The electric field due to a charged plane sheet is directly proportional to the surface charge density. Therefore, if σ is doubled, E also doubles.

The electric field inside a uniformly charged sphere is zero.

The electric field inside a charged conductor in electrostatic equilibrium is zero.

The total electric flux through a closed surface is given by Φ = ΣQ_enc/ε₀, where Q_enc is the total charge enclosed.

According to Gauss's law, the total electric flux through a closed surface is proportional to the total charge enclosed.

The electric field due to a point charge varies as E = kQ/r², where k is a constant.

The electric field outside a spherical charge distribution behaves as if all the charge were concentrated at the center, so E = Q/4πε₀r².

For r > R, the electric field behaves as if all the charge were concentrated at the center, given by E = Q/(4πε₀r²).

The electric field due to an infinite plane sheet of charge is constant and given by E = σ/2ε₀ on either side of the sheet.

Using Gauss's law, the electric field due to an infinite plane sheet of charge is E = σ/2ε₀ on either side of the sheet.

The total flux through the cube is Q/ε₀. Since the charge is at one corner, the flux through one face is Q/6ε₀.

The total electric flux through the cube is Q/ε₀, but since only 1/6 of the charge is enclosed by the cube, the flux is Q/6ε₀.

By Gauss's law, the electric field inside a uniformly charged spherical shell is zero.

Inside a uniformly charged spherical shell, the electric field is zero due to symmetry.

By symmetry, the flux through one face of the cube is Q/6ε₀, as the charge is shared equally among the six faces.

For r > R, the electric field behaves as if all the charge were concentrated at the center, given by E = Q/(4πε₀r²).

Inside a uniformly charged spherical shell, the electric field is zero.

For a point outside the sphere, the electric field behaves as if all the charge were concentrated at the center, hence E = Q/(4πε₀r²).

The total electric flux through the Gaussian surface is given by Φ = Q/ε₀, where Q is the charge enclosed.

The electric flux through a closed surface is given by Φ = Q_enc/ε₀. Here, Q_enc = Q, so Φ = Q/ε₀.

According to Gauss's law, the total electric flux through a closed surface is equal to the charge enclosed divided by ε₀, thus Φ = Q/ε₀.

The electric field inside a non-conducting sphere with linearly increasing charge density varies linearly with radius.

At the center of a non-uniform spherical charge distribution, the electric field is zero due to symmetry.

The electric field inside a non-uniform charge distribution varies quadratically with radius due to the increasing charge density.

According to Gauss's law, the electric flux is directly proportional to the enclosed charge, so it doubles.

The electric field due to an infinite plane sheet is constant and equal to E, regardless of the distance from the sheet.

The electric field due to an infinite plane sheet is constant and equal to E on both sides, thus the total field is 2E.

The fields due to the two sheets add up in the region between them, resulting in a total electric field of 2E.

The electric field due to a point charge varies inversely with the square of the distance, so at 2r, E becomes E/4.

In electrostatic equilibrium, excess charge resides on the surface of the conductor, leading to zero electric field inside.