflated number for a region where potential has stopped rising. Outside that surface it behaves exactly as though all charge sat at its centre, which is why this expression serves Van de Graaff domes so well.
How do I turn this into potential energy?
Multiply. A charge q₀ dropped where the potential is V carries energy U = q₀V joules, so an electron of −1.602 × 10⁻¹⁹ C at +100 V holds −1.602 × 10⁻¹⁷ J. Particle work abandons joules for this and quotes electronvolts instead: one eV is what a single elementary charge gains crossing one volt. Potential is measured in volts and belongs to a location; potential energy is measured in joules and needs a charge actually sitting there. Blurring those two ruins more exam answers than any other slip.
How would anyone measure a potential like this?
Not with an ordinary voltmeter, whose probe would bleed your charge away on contact. Electrostatic instruments read without drawing current: field mills chop incoming field with rotating earthed vanes and measure the induced alternating signal, while Kelvin probes vibrate a reference plate above your surface and null the resulting current to find contact potential. Semiconductor plants run both, since one technician crossing nylon carpet reaches several kilovolts and gate oxide can die on well under that.