⚡ Gauss's Law Calculator
Calculate electric flux from enclosed charge
Total charge enclosed by the Gaussian surface
How to Use This Calculator
Enter Enclosed Charge
Input the total electric charge enclosed by the Gaussian surface in Coulombs. This is the net charge inside the closed surface. Use positive for positive charge, negative for negative charge. Use scientific notation for small values (e.g., 1e-6 for 1 microcoulomb).
Calculate
Click the "Calculate Electric Flux" button to get the electric flux through the closed surface in N⋅m²/C. The flux depends only on the enclosed charge, not on the shape or size of the surface (for a given enclosed charge).
Formula
Φ = Qenclosed / ε₀
Where:
- Φ = Electric Flux (Newton-meters squared per Coulomb, N⋅m²/C)
- Qenclosed = Total Charge Enclosed by Gaussian Surface (Coulombs, C)
- ε₀ = Vacuum Permittivity = 8.85 × 10⁻¹² C²/(N⋅m²)
Gauss's Law Integral Form:
∮ E · dA = Qenclosed / ε₀
The integral of electric field E over a closed surface equals the enclosed charge divided by ε₀.
Example Calculation:
For a charge of 1 μC (1 × 10⁻⁶ C) enclosed by a surface:
Φ = 1 × 10⁻⁶ / 8.85 × 10⁻¹²
Φ = 1.13 × 10⁵ N⋅m²/C
Note: The flux depends only on the enclosed charge, not on the shape, size, or location of charges outside the surface. This is the power of Gauss's law!
About Gauss's Law Calculator
The Gauss's Law Calculator determines the electric flux through a closed surface based on the charge enclosed by that surface. Gauss's law is one of Maxwell's four fundamental equations of electromagnetism and provides a powerful method for calculating electric fields in symmetric situations. It states that the electric flux through any closed surface is proportional to the charge enclosed.
When to Use This Calculator
- Electrostatics: Calculate electric flux and field for symmetric charge distributions
- Physics Education: Solve problems involving Gauss's law and electric flux
- Field Analysis: Determine electric fields around charged objects and conductors
- Capacitor Analysis: Understand charge distribution and field in capacitors
- Conductor Properties: Analyze electric fields inside and around conductors
Why Use Our Calculator?
- ✅ Quick Calculation: Instantly determine electric flux from enclosed charge
- ✅ Fundamental Law: Based on one of Maxwell's equations
- ✅ Scientific Notation: Handles very large and very small charge values
- ✅ Free Tool: No registration or payment required
- ✅ Educational: Learn about Gauss's law and electric flux
Common Applications
Point Charge Fields: Calculate electric fields around point charges, where choosing a spherical Gaussian surface makes the calculation straightforward. Gauss's law shows that the electric field of a point charge follows the 1/r² relationship (Coulomb's law) in a elegant way.
Uniformly Charged Spheres and Cylinders: Determine electric fields inside and outside uniformly charged objects by choosing appropriate Gaussian surfaces. For spheres, use spherical surfaces; for infinite cylinders, use cylindrical surfaces. This makes calculating fields much easier than using Coulomb's law directly.
Conductors and Electrostatics: Understand that excess charge on a conductor resides on its surface (since electric field inside a conductor in equilibrium is zero). Gauss's law explains why conductors shield their interiors from external fields and why charge accumulates on sharp points.
Tips for Best Results
- Gauss's law works best for symmetric charge distributions (spheres, cylinders, planes)
- Flux depends only on enclosed charge, not on charges outside the surface
- For zero enclosed charge, flux is zero (but field may not be zero everywhere)
- Choose Gaussian surfaces that match the symmetry of the problem
- The shape and size of the surface don't matter if the enclosed charge is the same
Frequently Asked Questions
What is Gauss's Law?
Gauss's law states that the electric flux through any closed surface is equal to the net charge enclosed divided by ε₀: Φ = Qenclosed/ε₀. It's one of Maxwell's equations and relates electric fields to their sources (charges).
What is electric flux?
Electric flux is a measure of how much electric field passes through a surface. It's the surface integral of the electric field: Φ = ∮ E · dA. Positive flux means field lines leave the surface, negative flux means they enter.
Does the shape of the surface matter?
For a given enclosed charge, the total flux through any closed surface is the same, regardless of shape or size. However, the shape matters when using Gauss's law to calculate electric field - you choose surfaces that make the calculation easy (matching symmetry).
What if there's no enclosed charge?
If Qenclosed = 0, then Φ = 0. This means the net flux through the surface is zero. However, this doesn't mean the electric field is zero everywhere - field lines can enter and leave, canceling out the flux.
How is this related to Coulomb's law?
Gauss's law and Coulomb's law are equivalent - you can derive one from the other. Gauss's law is more powerful for symmetric situations (spheres, cylinders, planes) because it often makes calculations much easier than integrating Coulomb's law.
Why is Gauss's law useful?
Gauss's law simplifies electric field calculations for symmetric charge distributions. Instead of integrating over all charges (Coulomb's law), you can often find the field with a simple calculation by choosing the right Gaussian surface. It's particularly powerful for spheres, infinite planes, and infinite cylinders.