Gauss's Law and its Applications

Step 1: Gauss's law relates total flux to net enclosed charge. Step 2: If net charge is zero, total flux is zero even when local field exists. Step 3: Do not confuse zero total flux with zero field everywhere.

Step 1: Total flux depends on enclosed charge, not on the charge position. Step 2: The charge remains inside, so enclosed charge is unchanged. Step 3: Field distribution may change, but total flux remains the same.

Step 1: Gauss's law does not require the charge to be at the centre. Step 2: Since the charge is inside, total flux is decided by enclosed charge. Step 3: Off-centre position makes field non-uniform on the surface, so field calculation becomes harder.

Step 1: Total flux depends on enclosed charge. Step 2: Changing radius does not change the charge inside. Step 3: Hence total flux remains unchanged, though field on the surface decreases.

Step 1: Field due to a point charge is inversely proportional to square of distance. Step 2: Making radius four times makes square of distance sixteen times. Step 3: So field on the surface becomes one-sixteenth.

Step 1: Field due to a point charge decreases with distance. Step 2: At the same time, area of the Gaussian sphere increases. Step 3: If enclosed charge is unchanged, the total effect gives the same flux.

Step 1: Total flux is decided by the sign of net enclosed charge. Step 2: Five positive and three equal negative charges give net positive charge. Step 3: Therefore total outward flux is positive.

Step 1: In Gauss's law, total flux depends on net charge inside. Step 2: Two positive and six negative charges give net negative charge. Step 3: Hence outward flux is negative.

Step 1: Four positive and four equal negative charges give zero net charge. Step 2: Individual charges may create field on the surface. Step 3: But net enclosed charge is zero, so total flux is zero.

Step 1: The outside charge can make the field strong on the surface. Step 2: But it is not enclosed by the surface. Step 3: Its lines enter and leave equally, so total flux remains zero.

Step 1: Gauss's law gives total flux. Step 2: To find field, the field must be expressible simply on the surface. Step 3: Without symmetry, direct field calculation is difficult.

Step 1: Validity of the law and ease of calculation are different. Step 2: In asymmetry, field can vary from point to point. Step 3: Without symmetry, isolating field from flux becomes difficult.

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: These directions are perpendicular, so end flux is zero.

Step 1: An infinite line charge has cylindrical symmetry. Step 2: Every point on a coaxial cylindrical surface is equally distant from the axis. Step 3: Hence field magnitude is taken the same there.

Step 1: Field of an infinite line charge is inversely proportional to distance. Step 2: If distance becomes five times, field becomes five times smaller. Step 3: Hence it becomes one-fifth.

Step 1: Field of a point charge varies inversely with square of distance. Step 2: Making distance five times makes square of distance twenty-five times. Step 3: So field becomes one-twenty-fifth.

Step 1: For point charge, spherical area grows with square of distance. Step 2: For line charge, cylindrical curved area grows with distance. Step 3: For infinite sheet, field is independent of distance.

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

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

Step 1: Field of an infinite plane sheet does not depend on distance. Step 2: Its value is connected to surface charge density. Step 3: Since density is unchanged, increasing distance does not change the field.

Step 1: Field of an infinite sheet is proportional to surface charge density. Step 2: Distance effect does not appear in this ideal case. Step 3: If density becomes three times, field also becomes three times.

Step 1: The Gaussian surface inside encloses zero charge. Step 2: Zero total flux alone does not always prove zero field everywhere. Step 3: Spherical symmetry shows that the field inside is zero at every point.

Step 1: A spherical shell shows spherical symmetry from outside. Step 2: An outside Gaussian sphere encloses the entire charge. Step 3: Hence outside field is like that of a point charge at the centre.

Step 1: An inner Gaussian surface encloses only the charge within it. Step 2: Enclosed charge grows as cube of distance and surface area as square. Step 3: Therefore field inside is proportional to distance.

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