Gauss's Law and its Applications

Step 1: Equal positive and negative charges give zero net enclosed charge. Step 2: Individual charges may create field at the surface. Step 3: But total flux is zero because net enclosed charge is zero.

Step 1: Gauss's law relates total flux to enclosed charge, not to central position. Step 2: The charge is still inside, so enclosed charge is same. Step 3: But field on the spherical surface will not be uniform.

Step 1: Gauss's law gives total flux. Step 2: To find field, field magnitude and direction must be simple on the chosen surface. Step 3: Therefore a surface matching symmetry is most useful.

Step 1: Gauss's law is valid even for asymmetric cases. Step 2: The difficulty is in calculation, not in the law. Step 3: Without symmetry, field cannot be treated as simple on the surface.

Step 1: Field of an infinite line charge is radial. Step 2: Area vectors of flat ends of the cylinder are along the axis. Step 3: Field and area vector are perpendicular, so flux through the ends is zero.

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

Step 1: Field of an infinite line charge is inversely proportional to distance. Step 2: Tripling distance reduces an inverse relation by three times. Step 3: Hence field becomes one-third.

Step 1: For a point charge, the Gaussian surface is a sphere. Step 2: For a line charge, the Gaussian surface is a cylinder. Step 3: Different area dependence leads to different field-distance relations.

Step 1: Field of an infinite sheet is perpendicular to the sheet. Step 2: On the side surface of the pillbox, the field runs parallel to the surface. Step 3: Since it does not cross that surface, side flux is zero.

Step 1: An infinite uniform sheet has the same symmetry on both sides. Step 2: Field is perpendicular to the sheet on both sides. Step 3: Therefore both flat faces of the pillbox contribute to flux.

Step 1: Field of an infinite sheet does not depend on distance. Step 2: Its value is related to surface charge density. Step 3: Therefore doubling distance does not change the field.

Step 1: Field of an infinite sheet is proportional to surface charge density. Step 2: It changes with charge density, not distance. Step 3: If density doubles, field also doubles.

Step 1: Charge of a spherical shell lies on its surface. Step 2: A Gaussian surface inside the shell encloses no charge. Step 3: With symmetry, Gauss's law gives zero field inside.

Step 1: A spherical shell has symmetry about its centre. Step 2: An outside Gaussian sphere encloses the whole charge. Step 3: Hence outside field behaves as if the total charge were at the centre.

Step 1: Inside a uniformly charged solid sphere, a smaller Gaussian surface encloses only the charge within it. Step 2: Enclosed charge grows as cube of radius, while surface area grows as square of radius. Step 3: Hence field inside increases directly with distance from centre.

Step 1: Inside a uniformly charged solid sphere, field is proportional to distance from centre. Step 2: At the centre, distance is zero. Step 3: Therefore electric field at the centre is zero.

Step 1: Outside the sphere, the Gaussian surface encloses the whole charge. Step 2: Due to spherical symmetry, outside field behaves like a point charge at the centre. Step 3: Hence it follows inverse-square dependence.

Step 1: Inside the sphere, field increases with distance from centre. Step 2: Outside the sphere, field decreases with square of distance. Step 3: Therefore maximum value occurs at the surface.

Step 1: Inside a conductor in electrostatic equilibrium, electric field is zero. Step 2: Therefore total flux through an internal Gaussian surface is zero. Step 3: By Gauss's law, net enclosed charge inside it is zero.

Step 1: Free charges in a conductor arrange so that internal field becomes zero. Step 2: Flux through an internal closed surface is zero. Step 3: Thus excess net charge does not remain in the volume, it resides on the surface.

Step 1: Free charges in a conductor can move. Step 2: If a field component exists along the surface, they will not remain at rest. Step 3: Hence in electrostatic equilibrium, field can only be normal to the surface.

Step 1: Field just outside a conductor surface is related to surface charge density. Step 2: Larger field indicates larger local surface charge density. Step 3: This idea is often used for sharp regions of conductors.

Step 1: Field at a conductor surface is related to surface charge density. Step 2: Charge can be more concentrated near sharp regions. Step 3: Hence the nearby field can be stronger.

Step 1: In electrostatic equilibrium, field inside conducting material must be zero. Step 2: External influence rearranges charges on the conductor. Step 3: If cavity contains no charge, the field inside it remains zero.

Step 1: Field inside the conducting material must be zero. Step 2: A positive charge in the cavity tends to create field in the metal. Step 3: Equal negative charge is induced on the inner surface to cancel it in the conductor.