Gauth Calculator

Gauss's Law Electric Flux Calculator

Use this calculator to determine the electric flux (Φ) through a closed surface using Gauss's Law. You can calculate flux either by providing the total enclosed charge or by providing the electric field, surface area, and the angle between them.

Method 1: Using Enclosed Charge

Method 2: Using Electric Field, Area, and Angle

Calculated Electric Flux (Φ):

Understanding Gauss's Law and Electric Flux

Gauss's Law is a fundamental principle in electromagnetism that relates the distribution of electric charge to the resulting electric field. It is one of Maxwell's four equations and provides a powerful way to calculate electric fields, especially for systems with high degrees of symmetry.

What is Electric Flux (Φ)?

Electric flux is a measure of the electric field passing through a given surface. It quantifies how many electric field lines penetrate a surface. The unit for electric flux is Newton-meters squared per Coulomb (N·m²/C) or Volt-meters (V·m).

• If the electric field lines pass perpendicularly through a surface, the flux is maximum.
• If the electric field lines are parallel to the surface, no field lines pass through, and the flux is zero.

Gauss's Law Formulae

Gauss's Law can be expressed in two primary ways, which are interconnected:

1. Flux in terms of Enclosed Charge:

The total electric flux (Φ) through any closed surface (often called a Gaussian surface) is directly proportional to the total electric charge (Q) enclosed within that surface, divided by the permittivity of free space (ε₀).

Φ = Q / ε₀

• Φ (Phi): Electric Flux (N·m²/C)
• Q: Total electric charge enclosed by the surface (Coulombs, C)
• ε₀ (epsilon naught): Permittivity of free space, a fundamental physical constant approximately equal to 8.854 × 10⁻¹² F/m (Farads per meter). This constant represents the ability of a vacuum to permit electric field lines.

This form of Gauss's Law is particularly useful for calculating the electric field of charge distributions with spherical, cylindrical, or planar symmetry.

2. Flux in terms of Electric Field, Area, and Angle:

Alternatively, for a uniform electric field (E) passing through a flat surface of area (A), the electric flux can be calculated as:

Φ = E ⋅ A ⋅ cos(θ)

• Φ (Phi): Electric Flux (N·m²/C)
• E: Magnitude of the electric field (Newtons per Coulomb, N/C, or Volts per meter, V/m)
• A: Area of the surface (square meters, m²)
• θ (theta): The angle between the electric field vector and the area vector (which is a vector normal to the surface).

If the electric field is perpendicular to the surface, θ = 0°, and cos(0°) = 1, so Φ = E⋅A. If the electric field is parallel to the surface, θ = 90°, and cos(90°) = 0, so Φ = 0.

How to Use the Calculator

The calculator above allows you to compute electric flux using either of these methods:

1. Using Enclosed Charge: Enter the total electric charge (Q) in Coulombs. The calculator will use the standard value for ε₀ to find the flux.
2. Using Electric Field, Area, and Angle: Enter the magnitude of the electric field (E) in N/C, the surface area (A) in m², and the angle (θ) in degrees between the electric field and the surface normal.

The calculator prioritizes the enclosed charge method if a valid charge is provided. If not, it will attempt to use the electric field, area, and angle method.

Examples:

Example 1: Flux from a Point Charge

Imagine a point charge of +1.602 × 10⁻¹⁹ C (the charge of a single proton) enclosed by a spherical Gaussian surface. What is the electric flux through this surface?

• Enclosed Charge (Q): 1.602e-19 C
• Calculation: Φ = (1.602e-19 C) / (8.854e-12 F/m) ≈ 1.809 × 10⁻⁸ N·m²/C

Using the calculator: Enter 1.602e-19 in the "Enclosed Charge" field and click "Calculate".

Example 2: Flux through a Flat Surface in a Uniform Field

Consider a uniform electric field of 500 N/C passing through a flat surface with an area of 0.2 m². The electric field lines are perpendicular to the surface.

• Electric Field (E): 500 N/C
• Surface Area (A): 0.2 m²
• Angle (θ): 0° (since perpendicular to surface means parallel to surface normal)
• Calculation: Φ = (500 N/C) ⋅ (0.2 m²) ⋅ cos(0°) = 100 N·m²/C

Using the calculator: Enter 500 in "Electric Field", 0.2 in "Surface Area", and 0 in "Angle" fields, then click "Calculate".

Example 3: Flux at an Angle

What if the electric field from Example 2 (500 N/C) passes through the same 0.2 m² surface, but at an angle of 60° to the surface normal?

• Electric Field (E): 500 N/C
• Surface Area (A): 0.2 m²
• Angle (θ): 60°
• Calculation: Φ = (500 N/C) ⋅ (0.2 m²) ⋅ cos(60°) = 100 ⋅ 0.5 = 50 N·m²/C

Using the calculator: Enter 500 in "Electric Field", 0.2 in "Surface Area", and 60 in "Angle" fields, then click "Calculate".