Gauss's law: what is the electric flux through a closed surface that encloses a total charge Q?

Gauss's law: what is the electric flux through a closed surface that encloses a total charge Q?

• A. Φ_E = Q/ε0
• B. Φ_E = Q /(4π ε0)
• C. Φ_E = ε0 Q
• D. Φ_E = 4π ε0 Q

Answer: (A)

Gauss's law connects the net electric flux through any closed surface to the charge enclosed by that surface. The total flux through the surface is equal to the enclosed charge divided by ε0, written as Φ_E = Q_enc / ε0. If the surface encloses a total charge Q, the flux is Q/ε0. This result holds for any shape of the closed surface, not just spheres, because it’s based on the way electric field lines originate from charges and the fundamental constant ε0. For a point charge, the derivation with a spherical surface shows the same outcome: while the field strength falls off as 1/r^2, the area element grows as r^2, and the 4π in the derivation cancels to give Q/ε0. The other expressions don’t fit because they either misplace the 4π factor or mix up the dimensions; the correct relation must give the flux in units that match ε0 in the denominator, independent of surface shape.

Gauss's law connects the net electric flux through any closed surface to the charge enclosed by that surface. The total flux through the surface is equal to the enclosed charge divided by ε0, written as Φ_E = Q_enc / ε0. If the surface encloses a total charge Q, the flux is Q/ε0.

This result holds for any shape of the closed surface, not just spheres, because it’s based on the way electric field lines originate from charges and the fundamental constant ε0. For a point charge, the derivation with a spherical surface shows the same outcome: while the field strength falls off as 1/r^2, the area element grows as r^2, and the 4π in the derivation cancels to give Q/ε0.

The other expressions don’t fit because they either misplace the 4π factor or mix up the dimensions; the correct relation must give the flux in units that match ε0 in the denominator, independent of surface shape.