Gauss's Law and its Applications

Step 1: The charge is at the centre, so all six faces are equivalent. Step 2: Total flux is divided equally among six faces. Step 3: In such questions, first use symmetry to identify equal shares.

Step 1: Joining eight cubes makes the corner charge the centre of a larger cube. Step 2: Total flux of the larger cube is shared by eight identical cubes. Step 3: Hence the original cube gets one eighth of the total flux.

Step 1: Total flux depends only on net enclosed charge. Step 2: Since the charge is inside, total flux is unchanged. Step 3: Off-centre placement makes distances to surface points different, so field is non-uniform.

Step 1: Gauss's law relates total flux to net enclosed charge. Step 2: Zero total flux means net enclosed charge is zero. Step 3: Equal positive and negative charges may still be inside, so saying no charge is inside is wrong.

Step 1: An external charge can create field on the surface. Step 2: Its field lines enter and leave the closed surface. Step 3: Since net enclosed charge is zero, total closed flux is zero.

Step 1: Field entering through a small part may give local negative flux. Step 2: Total flux is the algebraic sum over the whole closed surface. Step 3: Since net enclosed charge is positive, total flux is positive.

Step 1: The outside positive charge may change the field on the surface. Step 2: Total closed flux is determined only by net enclosed charge. Step 3: Since enclosed charge is negative, total flux remains negative.

Step 1: A complete dipole has zero net charge. Step 2: Hence total flux through the closed surface is zero. Step 3: Field may enter or leave through different parts, so local field can be non-zero.

Step 1: Total flux through a closed surface depends on net enclosed charge. Step 2: Only the negative charge is inside. Step 3: Therefore total flux is negative, and the outside positive charge is not counted.

Step 1: Gauss's law is true for total flux through any closed surface. Step 2: If field is not uniform, it cannot be taken out simply. Step 3: Thus the law remains true, but calculation becomes difficult.

Step 1: A cube is also closed, so total flux can be found. Step 2: But field magnitude is not the same everywhere on a cube. Step 3: On a sphere, distance is constant, so finding field becomes simple.

Step 1: A spherical Gaussian surface inside the shell encloses no charge. Step 2: Symmetry allows a direct conclusion about the field. Step 3: Electric field inside the shell is zero.

Step 1: A Gaussian surface outside the shell encloses the whole charge. Step 2: Due to spherical symmetry, the outside field behaves like a point charge at the centre. Step 3: Remember inside and outside results separately.

Step 1: In an insulating sphere, charge is distributed through volume. Step 2: Enclosed charge inside a smaller Gaussian sphere grows as cube of radius. Step 3: Surface area grows as square of radius, so field increases linearly with distance.

Step 1: At the centre, all parts of the sphere surround the point symmetrically. Step 2: Every contribution is cancelled by an equal opposite contribution. Step 3: Therefore net electric field at the centre is zero.

Step 1: In a conductor, free charges move to the surface in electrostatic condition. Step 2: In an insulator, charge can remain within the volume. Step 3: Hence field is zero inside conductor but may vary inside insulator.

Step 1: For a line charge, enclosed charge depends on the length of line inside the cylinder. Step 2: Changing radius does not change that length. Step 3: Therefore enclosed charge remains unchanged.

Step 1: Total flux is proportional to net enclosed charge. Step 2: Doubling cylinder length doubles the enclosed line charge. Step 3: Therefore total flux doubles.

Step 1: An infinite line charge has cylindrical symmetry. Step 2: Every point on the curved surface is at the same distance from the line. Step 3: Therefore field magnitude is taken same there.

Step 1: Field of a long line charge is radial. Step 2: On the flat end caps, this field does not cross the surface. Step 3: Therefore flux through those end caps is zero.

Step 1: Charge on a plane sheet is spread on the surface. Step 2: Enclosed charge depends on the area cut from the sheet. Step 3: Changing height does not change this area, so total flux is unchanged.

Step 1: Field lines emerge from positive charge. Step 2: Symmetry of an infinite sheet makes the field perpendicular to the sheet. Step 3: Therefore field is away from the sheet on both sides.

Step 1: Field due to a positive sheet is away from the sheet on both sides. Step 2: Between two identical positive sheets, the two fields are opposite. Step 3: Equal magnitudes cancel each other.

Step 1: Field is away from the positive sheet and toward the negative sheet. Step 2: Between the sheets, these directions are the same. Step 3: Hence the fields add in the region between them.

Step 1: In the outside region, fields due to the two sheets are opposite. Step 2: Equal magnitude charge densities give equal field magnitudes. Step 3: Hence the outside field cancels to zero.