Gauss's Law can be proven by relating the electric flux through a closed surface to the charge enclosed within that surface. According to Gauss's Law, the total flux associated with a sealed surface equals 1/ε₀ times the charge encompassed by the closed surface.
Here's a breakdown of a proof using a point charge and a spherical Gaussian surface:
Proof using a Point Charge and Spherical Gaussian Surface
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Consider a point charge: Let's assume we have a point charge q located at a point in space.
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Construct a Gaussian surface: Imagine a spherical surface (also known as a Gaussian surface) centered on the point charge q. This sphere has a radius r.
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Electric field symmetry: The electric field E due to the point charge is radial and has the same magnitude at every point on the Gaussian surface. The magnitude is given by Coulomb's Law:
E = q / (4πε₀r²)
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Define the area vector: At any point on the sphere, define a small area element dA as a vector dA pointing outward, normal to the surface.
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Calculate the electric flux: The electric flux dΦ through the area element dA is the dot product of the electric field E and the area vector dA:
dΦ = E ⋅ dA
Since E and dA are parallel (both point radially outward), the dot product simplifies to:
dΦ = E dA = (q / (4πε₀r²)) dA
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Integrate the flux over the entire surface: To find the total electric flux Φ through the entire Gaussian surface, we integrate dΦ over the entire area A of the sphere:
Φ = ∫ dΦ = ∫ (q / (4πε₀r²)) dA = (q / (4πε₀r²)) ∫ dA
Since (q / (4πε₀r²)) is constant over the sphere, it can be taken out of the integral. The integral ∫ dA over the entire surface of the sphere is simply the surface area of the sphere, which is 4πr².
Φ = (q / (4πε₀r²)) * 4πr²
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Simplify to obtain Gauss's Law: The 4πr² terms cancel out, leaving:
Φ = q / ε₀
This shows that the total electric flux through the closed spherical surface is equal to the charge q enclosed by the surface divided by the permittivity of free space ε₀. This is Gauss's Law. Therefore, the total flux associated with a sealed surface equals 1/ε₀ times the charge encompassed by the closed surface.
Summary Table:
| Concept | Description |
|---|---|
| Gauss's Law | Φ = q / ε₀ |
| Electric Flux (Φ) | Measure of the electric field passing through a surface. |
| Gaussian Surface | An imaginary closed surface used to calculate the electric flux. |
| Point Charge (q) | A localized electric charge at a single point in space. |
| Permittivity (ε₀) | A physical constant representing the ability of a vacuum to permit electric fields. |
