Electrostatics

Electrostatics is the study of charges that are not moving, and it is the foundation on which current electricity, magnetism and even parts of modern physics are built. For NEET, this unit reliably contributes several questions each year — usually a mix of quick formula-based numericals (Coulomb's law, potential, capacitors) and one or two conceptual questions on field lines, Gauss's law or dielectrics.

Electric Charge and Coulomb's Law

Charge is an intrinsic property of matter, like mass, but unlike mass it comes in two varieties, conventionally labelled positive and negative. Rubbing two dissimilar insulators (glass and silk, ebonite and fur) transfers electrons from one to the other; neither body creates charge, it is merely redistributed. Three properties of charge should be memorised because they are frequently tested as statements:

• Additivity: the net charge of a body is the algebraic sum of the charges it carries (a body with +5 µC and −8 µC has net −3 µC).
• Conservation: charge can neither be created nor destroyed; in any isolated system the total charge stays fixed. In pair production, an electron and positron appear together so the net charge remains zero.
• Quantisation: any observable charge is an integer multiple of the elementary charge, q = ne, with e = 1.6 × 10⁻¹⁹ C. So a charge of 1.8 × 10⁻¹⁹ C is impossible.

The force between two point charges was measured by Coulomb using a torsion balance. In magnitude,

F = (1/4πε₀) · q₁q₂ / r²

where ε₀ = 8.85 × 10⁻¹² C² N⁻¹ m⁻² and the constant 1/4πε₀ ≈ 9 × 10⁹ N m² C⁻². The force acts along the line joining the charges, is repulsive for like charges, attractive for unlike ones, and obeys Newton's third law exactly (equal magnitudes even if the charges are wildly unequal). If the charges are immersed in a medium of dielectric constant K (relative permittivity), the force is reduced by a factor K:

F_medium = F_vacuum / K

Coulomb's law applies strictly to point charges at rest. For more than two charges, use the principle of superposition: compute each pairwise force independently, then add them as vectors. The presence of a third charge does not alter the force between the original pair.

A very common problem type: two charges q₁ and q₂ are fixed a distance d apart; where on the line joining them is the field or force on a test charge zero? For like charges the null point lies between them; for unlike charges it lies outside, on the side of the smaller magnitude charge.

Electric Field, Field Lines and the Electric Dipole

The electric field at a point is defined as the force per unit positive test charge placed there:

E = F/q₀ (units N C⁻¹ or V m⁻¹)

The test charge must be vanishingly small so that it does not disturb the source distribution. For a point charge Q, E = kQ/r², directed radially outward for positive Q and inward for negative Q. Fields from several sources superpose vectorially.

Field lines are a visualisation tool with strict rules:

• They start on positive charge and end on negative charge (or at infinity).
• The tangent at any point gives the direction of E; the density of lines indicates the magnitude.
• Two field lines never cross (that would give two directions for E at one point).
• Lines are always perpendicular to the surface of a conductor and can never form closed loops in electrostatics.

An electric dipole is a pair of equal and opposite charges ±q separated by a small distance 2a. Its dipole moment p = q(2a) points from the negative charge to the positive charge (a convention students often reverse). The field of a dipole falls off as 1/r³, faster than that of a point charge:

• On the axial line: E = 2kp/r³ (directed along p)
• On the equatorial line: E = kp/r³ (directed antiparallel to p)

Note the axial field is twice the equatorial field at the same distance.

In a uniform external field E, the two charges of a dipole feel equal and opposite forces, so the net force is zero but the net torque is not:

τ = pE sin θ, or τ = p × E

The dipole therefore rotates to align with the field. Its potential energy is U = −pE cos θ = −p·E, minimum (stable equilibrium) at θ = 0 and maximum (unstable) at θ = 180°. Work done in rotating from θ₁ to θ₂ is W = pE(cos θ₁ − cos θ₂). In a non-uniform field, a dipole experiences both a torque and a net force.

Electric Flux and Gauss's Law

Electric flux through a small flat area is φ = E·A = EA cos θ, where θ is the angle between the field and the area's outward normal (the normal, not the surface itself). Flux is a scalar, measured in N m² C⁻¹, and can be positive, negative or zero.

Gauss's law states that the total electric flux through any closed surface equals the net charge enclosed divided by ε₀:

φ = ∮ E·dA = q_enclosed / ε₀

Key features: the law holds for any closed surface of any shape; charges outside the surface contribute nothing to the net flux (their lines enter and exit); and the flux depends only on the enclosed charge, not on its position inside. However, E at a point on the surface does depend on all charges, inside and outside — so Gauss's law is only useful for calculating E when the charge distribution has enough symmetry (spherical, cylindrical or planar) to let you argue that E is constant over a conveniently chosen Gaussian surface.

Standard results you should know by heart:

1. Infinite straight charged wire, linear charge density λ: E = λ/(2πε₀r), radially outward, falls as 1/r.
2. Infinite charged plane sheet, surface density σ: E = σ/(2ε₀), independent of distance, perpendicular to the sheet.
3. Charged conducting plate (charge on both faces): field just outside = σ/ε₀.
4. Uniformly charged thin spherical shell, total charge Q: outside, E = kQ/r² (as if all charge were at the centre); inside, E = 0 everywhere.
5. Uniformly charged solid non-conducting sphere: inside, E = kQr/R³ (grows linearly with r); outside, E = kQ/r².

For a conductor in electrostatic equilibrium, the field inside the material is zero, all excess charge resides on the outer surface, and the field just outside is σ/ε₀ perpendicular to the surface. This is why a hollow conductor acts as an electrostatic shield (Faraday cage): the cavity is field-free regardless of external fields.

Electric Potential and Potential Energy

Because the electrostatic force is conservative, we can define a potential energy. The potential at a point is the work done per unit charge in bringing a test charge from infinity to that point, against the electrostatic force:

V = W/q₀ (units volt = J C⁻¹); V is a scalar

• Point charge: V = kQ/r (with sign of Q included — this is a signed scalar sum, much easier than adding field vectors).
• Dipole: V = kp cos θ/r², so V = 0 everywhere on the equatorial plane.
• Charged spherical shell: V = kQ/R inside and on the surface (constant), V = kQ/r outside. Note the field is zero inside but the potential is not.

Potential difference and field are linked by E = −dV/dr: the field points in the direction of steepest decrease of potential. Conversely V_A − V_B = ∫ E·dl from A to B. A useful consequence: field lines always run from high potential to low potential.

The potential energy of a pair of charges is U = kq₁q₂/r, taken as zero when they are infinitely apart. For a system of several charges, sum over all distinct pairs — three charges give three terms, four charges give six terms. The work an external agent must do to assemble the configuration equals this total U; the work done by the field is its negative. For a charge q moved through a potential difference, W_external = q(V_final − V_initial), and this is independent of the path taken.

Equipotential surfaces join points at the same potential. No work is needed to move a charge along an equipotential, so equipotentials are always perpendicular to field lines. They are concentric spheres for a point charge, planes parallel to the sheet for a charged plane sheet, and the surface of any conductor is itself an equipotential (including its interior volume).

Capacitors, Dielectrics and Stored Energy

A capacitor is two conductors separated by an insulator. Giving them charges +Q and −Q creates a potential difference V, and the ratio

C = Q/V

is the capacitance, measured in farad (1 F = 1 C V⁻¹). One farad is enormous, so practical values are in µF, nF, pF. Capacitance depends only on geometry and the medium — not on the charge or voltage applied.

For a parallel plate capacitor with plate area A, separation d and vacuum between the plates:

C = ε₀A/d

Insert a dielectric of constant K filling the gap and C becomes Kε₀A/d. The dielectric molecules polarise, producing an internal field opposing the applied field, so the net field (and hence V, at constant Q) drops by a factor K while C rises by K. If instead a slab of thickness t (t < d) is inserted, C = ε₀A/[d − t + t/K]; for a conducting slab (K → ∞), C = ε₀A/(d − t).

Combinations:

• Series: the same charge sits on each capacitor, voltages add. 1/C_eq = 1/C₁ + 1/C₂ + … The equivalent capacitance is smaller than the smallest member.
• Parallel: the same voltage across each, charges add. C_eq = C₁ + C₂ + … The equivalent is larger than the largest member.

Energy stored in a charged capacitor:

U = ½QV = ½CV² = Q²/2C

This energy resides in the field between the plates, with energy density u = ½ε₀E² per unit volume (½Kε₀E² in a dielectric). Two important scenarios:

1. Battery disconnected (Q constant): inserting a dielectric reduces V and U (U = Q²/2C, so U falls by K). The slab is pulled in — the field does work.
2. Battery connected (V constant): inserting a dielectric increases Q and U (U = ½CV², so U rises by K). The battery supplies the extra energy.

When two charged capacitors are connected together, charge redistributes until potentials equalise; the common potential is V = (C₁V₁ + C₂V₂)/(C₁ + C₂), and some energy is always lost as heat and radiation unless the initial potentials were already equal.

Common Mistakes and Exam Traps

1. Confusing "field is zero" with "potential is zero." Inside a charged shell E = 0 but V = kQ/R ≠ 0. On the equatorial plane of a dipole V = 0 but E ≠ 0. Also, at the midpoint between two equal unlike charges, V = 0 but E is maximum; between two equal like charges, E = 0 but V ≠ 0.

2. Adding potentials as vectors or fields as scalars. Potential is a scalar — add with signs (algebraically, including the sign of each charge); the electric field is a vector and must be combined by vector (component or triangle) rules, not by summing magnitudes. Doing it the other way round — summing potentials as if directional, or fields as if they were plain scalars — gives a wrong answer even when every individual term is calculated correctly.

3. Ignoring the medium in Coulomb's law versus in capacitance. A dielectric of constant K reduces the force and field between charges by a factor K but increases capacitance by the same factor K — the two effects run in opposite directions, and switching from 'force in a medium' to 'capacitance with a dielectric' within the same problem requires applying K the correct way each time, not applying it mechanically throughout.

4. Using the wrong energy formula when a dielectric is inserted or removed. With the battery disconnected, Q stays fixed and $U = Q^2/2C$ is the easiest form to track as C changes; with the battery connected, V stays fixed and $U = \tfrac{1}{2}CV^2$ is the right form. Using the formula that assumes the wrong fixed quantity gives the energy change the wrong sign or the wrong magnitude.