Gauss Law

Worked Example

Consider finding the enclosed charge, electric field, and outward flux for a uniform solid sphere using the standard rounded constants.

1. Given values: Total charge Q = 1 nC, charge radius R = 1 m, Gaussian radius r = 0.5 m, distribution = Uniform solid sphere.
2. Identify the enclosed charge Q_enc: Because r < R, the volume enclosed by the Gaussian surface is a fraction of the total volume. Q_enc = Q * (r / R)³ = 1 nC * (0.5 / 1)³ = 1 nC * 0.125 = 0.125 nC.
3. Calculate Electric Field E: Using Coulomb's constant k = 8.99e9 N·m²/C², the electric field at distance r is E = k * Q_enc / r². Substituting our values (converting nC to C exactly by 1e-9): E = 8.99e9 * (0.125e-9) / (0.5)² = 1.12375 / 0.25 = 4.495 N/C.
4. Calculate Outward Flux: Flux = 4 * π * k * Q_enc = 4 * π * 8.99e9 * 0.125e-9 = 4.495 * π N·m²/C.

Experiment

You can interact with the Gauss Law module to observe these relationships dynamically.

1. Change the Charge distribution to Uniform solid sphere.
2. Set the Total charge Q to 1.
3. Set the Charge radius R to 1.
4. Set the Gaussian radius r to 0.5. Observe the calculated Q_enc = 0.125 nC, E = 4.495 N/C, and flux = 4.495π ≈ 14.121459 N·m²/C.
5. Change the Gaussian radius r to 2. Since r > R, Q_enc now equals the total charge (1 nC), and E = 2.2475 N/C with flux = 35.96π ≈ 112.97167 N·m²/C.
6. Change the Charge distribution to Thin spherical shell and set r to 0.5. Notice that Q_enc = 0, E = 0, and flux = 0 despite the nonzero total shell charge.

Model Limitations

This experiment fixes the rounded teaching constant k = 8.99e9 N·m²/C² and defines ε₀ = 1/(4πk) consistently. Values are displayed with at least six significant digits; calculations retain full working precision before rounding.

Additionally, this mathematical model assumes idealized continuous charge distributions (perfect spheres and infinitely thin shells). At the exact boundary of a thin spherical shell (r = R) with nonzero charge, the electric field and enclosed-surface flux undergo a discontinuity. The module labels the inside and outside limits separately. The corresponding enclosed-charge limits are zero and Q. No single enclosed-charge, local-field or flux value is displayed on the nonzero charged sheet. When Q = 0, field and flux remain zero at the surface.

References

The following primary sources provide a deeper dive into the derivations and applications of Gauss's Law, as well as the idealized approximations used in introductory physics.

• OpenStax University Physics Volume 2: 6.3 Applying Gauss's Law - Covers the application of Gauss's Law to spheres and thin shells.