Gauss's Law and its Applications

Step 1: Excess charge on a conducting sphere resides on the surface. Step 2: For outside points, spherical symmetry is maintained. Step 3: Hence the outside field is same as if the total charge were concentrated at the centre.

Step 1: At the centre, all parts of the sphere surround the point symmetrically. Step 2: Field contribution in every direction has an equal opposite contribution. Step 3: Therefore the net electric field at the centre is zero by symmetry.

Step 1: For uniform volume charge density, charge is proportional to volume. Step 2: Volume of a smaller spherical Gaussian surface is proportional to cube of radius. Step 3: Hence enclosed charge increases as cube of radius.

Step 1: Inside the sphere, enclosed charge increases as cube of radius. Step 2: Spherical area increases as square of radius. Step 3: Their ratio is proportional to radius, so the field increases linearly.

Step 1: In a conductor, free charges move and settle on the surface. Step 2: In an insulator, charge can remain distributed throughout volume. Step 3: Therefore field is zero inside conductor but can 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 the enclosed length of the line. Step 3: Therefore enclosed charge remains unchanged if length is same.

Step 1: In Gauss's law, total flux depends on enclosed charge. Step 2: Doubling cylinder length doubles the enclosed line charge. Step 3: Therefore total flux also doubles.

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

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

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

Step 1: Positive sheets produce field away from themselves on both sides. Step 2: In an outer region of two sheets, the two fields point in the same direction. Step 3: Therefore they add there.

Step 1: A Gaussian surface inside the shell encloses no charge. Step 2: By symmetry, total flux on that surface is zero. Step 3: Therefore electric field inside the shell is zero.

Step 1: A Gaussian sphere outside the shell encloses the entire charge. Step 2: Outside, spherical symmetry makes it behave like a point charge at the centre. Step 3: Therefore field falls inversely as square of distance.

Step 1: Electric field just outside a conductor surface is related to surface charge density. Step 2: Where surface charge is denser, field lines are also denser. Step 3: Hence the electric field just outside is stronger.

Step 1: In an electrostatic conductor, excess charge stays on the surface. Step 2: Sharp parts can have higher charge density. Step 3: Higher charge density indicates stronger external electric field.

Step 1: In an electrostatic conductor, surface charges are at rest. Step 2: An oblique field has a component along the surface. Step 3: That component would move charges, so the field must be perpendicular to the surface.

Step 1: Electric field inside conductor material is zero. Step 2: With no charge inside the empty cavity, field lines cannot persist there. Step 3: In electrostatic condition, the conductor shields the cavity from external fields.

Step 1: Inside conductor material in electrostatic condition, electric field is zero. Step 2: Zero field gives zero total flux through a closed surface. Step 3: By Gauss's law, net enclosed charge for such a surface is zero.

Step 1: A dipole has equal positive and negative charges. Step 2: Since net charge is zero, total closed flux is zero. Step 3: Still, electric field can exist at different points.

Step 1: Total flux through a closed surface depends only on net enclosed charge. Step 2: Here only the positive charge is inside. Step 3: Therefore total flux is positive according to that enclosed positive charge.

Step 1: A complete dipole inside has zero net enclosed charge. Step 2: The external charge is not counted inside the closed surface. Step 3: Hence total flux remains zero even though field on the surface may change.

Step 1: Gauss's law is true for total flux through any closed surface. Step 2: If the field is not uniform, it cannot be easily taken out of the flux sum. Step 3: So the law is valid, but calculation may be difficult.

Step 1: Gauss's law is always true. Step 2: To find field, we need to write flux in a simple form. Step 3: Symmetry makes field uniform or perpendicular on chosen parts, making calculation easier.

Step 1: A cube is also a closed surface, so total flux can be found. Step 2: But distance from the point charge is not same everywhere on the cube surface. Step 3: Hence field cannot be treated as constant and taken out simply.

Step 1: Total flux remains fixed by the enclosed charge. Step 2: But an off-centre charge has different distances to different surface points. Step 3: Therefore field is not uniform and the simple multiplication method fails.