Electric Charges and Fields Class 12 Physics — Revision Notes

TL;DR: Electric charge is a fundamental property of matter. Understanding electric charge and electric fields is essential for comprehending electromagnetism…

Electric charge is a fundamental property of matter. Understanding electric charge and electric fields is essential for comprehending electromagnetism and

Electric Charge

• Electric charge is fundamental property of matter
• Two types of charge: positive and negative
• Protons are positively charged; electrons are negatively charged
• Elementary charge e equals 1.6 times 10 to power negative 19 coulombs
• Charge is conserved; total charge in isolated system remains constant

Coulomb's Law

• Force between two point charges is directly proportional to product of charges
• Force is inversely proportional to square of distance between them
• Formula: F equals k times q1 times q2 divided by r squared
• k is Coulomb's constant approximately 9 times 10 to power 9
• Force is repulsive for like charges; attractive for opposite charges

Electric Field

• Electric field is region around charge where electric force acts
• Field strength equals force per unit charge: E equals F divided by q
• Field lines show direction and strength of electric field
• Field lines originate from positive charges and terminate on negative
• Electric field is vector quantity with direction and magnitude

Electric Potential and Potential Difference

• Electric potential is work done per unit charge to bring charge to a point
• Potential difference is difference in potential between two points
• Unit of potential is volt (V)
• Potential difference: V equals W divided by q
• Equipotential surfaces are perpendicular to field lines

Gauss's Law

• Electric flux through closed surface equals charge enclosed divided by epsilon
• Useful for calculating electric field with symmetry
• Applies to any closed surface called Gaussian surface
• Simplifies calculations for uniform charge distributions
• Foundation for understanding electric fields

Key Terms

• Electric Charge: Fundamental property of matter that causes electric force
• Electric Field: Region of space where electric force acts on charged particles
• Coulomb's Law: Law describing force between two stationary electric charges
• Electric Potential: Work done per unit charge to bring charge from infinity to a point

The three properties of charge, and why quantisation is usually ignored

Charge has three basic properties that every numerical in this chapter quietly relies on. Additivity means charge is a scalar and charges add algebraically, with signs included — a system holding plus 2 and minus 5 microcoulombs has a total of minus 3. Conservation means the total charge of an isolated system remains constant; charge is never created or destroyed, only transferred.

Quantisation means charge exists only in integral multiples of a basic unit, so q equals n times e, where n is an integer and e is 1.602 times ten to the power minus nineteen coulombs. You cannot have half an electron's worth of charge.

Yet at the macroscopic scale quantisation is ignored, and the chapter says so explicitly. Because e is so tiny, a charge of one microcoulomb already involves about ten to the power thirteen electrons, and the steps between allowed values are far too small to detect. Charge is therefore treated as continuous in practical calculations, exactly as a stream of sand is treated as a fluid despite being made of grains.

What an electric field is for

Coulomb's law gives the force between two charges directly, so why introduce a field at all? The answer is that the field separates the source from the test object. The field at a point is defined as the force per unit positive test charge placed there, so it describes what the region of space will do to any charge, without naming a second charge in advance.

The definition carries a subtlety that is regularly examined: the test charge must be vanishingly small. A real test charge would itself push the source charges around and change the very field being measured, so the field is defined as the limit of force per unit charge as the test charge approaches zero.

There is also a physical reason the field is more than a bookkeeping device. Coulomb's law as written implies an instantaneous force at a distance, whereas electromagnetic effects actually travel at the speed of light. The field carries energy and momentum and mediates the interaction, and this becomes indispensable when charges are in motion.

Coulomb's law and the superposition principle

Coulomb's law states that the force between two stationary point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.

• In symbols, F equals one over four pi epsilon nought, times q one q two divided by r squared. The force acts along the line joining the two charges.
• The constant one over four pi epsilon nought has the value 9 times ten to the power 9 newton metre squared per coulomb squared in vacuum, and epsilon nought, the permittivity of free space, is 8.854 times ten to the power minus twelve coulomb squared per newton per metre squared.
• The force is repulsive for like charges and attractive for unlike charges, and it obeys Newton's third law: the forces on the two charges are equal in magnitude and opposite in direction.
• The law applies strictly to point charges at rest. For extended bodies it holds only when the separation is much greater than their size.
• The superposition principle states that the force on any charge due to a number of other charges is the vector sum of the forces exerted by each of them individually, and that the presence of the other charges does not alter any individual pair interaction.
• One coulomb is a very large unit: it is the charge which, placed at a distance of one metre from an equal charge in vacuum, experiences a repulsive force of 9 times ten to the power 9 newtons.

Electric field and field lines

The electric field at a point is the force experienced per unit positive test charge placed at that point, in the limit that the test charge tends to zero.

• Its SI unit is the newton per coulomb, and it is a vector directed along the force on a positive charge.
• For a point charge, the magnitude of the field at distance r is one over four pi epsilon nought times q divided by r squared, directed radially outward for a positive charge and radially inward for a negative charge.
• Field lines start from positive charges and end on negative charges. If there is a single charge, the lines start or end at infinity.
• Field lines never intersect. If two lines crossed, there would be two directions of the field at the point of intersection, which is impossible.
• In a charge-free region field lines are continuous curves without any breaks, and electrostatic field lines do not form closed loops.
• The relative closeness of the lines indicates the strength of the field: lines crowd where the field is strong and spread out where it is weak.
• The tangent to a field line at any point gives the direction of the field there.

The electric dipole

An electric dipole is a pair of equal and opposite point charges separated by a small distance, usually written as 2a.

• The electric dipole moment is the product of the magnitude of either charge and the separation, so p equals q times 2a. It is a vector whose direction is taken from the negative charge to the positive charge, and its SI unit is the coulomb metre.
• On the axial line, at a distance r from the centre and for r much greater than a, the field has magnitude one over four pi epsilon nought times 2p divided by r cubed, and it points in the direction of the dipole moment.
• On the equatorial line, under the same condition, the field has magnitude one over four pi epsilon nought times p divided by r cubed, and it points opposite to the dipole moment.
• So the axial field is twice the equatorial field at the same distance, and the two are opposite in direction. This ratio of two to one is a very common one-mark question.
• The dipole field falls off as the inverse cube of the distance, faster than the inverse square of a single point charge, because the fields of the two opposite charges increasingly cancel as you move away.
• In a uniform external field the two forces on a dipole are equal and opposite, so the net force is zero, but they do not act along the same line and therefore produce a torque. The torque equals p cross E, of magnitude pE sin theta.
• The torque tends to align the dipole with the field. In a non-uniform field the net force is not zero, so the dipole experiences both a torque and a translational force.

Electric flux and Gauss's law

Electric flux through a small area element is the product of the field, the area, and the cosine of the angle between the field and the outward normal to the area.

• Its SI unit is the newton metre squared per coulomb, and it is a scalar quantity. It is proportional to the number of field lines crossing the area.
• Gauss's law states that the total electric flux through any closed surface equals one over epsilon nought times the total charge enclosed by that surface.
• The closed surface used is called a Gaussian surface, and it is imaginary — chosen for convenience, not physically present.
• Charges lying outside the Gaussian surface contribute nothing to the total flux, since every line they send in also comes out. They do, however, contribute to the field at points on the surface.
• For an infinitely long straight uniformly charged wire, the field is lambda divided by two pi epsilon nought r, where lambda is the linear charge density. It falls off as one over r rather than one over r squared.
• For a uniformly charged infinite plane sheet the field is sigma divided by two epsilon nought, where sigma is the surface charge density. Remarkably, this does not depend on the distance from the sheet at all.
• For a uniformly charged thin spherical shell, the field at an outside point is the same as if the entire charge were concentrated at the centre, while the field at any point inside the shell is zero.

Solved Numericals — Step by Step

Worked Examples

• Field inside a charged spherical shell is exactly zero — A Gaussian sphere drawn inside encloses no charge, so by Gauss's law the flux is zero, and by symmetry the field itself must be zero everywhere inside.
• The field of an infinite charged sheet does not weaken with distance — E = σ/2ε₀ contains no r. Moving away brings more of the sheet into view at a shallower angle, exactly compensating the increased distance.
• A charge outside a closed surface contributes zero net flux — Every field line that enters the surface also leaves it, so the contributions cancel. It still contributes to the field at points on the surface, which is why Gauss's law is about total flux, not about E pointwise.
• A comb rubbed on dry hair picks up small pieces of paper — The charged comb induces a dipole in each neutral piece of paper, and because the comb's field is non-uniform the induced dipole experiences a net attractive force.

Common Mistakes Students Make

• Treating charge as a vector and adding magnitudes. Charge is a scalar and adds algebraically with signs; it is the FORCE that is a vector and requires vector addition under superposition.
• Forgetting to convert microcoulombs to coulombs, or centimetres to metres, before substituting into Coulomb's law. Because the distance is squared, an error in r is doubled in the answer.
• Assuming the field inside a spherical shell equals the surface value. It is exactly zero everywhere inside, because the enclosed charge is zero.
• Concluding that if the flux through a closed surface is zero then the field on it is zero. Zero net flux means only that the enclosed charge is zero; the field on the surface may well be non-zero.
• Using the inverse-square law for a dipole field. A dipole field falls as the inverse cube of the distance, because the two opposite charges increasingly cancel.
• Saying a dipole in a uniform field experiences a net force. The net force is zero and only a torque acts; a net force appears only in a non-uniform field.
• Drawing electrostatic field lines as closed loops or letting them intersect. They begin on positive and end on negative charge, and crossing would give two field directions at one point.
• Applying the wire and sheet formulas without checking the conditions. They are derived for an infinitely long wire and an infinite plane sheet, so they hold only near a finite object, far from its edges.

Where This Shows Up in Real Life

Lightning conductors and the action of points

Charge accumulates most densely where a conductor's surface is sharply curved, so the field just outside a pointed tip is far stronger than near a flat region. At a sharp point the field can become large enough to ionise the surrounding air, allowing charge to leak away steadily. A lightning conductor exploits this by presenting a sharp metal point to the sky, discharging the region continuously and, if a strike does occur, conducting it safely to earth.

Electrostatic precipitators in industry

Smoke leaving a chimney carries fine ash particles. In a precipitator the particles are given a charge as they pass through a region of strong field, and are then attracted to oppositely charged collecting plates where they stick and are periodically removed. The whole device is an application of the force on a charge in an electric field, and it is one of the main technologies for reducing particulate pollution from thermal power stations.

Why a car is a safe place in a thunderstorm

The field inside a closed conductor with no enclosed charge is zero, the same result obtained for the spherical shell. A metal car body approximates such an enclosure, so charge deposited by a strike resides on the outer surface and the field inside remains essentially zero. The same principle is used deliberately in electrostatic shielding, where sensitive instruments are enclosed in a metal case to keep external fields out.

Polar molecules and why water dissolves salt

A water molecule has a permanent dipole moment because its positive and negative charge centres do not coincide. When salt is added, these dipoles orient themselves around the sodium and chloride ions and screen the attraction between them, weakening the force holding the crystal together. The dipole in a field, and the field of a dipole, are therefore the basis of an everyday chemical fact.

Board-Style Questions with Model Answers

State Coulomb's law and express it in vector form. Mention two limitations. (3 marks)

Coulomb's law states that the force of interaction between two stationary point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them, and acts along the line joining them. In magnitude, F equals one over four pi epsilon nought times q one q two divided by r squared. In vector form the force on charge two due to charge one is this magnitude multiplied by the unit vector directed from charge one towards charge two, which automatically gives repulsion for like charges and attraction for unlike ones. Two limitations are that the law holds only for point charges at rest, so it cannot be applied directly to extended bodies unless their separation is much greater than their size, and that it applies to charges at rest, since moving charges also produce magnetic effects.

Define electric field intensity. Why must the test charge be vanishingly small? (2 marks)

The electric field intensity at a point is defined as the force experienced per unit positive test charge placed at that point, taken in the limit as the test charge tends to zero. Its SI unit is the newton per coulomb. The test charge must be vanishingly small because a charge of appreciable magnitude would itself exert forces on the source charges, causing them to redistribute, and the field measured would then not be the original field that was to be determined.

State Gauss's law and use it to find the electric field due to a uniformly charged infinite plane sheet. (5 marks)

Gauss's law states that the total electric flux through any closed surface is equal to one over epsilon nought times the net charge enclosed by that surface. To apply it to an infinite plane sheet with uniform surface charge density sigma, choose as the Gaussian surface a cylinder whose axis is perpendicular to the sheet, with its two flat circular faces of area A lying on either side and equidistant from the sheet. By symmetry the field is perpendicular to the sheet and has the same magnitude on both faces. No flux passes through the curved surface, since the field there is parallel to the surface, so the total flux is E times A through each of the two faces, giving 2EA in all. The charge enclosed by the cylinder is sigma times A. By Gauss's law, 2EA equals sigma A divided by epsilon nought, so E equals sigma divided by 2 epsilon nought. The result is independent of the distance from the sheet, so the field is uniform on either side.

Derive the expression for the torque on an electric dipole placed in a uniform electric field. (3 marks)

Consider a dipole of charges plus q and minus q separated by 2a, placed in a uniform field E with its axis making an angle theta with the field. The force on the positive charge is qE in the direction of the field and on the negative charge is qE opposite to it. These forces are equal in magnitude and opposite in direction, so the net force on the dipole is zero and it does not undergo translation. However they do not act along the same line, so they constitute a couple. The magnitude of the torque is one force multiplied by the perpendicular distance between their lines of action, which is 2a sin theta. Therefore the torque equals qE times 2a sin theta, that is pE sin theta, where p equals 2aq is the dipole moment. In vector form the torque is p cross E, and it acts to align the dipole with the field.

Using Gauss's law, show that the electric field inside a uniformly charged thin spherical shell is zero. (3 marks)

Consider a thin spherical shell of radius R carrying charge q distributed uniformly over its surface. To find the field at a point inside, choose as the Gaussian surface a concentric sphere of radius r less than R passing through that point. By symmetry the field at every point of this surface has the same magnitude and is directed radially, so the flux through it is E times four pi r squared. The Gaussian surface lies entirely inside the shell and therefore encloses no charge at all, since all the charge resides on the outer surface of the shell. By Gauss's law the flux equals the enclosed charge divided by epsilon nought, which is zero. Hence E times four pi r squared equals zero, and since the area is not zero, E must be zero at every point inside the shell.

State any three properties of electric field lines and explain why two field lines can never intersect. (3 marks)

First, field lines start from positive charges and end on negative charges, and in the case of a single charge they start or end at infinity. Second, in a charge-free region they are continuous curves without any breaks, and electrostatic field lines never form closed loops. Third, the relative closeness of the lines indicates the relative strength of the field, the lines being crowded where the field is strong. Two field lines can never intersect because the tangent to a field line at any point gives the direction of the electric field there; if two lines crossed, then at the point of intersection there would be two different tangents and hence two different directions of the field at the same point, which is impossible since the field at a point has a unique direction.

Assertion–Reason Questions (CBSE Format)

For each question choose: (a) both A and R are true and R is the correct explanation of A; (b) both are true but R is not the correct explanation; (c) A is true but R is false; (d) A is false but R is true.

Quick Self-Check

Key Terms Glossary

Quantisation of charge

The property that charge exists only in integral multiples of the elementary charge e, so q = ne.

Conservation of charge

The principle that the total charge of an isolated system remains constant; charge can be transferred but never created or destroyed.

Coulomb

The SI unit of charge; the charge which, placed 1 m from an equal charge in vacuum, experiences a repulsive force of 9 × 10⁹ N.

Permittivity of free space

The constant ε₀, of value 8.854 × 10⁻¹² C² N⁻¹ m⁻², appearing in Coulomb's law and Gauss's law.

Superposition principle

The rule that the force on a charge due to several others is the vector sum of the individual forces, each unaffected by the presence of the rest.

Electric field intensity

The force per unit positive test charge at a point, in the limit that the test charge tends to zero; unit N/C.

Electric field lines

Imaginary curves whose tangent at any point gives the direction of the field, drawn more densely where the field is stronger.

Electric dipole

A pair of equal and opposite point charges separated by a small distance 2a.

Electric dipole moment

The product of either charge and the separation, p = q × 2a, directed from the negative to the positive charge; unit C m.

Electric flux

The scalar quantity E ΔS cos θ, proportional to the number of field lines crossing an area; unit N m²/C.

Gaussian surface

An imaginary closed surface chosen for convenience when applying Gauss's law.

Linear charge density

Charge per unit length, denoted λ, used for a charged wire.

Surface charge density

Charge per unit area, denoted σ, used for a charged sheet or shell.

Electrostatic shielding

The use of a conducting enclosure to keep the field inside zero, since a closed conductor with no enclosed charge has no interior field.

One-Minute Revision

• Charge is additive (scalar, with signs), conserved, and quantised as q = ne with e = 1.6 × 10⁻¹⁹ C.
• Quantisation is ignored macroscopically because e is far too small for the steps to be noticed.
• Coulomb's law: F = (1/4πε₀)q₁q₂/r², with 1/4πε₀ = 9 × 10⁹ N m²/C², along the line joining the charges.
• Superposition: forces add as vectors, and no pair interaction is altered by the other charges.
• E = F/q₀ with the test charge tending to zero, so it does not disturb the source.
• Field lines: start on positive, end on negative, never cross, never form closed loops, crowding shows strength.
• Dipole moment p = q × 2a, directed from −q to +q, unit C m.
• Axial field 2p/4πε₀r³; equatorial field p/4πε₀r³ and opposite. Axial is twice equatorial.
• Dipole fields fall as 1/r³, faster than a point charge's 1/r².
• Uniform field: net force on a dipole is zero, torque = pE sin θ. Non-uniform field: both force and torque.
• Gauss's law: total flux through a closed surface = q_enclosed/ε₀. External charges add no net flux.
• Wire: E = λ/2πε₀r. Infinite sheet: E = σ/2ε₀, independent of distance. Shell: outside as a point charge, inside exactly zero.

Syllabus reference: NCERT Class 12 Physics — Electric Charges and Fields. It is the foundation for Electrostatic Potential and Capacitance and is assumed throughout Current Electricity.

Frequently Asked Questions

Why is electric field a vector quantity?

Electric field has both magnitude and direction. The direction is the direction a positive charge would be pushed, making it a vector quantity that can be added and resolved into components.

What is the relationship between electric field and potential?

Electric field is the negative gradient of potential. Mathematically, E equals negative dV divided by dx. The field points in the direction of decreasing potential.

If charge is quantised, why do we treat it as continuous in problems?

Because the elementary charge is extremely small compared with the charges met in practice. A charge of one microcoulomb already corresponds to roughly ten to the power thirteen electronic charges, so the jump from one allowed value to the next is utterly negligible on that scale. Treating charge as continuous is therefore an excellent approximation at the macroscopic level, in the same way that a stream of sand can be treated as a fluid even though it is made of discrete grains. Quantisation matters only when the number of charges involved is small.

Does Gauss's law mean charges outside a closed surface have no effect?

It means they make no contribution to the net flux through that surface, which is not the same thing. Every field line from an external charge that enters the closed surface must also leave it, so the inward and outward contributions cancel exactly. However, that external charge does contribute to the electric field at individual points on the surface. This is why Gauss's law is a statement about the total flux and the enclosed charge only, and why it can be used to find E only when symmetry guarantees that E is constant over the chosen surface.

Why does the field of an infinite sheet not decrease with distance?

Because moving further away brings a larger portion of the sheet into effective view. The contribution of each small element does weaken with distance as expected, but the number of elements at a comparable distance grows in exactly the compensating proportion, and the components parallel to the sheet cancel by symmetry. The two effects balance precisely for an infinite sheet, leaving E equal to sigma over two epsilon nought everywhere. For a real finite sheet the result holds only close to the sheet and far from its edges, where it looks effectively infinite.

Why does a dipole field fall off faster than a point charge field?

A dipole is made of two equal and opposite charges. At a point far away, the fields of the two charges are almost equal in magnitude but nearly opposite in direction, so they very nearly cancel. What survives is the small difference arising from their slightly different distances, and that difference shrinks faster than either field individually. Working this out gives a dependence on one over r cubed rather than one over r squared. The same reasoning explains why a neutral object with separated charge still produces a field at all, but only a weak and rapidly diminishing one.

More Class 12 Physics Revision Notes

• Current Electricity

🤖 Stuck on any of these? Ask Syllab's free AI Tutor to explain step by step →

More free resources for this chapter

• NCERT Solutions →
• Formula Sheet →