Gauss's Law and its Applications

Step 1: In Gauss's law, total flux depends only on enclosed net charge. Step 2: The charge is still inside, so enclosed charge is unchanged. Step 3: Distribution may change but total flux does not, so do not judge only from position.

Step 1: Net flux is the sum of all contributions over the closed surface. Step 2: Outgoing and incoming contributions can cancel. Step 3: Zero net flux should not be confused with zero field everywhere.

Step 1: A cube has six identical faces. Step 2: With charge at the centre, symmetry is same for all faces. Step 3: Total flux is equally divided into six parts, so one face gets one sixth.

Step 1: Eight such identical cubes can be joined so that the charge becomes the centre of a larger cube. Step 2: The full flux is shared equally among the eight small cubes. Step 3: Therefore one small cube gets one eighth of the full flux.

Step 1: Total flux depends on enclosed charge, so changing radius does not change it. Step 2: Field of a point charge varies inversely as square of distance. Step 3: Doubling distance makes the field one fourth.

Step 1: Gauss's law always relates net flux to enclosed charge. Step 2: To find field, symmetry is needed so the field can be treated simply. Step 3: Without symmetry, total flux does not give field at each point.

Step 1: Field of a long line charge is radial. Step 2: At the flat ends of the cylinder, this field runs along the surface. Step 3: A field parallel to a surface gives no flux through that part.

Step 1: Electric field of a long line charge is inversely proportional to distance. Step 2: Making distance four times makes field one fourth. Step 3: Do not apply inverse square rule to a line charge.

Step 1: Field of an infinite plane sheet does not depend on distance. Step 2: By Gauss's law and symmetry, the field is uniform on both sides. Step 3: Changing distance does not change the ideal infinite-sheet field.

Step 1: Field of an infinite sheet is normal to the sheet. Step 2: In the pillbox cylinder, this field runs parallel to the curved surface. Step 3: Field parallel to a surface gives zero flux.

Step 1: Equal opposite charges have zero net charge. Step 2: Since the charges are at different positions, local field can remain. Step 3: Keep net flux and local field separate.

Step 1: Net electric field inside a conductor in electrostatic equilibrium is zero. Step 2: Hence flux through an internal Gaussian surface is zero. Step 3: By Gauss's law, excess charge enclosed by it is zero.

Step 1: Free charges in a conductor can move on the surface. Step 2: A tangential field would make them move along the surface. Step 3: Electrostatic equilibrium is possible only when the field is normal to the surface.

Step 1: Free charges in the conductor rearrange under the influence of external field. Step 2: In equilibrium, field inside conducting material must be zero. Step 3: The empty closed cavity remains field-free due to electrostatic shielding.

Step 1: Charge inside the cavity induces charges on the surfaces. Step 2: Still, in electrostatic equilibrium, net field inside conducting material must be zero. Step 3: Think separately about conducting material and the cavity region.

Step 1: Charge of the shell is on its surface. Step 2: A Gaussian surface chosen inside the shell encloses no charge. Step 3: Therefore net flux is zero and by symmetry the inside field is zero.

Step 1: An outside spherical Gaussian surface encloses the full charge. Step 2: Spherical symmetry makes the field equal in magnitude and radial on the surface. Step 3: The result is like a point charge at the centre.

Step 1: Inside a uniformly charged solid non-conducting sphere, field is proportional to distance from the centre. Step 2: Doubling distance doubles the field as long as the point remains inside. Step 3: Do not confuse it with zero field inside a conductor.

Step 1: At the centre all directions are equivalent. Step 2: Effect from every direction is balanced by the opposite direction. Step 3: Hence resultant electric field at the centre is zero.

Step 1: Outside the sphere, field behaves like total charge at the centre. Step 2: Field of a point charge varies inversely as square of distance. Step 3: Tripling distance makes field one ninth.

Step 1: Flux comes from the component crossing the surface. Step 2: In an oblique field, the full field does not cross the surface. Step 3: Use simple multiplication only when field is uniform and normal to the surface.

Step 1: In Gauss's law, net flux is decided only by enclosed net charge. Step 2: External positive charges can change field on the surface but not the sign of net flux. Step 3: Since enclosed net charge is negative, flux remains negative.

Step 1: Initially enclosed net charge was positive, so flux was positive. Step 2: Including an equal negative charge makes enclosed net charge zero. Step 3: By Gauss's law, net flux becomes zero.

Step 1: A spherical Gaussian surface is chosen with the point charge at its centre. Step 2: Every point on the sphere is at equal distance from the charge. Step 3: Therefore field magnitude is same everywhere.

Step 1: Field of an infinite sheet has equal magnitude on both sides. Step 2: The two flat faces of the pillbox lie on these two sides. Step 3: Therefore contributions from both faces are added.