Gauss's Law and its Applications

Step 1: If there were a tangential field at the surface, charges would move. Step 2: This cannot happen in electrostatic equilibrium. Step 3: Hence field just outside is perpendicular to the surface.

Step 1: Field near a conductor surface is related to surface charge density. Step 2: Larger surface charge density indicates stronger field. Step 3: Remember the direct relation between surface charge density and field.

Step 1: An infinite sheet looks the same everywhere. Step 2: Using a pillbox Gaussian surface gives a field independent of distance. Step 3: This result is for the ideal infinite sheet.

Step 1: For an infinite line charge, field is inversely proportional to distance. Step 2: Doubling distance makes the field half. Step 3: Keep this different from point charge inverse-square relation.

Step 1: Field of a point charge varies inversely with square of distance. Step 2: Doubling distance makes square of distance four times. Step 3: Hence field becomes one-fourth.

Step 1: Field due to an infinite charged sheet is independent of distance. Step 2: Doubling distance does not change the field value. Step 3: This is a special result for an ideal infinite sheet.

Step 1: For a point charge, the Gaussian surface is a sphere. Step 2: Area of sphere increases with square of distance. Step 3: Hence field varies inversely with square of distance.

Step 1: A cylindrical surface is chosen for a line charge. Step 2: The curved area of the cylinder increases with distance. Step 3: Therefore field varies inversely with distance for a uniform line charge.

Step 1: An infinite sheet has the same plane symmetry on both sides. Step 2: Gauss's law gives equal magnitude of field on both sides. Step 3: Direction depends on the sign of charge on the sheet.

Step 1: Field lines go outward from positive charge. Step 2: An infinite sheet has symmetry on both sides. Step 3: Therefore field points away from the sheet on both sides.

Step 1: Field lines go toward negative charge. Step 2: Symmetry is the same on both sides of the sheet. Step 3: Hence field points toward the sheet on both sides.

Step 1: In electrostatic equilibrium, electric field inside a conductor is zero. Step 2: If field is zero, flux through every part is zero. Step 3: Therefore total flux is zero.

Step 1: Electric field inside conductor material is zero. Step 2: Thus total flux through an internal Gaussian surface is zero. Step 3: By Gauss's law, net enclosed charge is zero.

Step 1: Conductors contain free charges. Step 2: A tangential field would make them move along the surface. Step 3: In electrostatic equilibrium this cannot occur, so field is normal.

Step 1: Total flux is the sum over the entire closed surface. Step 2: Positive and negative contributions on different parts may cancel. Step 3: So zero total flux does not require zero field everywhere.

Step 1: An external charge creates an electric field around it. Step 2: Its field may exist on the closed surface. Step 3: But lines entering and leaving balance, so net flux contribution is zero.

Step 1: In Gauss's law, total flux depends on enclosed charge. Step 2: Changing radius still encloses the same point charge. Step 3: Therefore total flux remains unchanged.

Step 1: Field of a point charge is inversely proportional to square of distance. Step 2: Doubling the sphere radius doubles distance from the charge. Step 3: Hence field on the surface becomes one-fourth.

Step 1: As distance increases, field of a point charge decreases. Step 2: At the same time, sphere area increases. Step 3: Since enclosed charge is same, total flux remains same by Gauss's law.

Step 1: Field of a line charge is radial. Step 2: Area vectors of the flat ends of the cylinder are along the axis. Step 3: These directions are perpendicular, so flux through the ends is zero.

Step 1: Field of an infinite sheet is perpendicular to the sheet. Step 2: For the side surface of the pillbox, the field does not cross the surface. Step 3: Hence flux through the side surface is zero.

Step 1: A pillbox Gaussian surface is taken across the sheet. Step 2: Field is perpendicular to the sheet on both sides. Step 3: Thus both flat faces contribute to flux.

Step 1: Zero total flux directly means net enclosed charge is zero. Step 2: Field may still exist at different points on the surface. Step 3: Therefore zero field everywhere is not always correct.

Step 1: Lines leaving the surface give positive outward flux. Step 2: If more lines leave than enter, total flux is positive. Step 3: By Gauss's law, net enclosed charge is positive.

Step 1: Lines entering the surface give negative contribution to outward flux. Step 2: If this contribution is larger, total flux is negative. Step 3: Hence net enclosed charge is negative.