How should the integral in Gauss’s law be evaluated?
around the perimeter of a closed loop
over the surface bounded by a closed loop
over a closed surface
The Correct Answer and Explanation is:
Correct Answer:
Over a closed surface
Explanation:
Gauss’s Law is one of Maxwell’s equations and is a fundamental law in electromagnetism. It relates the electric flux through a closed surface to the electric charge enclosed by that surface. Mathematically, it is expressed as: ∮closed surfaceE⃗⋅dA⃗=Qinsideε0\oint_{\text{closed surface}} \vec{E} \cdot d\vec{A} = \frac{Q_{\text{inside}}}{\varepsilon_0}
Here,
- E⃗\vec{E} is the electric field vector,
- dA⃗d\vec{A} is the differential area vector on the closed surface (pointing outward),
- QinsideQ_{\text{inside}} is the total electric charge enclosed within the surface, and
- ε0\varepsilon_0 is the vacuum permittivity constant.
The integral symbol with a circle (∮) indicates a surface integral over a closed surface. This distinguishes Gauss’s Law from other types of integrals such as line integrals (around a loop) or surface integrals over open surfaces.
A closed surface is any surface that completely encloses a volume—like the surface of a sphere, cube, or cylinder with end caps. The key is that there is no boundary; the surface forms a complete shell.
This law is especially powerful when applied to systems with high symmetry (spherical, cylindrical, or planar), where it can simplify the calculation of electric fields. For example, using a spherical Gaussian surface around a point charge allows easy computation of the radial electric field using symmetry and the inverse-square law.
If the integral were taken “around the perimeter of a closed loop,” it would represent a line integral, not a surface integral, and would apply to Faraday’s Law of Induction—not Gauss’s Law. Similarly, “over the surface bounded by a closed loop” refers to an open surface, which is not applicable to Gauss’s Law.
Thus, Gauss’s Law always involves integrating over a closed surface.
