Electric Charges, Fields, Potential & Capacitance

Charge · Coulomb's law · field · dipole · Gauss's law · potential · capacitance · Class 12 Physics

1What electric charge is

Charge is the property of matter that makes it respond to — and produce — electric forces, exactly as mass is the property that makes matter respond to gravity. There are two kinds, which we label positive and negative, and the founding observation of the whole subject is simple: like charges repel, unlike charges attract.

Three facts about charge run through every problem you will meet:

• It is conserved. Charge is never created or destroyed, only moved around. Rub a glass rod with silk and the rod becomes positive by exactly the amount the silk becomes negative — the electrons simply changed address.
• It is quantised. Every observed charge is a whole-number multiple of the elementary charge e = 1.6 × 10⁻¹⁹ C, so q = n e. You never find half an electron's worth.
• It is additive. The charge of a body is the plain algebraic sum of its parts, signs included.

Because e is so tiny, one coulomb is an enormous amount of charge (about 6 × 10¹⁸ electrons), which is why everyday charged objects carry only microcoulombs or nanocoulombs.

2Coulomb's law

Coulomb measured the force between two small charged spheres and found it behaves just like gravity's inverse-square law, but with charge in place of mass and with the freedom to be attractive or repulsive. For two point charges q₁ and q₂ a distance r apart:

The constant is k = 14πε₀ ≈ 9 × 10⁹ N m² C⁻², where ε₀ = 8.85 × 10⁻¹² C² N⁻¹ m⁻² is the permittivity of free space. Writing k as 1/4πε₀ looks like extra baggage, but that 4π is deliberately parked here so that Gauss's law later comes out clean.

Three things to hold on to. The force is along the line joining the charges. Its sign takes care of direction automatically: put in the actual signs of q₁ and q₂ and a positive F means repulsion, a negative F means attraction. And when many charges are present, the total force on one of them is the vector sum of the individual pair forces — this is the principle of superposition, and it is what lets us handle everything from a dipole to a charged sheet.

3The electric field & field lines

Rather than talk about action at a distance, we say a charge fills the space around it with an electric field. Any other charge then simply responds to the field at its own location. The field is defined as the force per unit positive test charge:

Divide Coulomb's law by the test charge q and the field of a single point charge Q falls straight out:

It points radially outward from a positive charge and radially inward toward a negative one. The great advantage of the field idea is that it is a property of space that exists whether or not a second charge is there to feel it.

Field lines make the field visible. A field line is drawn so that the tangent at every point gives the direction of E there, and the lines are packed close where the field is strong. Their rules are worth memorising:

• They start on positive charges and end on negative charges (or run off to infinity).
• They never cross — the field has one definite direction at each point.
• They are continuous, never form closed loops, and always meet the surface of a conductor at right angles.

4The electric dipole

A pair of equal and opposite charges +q and −q held a small distance 2a apart is an electric dipole — the model for a water molecule, a polarised atom, an antenna. Its strength and direction are packaged into the dipole moment

The field of a dipole is not inverse-square — the two charges nearly cancel, and what survives falls off faster, as 1/r³. Two positions matter for the exam.

On the axis (end-on)

At a point distance r from the centre along the axis, add the two point-charge fields (they point the same way here):

For a short dipole (r ≫ a) the a² is negligible and, with p = 2aq, this collapses to the memorable form

On the perpendicular bisector (broadside)

Off to the side, the two fields' components along p add while their sideways parts cancel. The result is half as large and points opposite to p:

So axial is twice equatorial at the same distance — a favourite one-mark comparison.

Torque in a uniform field

Place a dipole in a uniform field E at angle θ. The two charges feel equal and opposite forces qE, so there is no net force — but the forces are offset, so they twist. That twist (torque) is

The torque is zero when the dipole lines up with the field (θ = 0, stable) and maximum when it sits across it (θ = 90°). This is precisely why a dielectric's molecules swing to align with an applied field — the idea we cash in at the end of the chapter.

5Gauss's law & its standard results

Superposition works, but adding up fields charge-by-charge is brutal for anything spread out. Gauss's law is the shortcut, and it trades summation for symmetry. First, electric flux Φ counts the field lines threading a surface: Φ = E·A for a flat area perpendicular to E, and in general the surface integral Φ = ∮ E · dA.

In words: the total flux out of any closed surface depends only on the charge sealed inside it — not on where that charge sits, and not at all on charges outside. The recipe is always the same: pick a "Gaussian surface" that matches the symmetry so that E is constant and either along or across the surface, then read E off. Three results follow in a couple of lines each.

Infinite line of charge (linear density λ)

Wrap a cylinder of radius r and length ℓ around the wire. By symmetry E points radially and pierces only the curved side, area 2πrℓ. The enclosed charge is λℓ:

Notice this one falls off as 1/r, not 1/r² — a line source spreads its lines over a cylinder, not a sphere.

Infinite charged sheet (surface density σ)

Use a "pillbox" cylinder poking through the sheet, with end caps of area A on each side. Field lines leave both faces, so the flux is 2EA and the enclosed charge is σA:

Remarkably the field does not depend on distance from the sheet — it is uniform on each side.

Surface of a charged conductor

Inside a conductor the field is zero (charges rearrange until it is), so a pillbox has flux through its outer cap only. That single face gives EA = σA/ε₀, hence

This is exactly twice the field of an isolated sheet, because a conductor's charge sits on one face with all the field pushed to the outside.

6Electric potential & equipotentials

Force has a partner called energy; field has a partner called potential. The electric potential V at a point is the work done per unit charge in bringing a small positive test charge from infinity to that point. For a point charge Q, integrating the field kQ/r² in from infinity gives the clean result

Potential is a scalar — no direction to track — so for several charges you just add the numbers (with their signs). That makes potential far easier to work with than the field, and the two are linked by E = −dV/dr: the field points "downhill", from high potential to low. The potential energy of a charge q sitting at potential V is simply U = qV.

An equipotential surface joins all points at the same potential. Because it takes no work to move a charge along such a surface, the field can have no component along it — so field lines always cross equipotentials at right angles. Around a point charge the equipotentials are spheres; near a large charged plate they are flat sheets parallel to it.

7Capacitance

A capacitor stores charge, and with it energy. Give a conductor charge Q and its potential rises to V; the two are proportional, and the constant of proportionality is the capacitance:

Big capacitance means the body soaks up a lot of charge for only a small rise in potential. A farad is huge, so real capacitors are rated in microfarads or picofarads.

The parallel-plate capacitor

Two plates of area A a distance d apart, carrying +Q and −Q. Between them the fields of the two sheets add, giving a uniform E = σ/ε₀ = Q/ε₀A. The potential difference is field times gap, V = Ed = Qd/ε₀A. Divide:

So you get more capacitance from bigger plates or a narrower gap — exactly the two knobs a real capacitor is built around.

Combining capacitors — the mirror image of resistors

In parallel, both capacitors sit across the same voltage V and their charges add, so Q = Q₁ + Q₂ = (C₁ + C₂)V:

In series, each carries the same charge Q and the voltages add, V = Q/C₁ + Q/C₂:

Note the deliberate swap: capacitors add in parallel and reciprocate in series — the exact opposite of resistors. If you have already learnt resistor rules, just flip them.

Energy stored

Charging a capacitor means pushing charge onto a plate that already resists it, so the work builds up. Adding a sliver dq when the voltage is q/C costs (q/C) dq; integrating from 0 to Q:

Filling the gap: dielectrics

Slide an insulator (a dielectric) between the plates and its molecules polarise — their tiny dipoles swing to align with the field (the torque from section 4), setting up an internal field that opposes the applied one. The net field weakens, so for the same charge the voltage drops, and capacitance rises by the dielectric constant κ (often written K):

Since κ > 1 for every real material (about 80 for water, 2–5 for common plastics), inserting a dielectric always increases the capacitance — and lets the capacitor survive a higher voltage before breaking down.

8Common confusions to clear up

• Field vs force. The field E is a property of space set up by source charges; the force F = qE is what a charge placed there feels. The field exists even with no charge to push on.
• Potential vs potential energy. Potential V is energy per unit charge (a property of the location); potential energy U = qV belongs to a specific charge sitting there.
• Zero field ≠ zero potential. Inside a charged hollow conductor the field is zero but the potential is a constant non-zero value. At the midpoint of a dipole it is the reverse — zero potential, non-zero field.
• Gauss's law uses only enclosed charge. Charges outside the surface still bend the field lines, but every line they send in comes back out, so they add nothing to the net flux.
• Capacitors are the mirror of resistors. They add in parallel and combine reciprocally in series — precisely opposite to resistors. Mixing the two rules is the classic exam slip.
• Dielectric always helps. Because κ > 1, inserting a dielectric raises capacitance and never lowers it.

9Check yourself

Class 12 Physics · Electric Charges, Fields, Potential & Capacitance · aligned to NCERT Chapters 1–2 · SmartStudy.School