Gauss's Law Calculator (Charged Sphere)
Electric field, potential and flux at any distance from a charged sphere, inside or outside it, with the field profile drawn.
Results
The dashed line marks the value you entered.
What this tool does
Gauss's law is the first of Maxwell's four equations: the electric flux through any closed surface equals the charge inside divided by ε₀. For a sphere the symmetry does all the work — outside, the field is exactly that of a point charge at the centre, however big the sphere. Inside is where the two cases part ways: in a conductor the charge sits on the surface and the field is zero, while in a uniformly charged solid it grows in a straight line from the centre and peaks at the surface.
Formula
∮E·dA = Q_enc ÷ ε₀ · r ≥ R: E = kQ ÷ r² · r < R: E = kQr ÷ R³ (solid) or 0 (conductor)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
q | Electric charge | nC |
rs | Sphere radius | cm |
r | Distance from the centre | cm |
kd | Where the charge sits | — |
E | Electric field | V/m |
QE | Charge enclosed | nC |
FL | Electric flux through that sphere | V·m |
V | Electric potential | V |
ES | Field at the surface | V/m |
Worked example
- Electric charge10 nC
- Sphere radius5 cm
- Distance from the centre10 cm
- Where the charge sits1
- Electric field8987.55 V/m
- Charge enclosed10 nC
- Electric flux through that sphere1129.41 V·m
- Electric potential898.755 V
- Field at the surface35,950.2 V/m
Limitations
- The formula assumes ideal geometry — an infinitely long wire or coil, a perfectly uniform sphere — in empty space or a uniform material. Real parts come close, but ends, nearby metal and magnetic materials change the field.
- Mixing units is the most common source of error. Convert every input to the units shown next to each field before calculating.