Class 12 Physics · Chapter 1 NotesElectric Charges and Fields

Rubbing a glass rod with silk, or a plastic rod with fur, makes the rod attract small bits of paper. Careful experiments showed that there are only two kinds of electrification, and that like charges repel while unlike charges attract. This difference is called the polarity of charge. By convention, the charge on a glass rod rubbed with silk is taken as positive and that on a plastic rod rubbed with fur as negative. A body with no net charge is electrically neutral. Charging happens because electrons, which are less tightly bound in atoms, are transferred from one body to another. A body that loses electrons becomes positive, and one that gains electrons becomes negative. No new charge is created during rubbing.

• Like charges repel, unlike charges attract
• Positive and negative are the two polarities of charge
• A gold-leaf electroscope detects charge; the leaves diverge when charged
• Charging by rubbing involves transfer of electrons, not creation of charge

Substances that allow electricity to pass through them easily are called conductors. In metals, human and animal bodies, and the earth, charges are comparatively free to move. Substances such as glass, porcelain, plastic, nylon and wood offer high resistance to the passage of electricity and are called insulators. When charge is placed on a conductor, it spreads over the entire surface, whereas on an insulator it stays where it was placed. This is why a plastic comb gets charged on combing dry hair, but a metal spoon does not: charge on the metal leaks through our body to the ground, since both are conductors. A metal rod with a wooden or plastic handle can be charged if its metal part is not touched.

• Conductorscharges free to move, e.g. metals, human body, earth
• Insulatorscharges not free to move, e.g. glass, plastic, nylon, wood
• Charge spreads over the surface of a conductor but stays in place on an insulator

Charge has three basic properties. Additivity means the total charge of a system is the algebraic sum of individual charges, with proper signs; charge is a scalar and has magnitude but no direction. Conservation means the total charge of an isolated system remains unchanged with time, even though charges may be redistributed or particles created and destroyed. Quantisation means the charge q on any body is always an integral multiple of a basic unit e, that is q = ne, where n is any integer. The charge on an electron is –e and on a proton is +e, with e = 1.602192 × 10⁻¹⁹ C. The SI unit of charge is the coulomb (C). At the macroscopic level, e is so small that quantisation can be ignored.

• Additivityq total = q₁ + q₂ + q₃ + ... (with signs)
• Conservationtotal charge of an isolated system stays constant
• Quantisationq = ne, where n = 0, ±1, ±2, ...
• e = 1.602192 × 10⁻¹⁹ C; 1 mC = 10⁻⁶ C, 1 mC (milli) = 10⁻³ C

Coulomb's law gives the force between two point charges. If two point charges q₁ and q₂ are separated by a distance r in vacuum, the magnitude of the force is F = k q₁q₂ / r², where k = 1/(4πε₀) and ε₀ is the permittivity of free space, equal to 8.854 × 10⁻¹² C² N⁻¹ m⁻². In SI units k is about 9 × 10⁹ N m² C⁻². The force acts along the line joining the charges; it is repulsive for like charges and attractive for unlike charges. In vector form F₂₁ = (1/4πε₀)(q₁q₂/r²) r̂₂₁, and F₁₂ = –F₂₁, so Coulomb's law agrees with Newton's third law. When several charges are present, the principle of superposition states that the total force on a charge is the vector sum of the forces due to the other charges taken one at a time.

• F = k q₁q₂ / r², with k = 1/(4πε₀) ≈ 9 × 10⁹ N m² C⁻²
• ε₀ = 8.854 × 10⁻¹² C² N⁻¹ m⁻²
• Force is along the line joining the charges; like charges repel, unlike attract
• SuperpositionF₁ = F₁₂ + F₁₃ + ... + F₁ₙ (vector sum)

A charge Q produces an electric field everywhere around it. The electric field E at a point is the force experienced by a unit positive test charge placed at that point, E = F/q, and its SI unit is N/C. For a point charge Q at a distance r, E = (1/4πε₀)(Q/r²) r̂. The field points radially outward for a positive charge and radially inward for a negative charge, and its magnitude depends only on the distance r. For a system of charges, the field at a point is the vector sum of the fields due to individual charges. Electric field lines are curves whose tangent at each point gives the direction of the field. Field lines start from positive charges and end at negative charges, are continuous in charge-free regions, never cross each other, and do not form closed loops.

• E = F/q; unit is N/C (also V/m)
• E due to a point chargeE = (1/4πε₀)(Q/r²) r̂
• Field lines crowd where the field is strong and are spaced apart where it is weak
• Two field lines never cross; electrostatic field lines do not form closed loops

Electric flux through an area element is Δφ = E·ΔS = E ΔS cos θ, where θ is the angle between E and the area vector ΔS. For a closed surface, the area vector is taken along the outward normal. Its unit is N C⁻¹ m². Gauss's law states that the total electric flux through a closed surface equals q/ε₀, where q is the total charge enclosed by the surface. The law holds for any closed surface of any shape or size. The surface chosen is called a Gaussian surface, and it should not pass through a discrete charge. Gauss's law is useful for finding the field when the charge distribution has symmetry, and it rests on the inverse-square dependence of Coulomb's law.

• Δφ = E·ΔS = E ΔS cos θ
• Gauss's lawφ = q/ε₀, q = total charge enclosed
• Flux through a closed surface is zero if no charge is enclosed
• Gaussian surface should not pass through a discrete point charge

An electric dipole is a pair of equal and opposite point charges q and –q separated by a distance 2a. Its dipole moment p = q × 2a, directed from –q to q, and its SI unit is C m. The total charge of a dipole is zero, but its field is not zero because the charges are separated. On the dipole axis at a large distance r, E = 2p/(4πε₀r³) along p. On the equatorial plane, E = –p/(4πε₀r³), opposite to p. The dipole field falls off as 1/r³, faster than the 1/r² dependence for a single charge. In a uniform external field E, the net force on a dipole is zero but it experiences a torque τ = p × E, of magnitude pE sin θ, which tends to align p with E.

• Dipole moment p = q × 2a, directed from –q to q; unit C m
• Axial fieldE = 2p/(4πε₀r³); equatorial field: E = –p/(4πε₀r³), for r >> a
• Dipole field varies as 1/r³
• Torque in uniform fieldτ = p × E, magnitude pE sin θ; net force is zero

Gauss's law gives simple results for symmetric charge distributions. For an infinitely long straight wire with uniform linear charge density λ, the field at a perpendicular distance r is E = λ/(2πε₀r), directed radially outward for positive λ. For an infinite plane sheet with uniform surface charge density σ, the field is E = σ/(2ε₀), normal to the sheet and independent of distance. For a uniformly charged thin spherical shell of radius R and total charge q, the field outside is E = q/(4πε₀r²), as if all the charge were at the centre, while the field inside the shell is zero. For a continuous distribution, charge density may be defined as linear (λ, C/m), surface (σ, C/m²) or volume (ρ, C/m³).

• Long straight wireE = λ/(2πε₀r)
• Infinite plane sheetE = σ/(2ε₀), independent of distance
• Thin spherical shellE = q/(4πε₀r²) outside (r ≥ R), E = 0 inside (r < R)
• Charge densitiesλ = ΔQ/Δl, σ = ΔQ/ΔS, ρ = ΔQ/ΔV