Electric Potential Calculator

Electric potential in one definition

Electric potential is the potential energy held per unit of charge at a point in space, and its unit is the volt, which is one joule per coulomb. The OpenStax College Physics 2e section on the potential of a point charge gives the definition as V = U/q and stresses that potential is a scalar while the electric field is a vector.

Two properties make potential easier to work with than field strength. Contributions add as plain arithmetic because scalars add, and the value fades with distance as one over r rather than one over r squared. The reference point is placed at infinity, where the influence of a lone charge has died away, so the potential of one charge is always written relative to that zero.

A voltmeter does not read an absolute potential, it reads a difference between two points. Saying a battery is 12 V means its terminals sit 12 V apart. The calculator here produces the potential at one point, and those single-point values are the pieces you combine when you want a difference.

The point charge formula and its constant

For a charge Q placed at the origin, the potential a distance r away is V = kQ/r, where k is the Coulomb constant. The NIST reference on fundamental constants lists k together with its uncertainty, and the rounded value that appears in textbooks and in this calculator is 8.99 x 10^9 N m2/C2.

Two unit traps account for most wrong answers here. The distance has to be in metres, so 5 cm is entered as 0.05 m or chosen from the unit menu, and the charge has to be in coulombs, so a microcoulomb is 0.000001 C and a nanocoulomb is 0.000000001 C. Miss either conversion and the answer moves by a factor of a thousand or more.

The HyperPhysics page on point charge potential reaches the same expression by working out the effort needed to carry a test charge in from infinity, and it points out that because the result is a scalar, the potentials of several charges can be added together.

A micro-coulomb at one metre

Take a charge of 1 microcoulomb and ask what the potential looks like as you walk away from it. At 1 metre the calculation is 8.99 x 10^9 times 1 x 10^-6 divided by 1, which is 8,990 V. Step out to 2 metres and the value halves to 4,495 V. Come in to 10 cm and it climbs to 89,900 V, and at 1 cm it reaches 899,000 V.

A microcoulomb is a small amount of charge, yet a metre away it sits at nearly nine thousand volts. Bring a charged object within a centimetre and the same charge pushes the potential to hundreds of kilovolts, which is the region where air starts to break down and sparks jump. Distance is doing nearly all of the work in these numbers.

Two published cases you can reproduce

The OpenStax problem set works a metal sphere 1 cm across that carries a static charge of -3.00 nC, and asks for the potential 5.00 cm away from its centre. Since charge on a metal sphere spreads out evenly and behaves like a point charge at the centre, the formula applies directly: 8.99 x 10^9 times -3.00 x 10^-9 divided by 0.0500 m, which is -539.4 V.

The second case comes from atomic physics. A proton carries 1.602 x 10^-19 C, and in a hydrogen atom it sits about 5.29 x 10^-11 m from the electron. The potential of the proton at that separation is 8.99 x 10^9 times 1.602 x 10^-19 divided by 5.29 x 10^-11, which is 27.2 V.

The hydrogen number has a second life. An electron sitting at that 27.2 V point holds potential energy of -27.2 eV, its kinetic energy adds +13.6 eV, and the two together give -13.6 eV, the binding energy that must be paid to ionise hydrogen. Voltage is not an abstract idea there, it is the energy budget of the atom.

One over r against one over r squared

The distance appears once in the denominator for potential and twice for the field, so the two quantities fall at different speeds. Double the distance and the potential halves while the field falls to a quarter. Ten times the distance leaves a tenth of the potential and a hundredth of the field. Halve the distance and the potential doubles while the field quadruples.

Every point at the same distance from a lone charge sits at the same potential, so the level sets are spheres around it. Walk along one of those spheres and the potential never changes, no matter which way you face. That is the cleanest picture of what voltage is, a height on a landscape that slopes down as you move away from the charge.

Superposing potentials from several charges

Two charges make the scalar nature of potential obvious. Put +2.00 µC and -2.00 µC 0.600 m apart and stand at the midpoint, 0.300 m from each. The positive charge contributes 59,933 V and the negative one contributes -59,933 V, so the total there is exactly zero volts.

Now stand 0.200 m from the positive charge, which puts you 0.400 m from the negative one. The positive contribution is 89,900 V and the negative one is -44,950 V, giving 44,950 V overall. The field at that spot is not zero, because the field is a vector and the two contributions point in the same direction: 449,500 V/m from the positive charge and 112,375 V/m from the negative one, so the net field is 561,875 V/m aimed at the negative charge.

Adding the numbers by hand is the standard method, and the calculator on this page handles one charge at a time, which is enough for any number of them as long as you keep each distance straight.

Turning volts into joules and electron volts

Potential is energy per charge, so the energy on a test charge follows from U = qV. A 1 nC test charge placed at the 8,990 V point holds 1 x 10^-9 times 8,990, which is 8.99 x 10^-6 J. The same point with a 1 µC test charge holds 8.99 x 10^-3 J, a thousand times as much, because the charge is a thousand times larger.

Electron volts are the unit used for this energy at atomic scale, and the conversion is fixed: one electron volt is exactly 1.602176634 x 10^-19 J, the elementary charge expressed in joules per volt. The 8.99 x 10^-6 J figure is therefore 5.61 x 10^13 eV, and an electron crossing a 1 V difference picks up 1 eV on the way.

The practical reading of all this is that high voltage and small charge go together. A charged balloon touching a metal door moves a fraction of a microcoulomb across a gap of a few millimetres, and the potential at that gap is high enough for the air to conduct. The energy released stays modest because the charge involved is so small.

Particle accelerators push the definition to extreme numbers. The LHC ramps protons through radiofrequency cavities until each one carries tera electron volts of energy, which a voltmeter can never display but the same joule per coulomb accounting describes. On the opposite end, a thundercloud bottom can sit tens of megavolts above the ground, which is why lightning jumps kilometers of air. Battery engineering uses the concept too: stacking alkaline cells in series adds their terminal potentials, and a capacitor stores charge until the potential across its plates matches the charger. Voltmeter probes, oscilloscope inputs, grounding straps and even the ground pin on a household outlet are all references to some chosen zero of potential.

Reading the output panel

The panel starts on a 1 µC charge 1 m away with a 1 nC test charge loaded for the energy row, so the numbers on screen are the ones quoted throughout this page. Change the charge, change the distance, change the unit menus, and every row updates at once.

Two checks keep the reading honest. Confirm the sign line, since a negative potential means the point sits below the zero chosen at infinity, which is normal for a negative charge. And confirm the scale, since a potential of thousands of volts from a handheld charge means the distance is small or the charge is large, and both are worth a second look before the number gets used in a calculation.