Revision: Electrostatics >> Electric Charges and Fields Physics (Theory) ISC (Science) ISC Class 12 CISCE

Electric charge is an intrinsic property of certain fundamental particles (like electrons and protons) that gives rise to electric and magnetic forces, causing them to experience a force when placed in an electromagnetic field.

OR

Electric charge is the physical property of matter that causes it to experience a force when placed in an electric field.

Key facts:

• SI Unit: Coulomb (C)
• Dimensional Formula: [A T] (Ampere × Time)
• Two types exist: positive and negative

The smallest unit of electric charge, denoted by e, is called the elementary charge.

OR

The smallest unit of free electric charge, denoted e, with value e ≈ 1.602 × 10⁻¹⁹ C.

A charged body whose size is negligibly small compared to the distance between the charges under consideration, is called a point charge.

OR

A charged body is treated as a point charge when its physical size is negligible compared to the distance between it and other charges under consideration.

Atoms of the same element that have the same atomic number but different numbers of neutrons (and hence different mass numbers) are called isotopes.

A charged atom, formed when a neutral atom gains or loses electrons, is called an ion.

The process by which a free neutron (outside the nucleus) transforms, occurring with a defined half-life, is called beta decay.

The force that binds protons and neutrons together in the nucleus, overcoming the electrostatic repulsion between protons, is called the strong nuclear force.

A negatively charged ion, formed when an atom gains one or more electrons, is called an anion.

A positively charged ion, formed when an atom loses one or more electrons, is called a cation.

The dense central core of an atom, containing protons and neutrons and holding nearly all of the atomic mass, is called the nucleus.

A positively charged particle found inside the nucleus, composed of quarks, is called a proton.

An electrically neutral particle found inside the nucleus, comparable in mass to the proton, is called a neutron.

A negatively charged fundamental particle that orbits the nucleus in shells/orbitals and determines the atom's chemical behaviour is called an electron.

The number of protons present in the nucleus of an atom, which uniquely identifies a chemical element, is called the atomic number (Z).

The total number of protons and neutrons present in the nucleus of an atom is called the mass number (A).

Substances whose resistance to the movement of charges is intermediate between conductors and insulators, are called semiconductors.

Conductors are those through which electric charge can easily flow. Metals, human body, earth, mercury and electrolytes are conductors of electricity.

OR

The material through which electric charge can flow easily is called a conductor.

Those substances in which electric charge cannot flow are called ‘insulators' (or dielectrics). Glass, hard-rubber, plastics and dry wood are insulators. Insulators have practically no free electrons.

OR

The material in which electrons are tightly bound to the nucleus and thus not available for conductance is called an insulator.

OR

Substances which offer high resistance to the passage of electricity and do not allow electricity to pass through them easily, are called insulators.

Charging that occurs when two different materials are rubbed together, causing electron transfer.

Process of creating an imbalance between protons and electrons in a body, giving it a net electric charge.

A ranked list of materials based on their tendency to gain or lose electrons on contact.

Charging without direct contact, caused by the redistribution of charge due to a nearby charged body.

Charging by direct physical contact between a charged and an uncharged conductor.

Charging by friction is the process by which two different insulating (dielectric) materials become electrically charged — equal in magnitude but opposite in sign — when rubbed against each other, due to the transfer of electrons from one surface to the other.

The process of transferring electric charge from a charged object to a neutral (or oppositely charged) conductor through direct physical contact, until electrostatic equilibrium is reached.

Key idea: Electrons flow from the region of higher electron concentration to lower, until both bodies reach the same electric potential.

The process of charging a neutral conductor without any direct contact with a charged object, by redistributing charges within the conductor through electrostatic influence.

One coulomb is the amount of charge which, when placed at a distance of one metre from another charge of the same magnitude in vacuum, experiences a force of 9.0 × 10⁹ N.

The total electrostatic force on any charge in a system of multiple charges is the vector sum of the forces exerted on it individually by each of the other charges, with each pairwise force being unaffected by the presence of the remaining charges.

A force that acts to bring a displaced charge back toward its original equilibrium position.

A charge is said to be in equilibrium when the net electrostatic force acting on it due to all other charges in the system is zero.

A group of two or more point charges arranged in space such that they exert forces on one another.

The space surrounding an electric charge q in which another charge q₀ experiences a (electrostatic) force of attraction or repulsion, is called the electric field of the charge q.

OR

Electric field due to a charge Q at a point in space may be defined as the force that a unit positive charge would experience if placed at that point.

OR

The region surrounding an electric charge or a group of charges in which another charge experiences a force is called an electric field.

The charge q that tests the effect of the source charge is called the test charge.

The region in which the charge experiences an electric force is the electric field around the charge.

The charge Q that produces the electric field is called the source charge.

The electric field intensity at any point is the strength of the electric field at that point.

• It is defined as the force experienced by a unit positive charge placed at that point.

\[\vec{E}=\frac{\vec{F}}{q_0}=\frac{kq}{r^2}\hat{r}=\frac{kq}{r^3}\vec{r}\]

• The SI unit of E is NC⁻¹ (newtons per coulomb).

Electric line of force is an imaginary curve drawn in an electric field such that the tangent at any point on it gives the direction of the electric field at that point.

• It represents the path along which a unit positive test charge would tend to move if free to do so.
• Lines are imaginary — they have no physical existence; they are a visualization tool only.

Electric dipole moment \[\vec p\] is a vector quantity defined as the product of the magnitude of either charge and the separation between them.

Mathematical definition: \[\vec p\] = q × 2a

The electric dipole moment is defined as the product of the magnitude of one of the charges and the distance between the two equal and opposite charges.

“The line joining the two charges, pointing from the negative charge to the positive charge. This is known as the ‘direction of dipole axis’.”

OR

The line passing through both charges +q and −q is called the dipole axis (also called the axial line or axis of the dipole).

An electric dipole is a pair of equal and opposite point charges placed at a short distance apart.

OR

A system formed by two equal and opposite point charges placed at a small distance apart is called an electric dipole.

OR

A system of two equal and opposite point charges +q and −q separated by a small fixed distance 2a is called an electric dipole.

• The total charge of an electric dipole is zero
• Zero net charge does not mean zero electric field - the field exists because the charges are spatially separated​
• The midpoint of the line joining −q and +q is called the centre of the dipole

The midpoint of the line joining the two charges is called the centre of the dipole.

The line passing through the centre of the dipole and perpendicular to the dipole axis is called the equatorial line.

OR

The plane passing through the centre of the dipole and perpendicular to the dipole axis is called the equatorial plane; the line along which the equatorial field is evaluated is the equatorial line (perpendicular bisector).

A system consisting of two equal and opposite point charges (+q and −q) separated by a small distance (2a) is called an electric dipole.

The vector quantity equal to the product of the magnitude of either charge and the distance between the two charges, directed from −q to +q, is called the dipole moment.

The straight line passing through both charges, +q and −q, of a dipole is called the dipole axis.

Equilibrium of a charged body in a uniform electric field occurs when the electric force on the charge exactly balances the gravitational force (weight) acting on it, resulting in zero net force.

Area Vector is a vector quantity used to represent a planar (or elemental) surface area, having a magnitude equal to the area and a direction along the normal to the surface.

Magnetic flux (Φ_B​) is the total number of magnetic field lines passing normally through a given area.

Flux of a vector field through a surface is defined as the total number of field lines passing normally through that surface. Mathematically, it is the scalar (dot) product of the field vector and the area vector.

Electric flux (Φ_E​) through a surface is the measure of the electric field lines passing through that surface.

An imaginary, arbitrary closed three-dimensional surface chosen to calculate the electric flux due to a charge distribution using Gauss's law.

Valid Gaussian surfaces: sphere, cylinder, cube (all closed surfaces).
Invalid Gaussian surfaces: disc, square, open hemisphere (not closed).

Electric field intensity just outside a charged conductor is the strength of the electric field at a point infinitesimally close to (but outside) the surface of a charged conductor, arising due to the surface charge density on the conductor.

A uniformly charged sphere is a sphere in which charge q is distributed evenly throughout its volume (non conducting/insulating) or over its surface (conducting), with uniform volume charge density ρ = \[\frac{q}{\frac{4}{3}\pi R^3}.\]

“1 steradian is the solid angle subtended by a part of the surface of a sphere at the centre of the sphere, when the area of the part is equal to the square of the radius of the sphere.”

The arc of a circle subtends an angle at the centre of the circle. This angle is called a 'plane angle'.

In an electric field, the ratio of electric flux through a surface to the area A of the surface is called the 'electric flux density' at the location of the surface.

Mathematical Definition:

Electric flux density = \[\frac {Φ_E}{A}\]

For a plane surface normal to the electric field:

Electric flux density = \[\frac {E A}{A}\] = E

The electric flux linked with a surface in an electric field may be defined as the surface integral of the normal component of the electric field over that surface.

The electric flux is a measure of the number of lines of force passing through some surface held in the electric field. It is denoted by Ф_E.

OR

The electric flux through a surface is the dot product of the electric field and the area vector of the surface, is called electric flux.

“1 radian is the angle which an arc of length equal to the radius of a circle subtends at the centre of the circle.”

Relationship between mass number, atomic number, and neutron number:

A = Z + N

where

• A = mass number,
• Z = atomic number (number of protons),
• N = number of neutrons.

\[\rho=\frac{\text{mass of nucleus}}{\text{volume of nucleus}}\]

\[\vec{E}=\frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2}\hat{r}\]

The dimensional formula of the electric field E is:

E = \[\frac {F}{q_0}\] = \[\frac{[LMT^{-2}]}{[IT]}=[MLT^{-3}I^{-1}]\]

For a point charge q at distance r:

E = \[\frac{1}{4\pi\varepsilon_0}\cdot\frac{q}{r^2}\]

where \[\dfrac{1}{4\pi\varepsilon_0} = 9 \times 10^9 \, \text{N·m}^2/\text{C}^2\]

\[\vec A\] = A\[\hat n\]

• \[\vec A\] = area vector
• A = magnitude of the surface area (scalar)
• \[\hat n\] = unit vector normal (perpendicular) to the surface

For a uniform field \[\vec F\] passing through a flat surface of area A, inclined at angle θ to the field:

For a non-uniform field or curved surface, flux is expressed as a surface integral:

\[\Phi_E=\int_A\vec{E}\cdot d\vec{A}\]

For a closed surface (used in Gauss's Law):

\[\Phi_E=\oint\vec{E}\cdot d\vec{A}=\frac{q_{enc}}{\varepsilon_0}\]

E = \[\dfrac{\sigma}{\varepsilon_0} \]

• Directed outward (away from surface) if the conductor is positively charged
• Directed inward (toward the surface) if the conductor is negatively charged
• σ = surface charge density (C/m²); ε0 = permittivity of free space

Ф_Е = E A cos θ

• If the plane surface is normal to the electric field (θ = 0):
  ФЕ = ЕА cos 0 = ЕА
• If the plane surface is parallel to the electric field (θ = 90°):
  Or = E A cos 90° = 0
• For field lines entering the plane surface normally (θ = 180°):
  ФЕ = ЕА cos 180° = -EA

\[\Phi_{E}=\int_{A}\vec{\mathbf{E}}\cdot d\vec{\mathbf{A}}\]

where,
∫_A = is the (surface) integral over the entire surface
Φ_E =  is positive when lines leave the surface, negative when they enter.

OR

Electric flux through a small area element:

dΦ = E ⋅ dS = E dS cos θ

Total electric flux through a surface:

Φ = \[\int_S\vec{E}\cdot d\vec{S}\]

Coulomb's Law: The electrostatic 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. The force acts along the straight line joining the two charges.

Statement

The electrostatic force acting between two stationary point charges is given by a vector quantity whose magnitude obeys Coulomb’s law and whose direction is along the line joining the two charges. The force on each charge is equal in magnitude and opposite in direction.

Explanation / Mathematical Form

Let two point charges q_1​ and q₂ be located at position vectors \[\vec {r_1}\]​ and \[\vec {r_2}\]​ respectively.

The force on charge q₁ due to charge q₂​ is:

\[\vec F_{12}\] = \[\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r_{12}^2}\hat{r}_{12}\]​

Similarly, the force on q₂​ due to q₁ is:

\[\vec F_{21}\] = \[\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r_{12}^2}\hat{r}_{21}\]​

where
\[\hat r _{12}\]​ and \[\hat r_{21}\] are unit vectors along the line joining the charges and

​\[\hat r_{21}\] = -\[\hat r _{12}\]​

Hence,

\[\vec F_{21}\] = −\[\vec F_{12}\] ​

This relation is valid for both like and unlike charges, representing repulsion or attraction respectively.

Conclusion

The vector form of Coulomb’s law shows that:

• Electrostatic force is a central force acting along the line joining the charges.
• Forces between two charges are equal and opposite, satisfying Newton’s third law.
• The direction of force is clearly specified, unlike the scalar form.

Statement

The principle of superposition states that the net electric force acting on a given charge due to a number of other charges is equal to the vector sum of the individual forces exerted on it by each charge taken separately, assuming the other charges are absent.

Explanation / Mathematical Form

Consider a system of nnn point charges q₁,q₂,q₃,…,q_n.

The force acting on charge q₁ due to the other charges is:

\[\vec{F}_1=\vec{F}_{12}+\vec{F}_{13}+\cdots+\vec{F}_{1n}\]​

where
\[\vec F_{12}\]​ is the force on q_1​ due to q₂​,
\[\vec F_{13}\]​​ is the force due to q₃​, and so on.

According to Coulomb’s law, the force on q₁​ due to q₂​ is:

\[\vec F_{12}\]​ = \[\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r_{12}^2}\hat{r}_{12}\]​

Similarly, forces due to other charges can be written, and their vector sum gives the resultant force on q₁​.

Thus, the force between any two charges is independent of the presence of other charges.

Conclusion

The principle of superposition shows that:

• Electric forces obey vector addition.
• Each pair of charges interacts independently.
• The net force on a charge in a multi-charge system is found by adding all individual Coulomb forces vectorially.

Statement:

The total electric flux through any closed surface is equal to \[\frac{1}{\varepsilon_0}\]​ times the net electric charge enclosed by that surface.

This holds true regardless of the shape or size of the closed surface, and irrespective of the position of the charge within it.

Mathematical Formulation

In free space (vacuum/air):

The electric flux (Φ_E) through any closed surface is equal to `1/in_0` times the ‘net’ change q enclosed by the surface.

Φ_E = `oint  vec E d vec A`

= `q/in_0`

∈₀ = Permittivity of free space.

Gauss’ theorem states that the net electric flux over a closed surface is `1/epsilon_0` times the net electric charge enclosed by the surface.

Φ = `oint vec E * d vec A` 

= `q/epsilon_0`

Statement

Gauss’s theorem in electrostatics states that the total electric flux through any closed surface (called a Gaussian surface) is equal to \[\frac {1}{ε_0}\] times the net charge enclosed by the surface, irrespective of the shape and size of the surface.

Mathematical Form

Φ_E = \[\oint\vec{E}\cdot d\vec{A}=\frac{q_{\mathrm{enc}}}{\varepsilon_0}\]​​

where

• \[\vec E\] = electric field intensity
• d\[\vec{A}\] = outward normal area element
• q_enc = net charge enclosed
• ε_0​ = permittivity of free space

Proof (Outline)

Consider a point charge +q placed inside a closed surface.
The electric field at a point on the surface is

E = \[\frac{1}{4\pi\varepsilon_0}\frac{q}{r^2}\]​

The flux through a small area element dA is

dΦ_E = \[\vec{E}\cdot d\vec{A}=EdA\cos\theta\]

Since dAcos⁡θ = r²dΩ,

dΦ_E = \[\frac{q}{4\pi\varepsilon_0}d\Omega\]

Integrating over the entire closed surface,

Φ_E = \[\frac{q}{4\pi\varepsilon_0}\int d\Omega\]

But the total solid angle subtended by a closed surface is 4π,

Φ_E = \[\frac  {q}{ε_0}\]​

Hence proved.

Key Note

Gauss’s law is most useful for symmetric charge distributions (spherical, cylindrical, planar) and is valid for all inverse-square law fields.

• Quantisation of charge: Electric charge exists in discrete packets, and the charge on any body is given by
  q = ±ne
  where n is an integer and e = 1.6 × 10⁻¹⁹ C is the elementary charge.
• No fractional charge: Charge cannot exist as a fraction of e (like 0.5e or 2.3e); hence, electric charge is atomic in nature.
• Conservation of charge: The total electric charge of an isolated system remains constant; charge can neither be created nor destroyed, only transferred.
• Experimental support: Processes such as rubbing, pair production and annihilation, and radioactive decay always conserve the net charge of the system.
• Invariance of charge: The value of electric charge does not change with velocity, unlike mass, which varies with speed.

• Atom = Nucleus (protons + neutrons) + Electrons (in shells/orbitals around the nucleus).
• Proton: positive charge, composed of quarks, stable.
• Neutron: neutral charge, comparable mass to proton; free neutron undergoes beta decay.
• Electron: negative charge, fundamental particle, governs chemical behaviour.
• Atomic number (Z) = number of protons.
• Mass number (A) = number of protons + number of neutrons (A = Z + N).
• Isotopes: same Z, different N (and hence different A).
• Ions: cation formed by electron loss; anion formed by electron gain; proton count unchanged.
• Nuclear density is extremely high, e.g., copper nucleus ≈ 3.72 × 10¹⁸ kg m⁻³.

• Conductors allow charge to flow easily; insulators resist it strongly.
• In metals, free electrons carry charge; in electrolytes, ions carry it.
• Charge spreads across a conductor's surface but stays localized on an insulator.
• Semiconductors lie between the two and are sensitive to heat/doping.
• Insulators can still polarize under an electric field without conducting current.

• Charging results from electron transfer, not proton movement.
• Three methods: Friction, Conduction, Induction.
• Charge follows the Law of Conservation of Charge — total charge in an isolated system remains constant.
• Triboelectric series predicts polarity of charge when two materials are rubbed.
• Losing electrons → positive charge, slight mass decrease.
• Gaining electrons → negative charge, slight mass increase.

• Conduction requires direct contact; induction does not.
• The neutral body acquires the same sign of charge as the charging body.
• Charge is shared, not duplicated — total charge before and after contact is conserved.
• For identical spheres, final charge splits equally; for unequal spheres, distribution depends on size/capacitance.
• Grounding (earthing) can neutralize the object by providing/removing electrons via the earth.

• Induction charges a conductor without contact, via internal charge redistribution.
• The conductor always acquires a charge opposite in sign to the inducing charge.
• Order of operations matters: ground → disconnect earth → remove charged object.
• Charging by induction is permanent only if grounding is done correctly.
• Distinguish clearly from conduction (contact, same-sign charge) and friction (rubbing, opposite charges on two bodies).

• Electric field intensity: \[\vec{E}=\frac{1}{4\pi\varepsilon_0}\frac{q}{r^2}\hat{r}\].
• It is a vector quantity, directed outward for +q and inward for −q.
• Follows an inverse-square law with distance.
• Independent of the test charge used to measure it.
• For multiple charges, use vector superposition.
• In a medium, divide by dielectric constant K.

• Continuous charge distributions require integration, not simple addition.
• Three types exist: linear (λ), surface (σ), and volume (ρ).
• The general method: apply Coulomb's Law to an element dq, then integrate over the full body.
• \[\hat r_{21}\] varies across the body — always check symmetry before integrating.
• Gauss's Law is the preferred shortcut for symmetric distributions.

• Charged particle in uniform field → constant force → constant acceleration
• Velocity along field → straight-line, uniformly accelerated motion
• Velocity perpendicular to field → parabolic path (projectile analogy)
• Master formula for trajectory: y = \[\frac {qE}{2mv_0^2}\]x²
• Gravity is neglected unless explicitly stated (charge-to-mass ratio is usually very high)

• ΦE ​= q/ε₀​ (vacuum); Φ_E = q/(ε₀K) (dielectric medium)
• Flux depends only on net enclosed charge, not surface shape/size.
• Useful for symmetric bodies: sphere, cylinder, infinite sheet.
• Derived from Coulomb's Law — a generalized statement of it.

• Point Charge:
  Using a spherical Gaussian surface, the electric field due to a point charge is
  E = \[\frac{1}{4\pi\varepsilon_0}\frac{q}{r^2}\]​
  which directly leads to Coulomb’s law.
• Infinite Line of Charge:
  For a uniformly charged infinite wire with linear charge density λ,
  E = \[\frac{\lambda}{2\pi\varepsilon_0r}\]​
  The field is radial and varies inversely with distance r.
• Infinite Plane Sheet of Charge:
  For a sheet with surface charge density σ,
  E = \[\frac{\sigma}{2\varepsilon_0}\]​
  The field is independent of distance from the sheet.
• Two Parallel Charged Sheets:
  The electric field is uniform between the sheets and zero outside when the sheets carry equal and opposite charges.
• Charged Conductor:
  The electric field inside a conductor is zero, and just outside the surface,
  E = \[\frac{\sigma}{\varepsilon_0}\]​
  where σ is surface charge density.
• Uniformly Charged Spherical Shell / Conducting Sphere:
  Outside the shell: behaves like a point charge at the centre
  Inside the shell: the electric field is zero
• Uniformly Charged Non-conducting Sphere:
  Outside: E ∝ \[\frac {1}{r^2}\]​
  Inside: E ∝ r, increasing linearly from centre to surface

• A Gaussian surface is an imaginary closed surface used to calculate the electric flux of a vector field.
• It must be a closed surface (e.g., a sphere, cylinder, or cube); open surfaces such as discs or squares are not valid.
• The shape of the Gaussian surface should match the symmetry of the charge distribution so that the electric field is uniform or normal to the surface.
• The surface must not pass through any discrete charge, though it may pass through a continuous charge distribution.
• Electric flux through a Gaussian surface depends only on the charges enclosed, even though the electric field on the surface is due to both internal and external charges.

• Gauss’ theorem establishes a connection between the electric flux through a closed surface and the charge enclosed, and is especially useful for highly symmetric charge distributions.
• An area can be treated as a vector quantity, with both magnitude (area) and direction (the outward normal to the surface).
• A solid angle is the three-dimensional analogue of a plane angle and describes how a surface appears from a point.
• The total solid angle subtended at a point by a closed surface, irrespective of its shape, is always the same.
• The solid angle subtended by an area element depends on its orientation and position relative to the point, being maximum when the area faces the point directly.

• SI unit of Electric flux = N·m²·C⁻¹ or V·m (since E = N / C = V/m).
• Dimensions of electric flux: [Φ_E] = [E] [A] = [ML³T⁻³A⁻¹]