Revision: Electrostatics >> Electric Charges and Fields Physics Science (English Medium) Class 12 CBSE

An electric charge which can be considered to exist at a single point is called a point charge.

A unit positive charge used to test the strength of electric fields is called a test charge.

The basic property of matter due to which it experiences electric force and shows attraction or repulsion, is called electric charge.

OR

The fundamental property of subatomic particles that gives rise to the phenomenon of experiencing force in the presence of electric and magnetic fields is called electric charge.

• Positive charge: Deficiency of electrons
• Negative charge: Excess of electrons
• SI unit: Coulomb (C)
• Dimension: [M⁰L⁰T¹A¹]

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.

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.

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

The ratio of the force between two point charges placed at a certain distance apart in free space or vacuum to the force between the same two point charges when placed at the same distance in the given medium is called relative permittivity (K) or dielectric constant.

The force of attraction or repulsion acting between two electric charges is called the electric force.

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).

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

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

A field whose magnitude and direction is the same at all points is called a uniform electric field.

A field whose magnitude and direction are not the same at all points is called a non-uniform electric field.

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

A time-dependent combination of electric and magnetic fields that propagates through space and can transport energy is called an electromagnetic field.

Electric Field \[\vec E\] at a point is the electrostatic force \[\vec F\] experienced by a vanishingly small positive test charge q₀ placed at that point:

\[\vec E\] = \[\frac {\vec F}{q_0}\]

An electric field line is an imaginary curve (straight or curved) drawn in a region of electric field such that:

• The tangent at any point on the curve gives the direction of the electric field \[\vec E\] at that point.
• The density (closeness) of field lines at any region represents the relative magnitude of the electric field at that region.

A measure of electric field through a surface, given by the number of electric lines of force per unit area enclosing the electric lines of force, is called electric flux.

OR

Electric flux through a surface is defined as the dot product of the electric field vector and the area vector of the surface.

For any general surface,

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).

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.

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 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).

When charge is distributed over a surface, the charge distribution is called surface charge distribution.

OR

The surface charge density σσ is the charge per unit area at any point on the surface.

When charge is distributed over the volume of an object, it is called volume charge distribution.

OR

The volume charge density ρρ is the charge per unit volume at any point inside the body.

The charge per unit area on a surface, is called surface charge density.

The charge per unit volume in a region of space, is called volume charge density.

OR

When charge is distributed over the volume of an object, it is called volume charge distribution.

When charge is distributed along a line, the charge distribution is called a linear charge distribution.

OR

The linear charge density λ is the charge per unit length at any point on the line.

The charge per unit length along a line (such as a wire), is called linear charge density.

OR

When charge is distributed along a line, the charge distribution is called linear charge distribution.

A charge distribution in which charge is treated as continuously spread over a line, surface, or volume (ignoring microscopic discreteness), is called continuous charge distribution.

A Gaussian surface is an imaginary, closed mathematical surface chosen to apply Gauss's Law conveniently.

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

The dimensional formula of the electric field E is:

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

For a system of n point charges q₁, q₂, q₃,…, qn, the total electric field at point P is:

E(r) = \[{\frac{1}{4\pi\varepsilon_0}\sum_{i=1}^n\frac{q_i}{r_{iP}^2}\hat{\mathbf{r}}_{iP}}\]

Symbol Reference

E = \[\frac {\text {Number of electric lines of force}}{\text {Area enclosing the electric lines of force}}\]

OR

Φ = EA cos θ

where:

• Φ = electric flux
• E = magnitude of the electric field
• A = area of the surface
• θ = angle between \[\vec{E}\] and the area vector \[\vec{E}\]

SI Unit

• SI unit of electric flux = N m² C⁻¹
• Equivalent SI unit = V m

Dimensional Formula: [ML³T^-3A^-1]

ρ = \[\frac {ΔQ}{ΔV}\] ⇒ dq = ρ dV

where ΔQ is the charge distributed over a small volume ΔV of the material.

• SI Unit: C m⁻³ (coulomb per cubic metre)
• Nature: Scalar quantity

\[\vec{E}=\frac{1}{4\pi\varepsilon_0}\sum\frac{\rho\Delta V}{r^{\prime2}}\hat{r}^{\prime}\]

λ = \[\frac {ΔQ}{Δl}\] ⇒ dq = λdl

where ΔQ is the charge distributed over a small length Δl of the wire.

• SI Unit: C m⁻¹ (coulomb per metre)
• Nature: Scalar quantity

σ = \[\frac {ΔQ}{ΔS}\] ⇒ dq = σ dS

where ΔQ is the charge distributed over a small surface area ΔS.

• SI Unit: C m⁻² (coulomb per square metre)
• Nature: Scalar quantity

The force of attraction or repulsion between two point charges at rest is directly proportional to the product of the magnitude of the charges and inversely proportional to the square of the distance between them.

Scalar Form:

Vector Form:

where q₁​ and q₂ are charges separated by distance r, and \[\hat r_{12}\] is the unit vector from q_1​ to q₂​.

• Since force is a vector, Coulomb's Law is better expressed in vector notation.
• The vector leading from charge 1 to charge 2 is r₂₁ = r₂ − r₁, and from charge 2 to charge 1 is r₁₂ = r₁ − r₂ = −r₂₁.
• The corresponding unit vectors are \[\hat r_{21}\] = \[\frac {r_{21}}{r_{21}}\]​​ and \[\hat r_{12}\] = \[\frac {r_{12}}{r_{12}}\]​​, with \[\hat r_{21}\] =−\[\hat r_{12}\]​.
• The vector form of Coulomb's Law is:
  F₂₁ = \[\frac {1}{4πε_0}\] ⋅ \[\frac {q_1q_2}{r^2_{21}}\]\[\hat r_{21}\] ...(3)
• If q₁​ and q₂​ are of the same sign, F₂₁​ is along \[\hat r_{21}\]​, representing repulsion.
• If q₁ and q_2​ are of opposite signs, F_21​ is along −\[\hat r_{21}\]​, representing attraction.
• Eq. (3) handles both like and unlike charges correctly within a single equation — no separate formulas are needed.
• The force F₁₂​ on q₁​ due to q_2​ is obtained by interchanging 1 and 2: F₁₂ = −F₂₁, confirming agreement with Newton's Third Law.
• Eq. (3) gives the force in vacuum; when charges are placed in matter, the situation becomes more complex due to the charged constituents of the medium.

• It is experimentally verified that the force on any charge due to a number of other charges is the vector sum of all the forces on that charge, taken one at a time.
• The individual forces are unaffected by the presence of other charges.
• This is termed the Principle of Superposition.

Mathematical Formulation:

• Consider a system of three charges q₁, q₂, and q₃.
• The force on q₁ due to q₂ and q₃ is obtained by performing a vector addition of the forces due to each one of these charges.
• The force on q₁ due to q₂, denoted F₁₂, is given by Coulomb's law even though other charges are present:
  F₁₂ = \[\frac{1}{4\pi\varepsilon_0}\cdot\frac{q_1q_2}{r_{12}^2}\hat{\mathbf{r}}_{12}\]
• The force on q₁ due to q₃, denoted F₁₃, is again the Coulomb force even though q₂ is present:
  F₁₃ = \[\frac {1}{4πε_0}\] ⋅ \[\frac{q_1q_3}{r_{13}^2}\hat{\mathbf{r}}_{13}\]
• The total force F₁ on q₁ due to both q₂ and q₃ is (Equation 1.4):
  F₁ = F₁₂ + F₁₃ = \[\frac {1}{4πε_0}\]\[\begin{bmatrix} \frac{q_1q_2}{r_{12}^2}\hat{\mathbf{r}}_{12}+\frac{q_1q_3}{r_{13}^2}\hat{\mathbf{r}}_{13} \end{bmatrix}\]
• For a system of n charges, the force on q₁ due to all other charges is (Equation 1.5):
  F₁ = \[\frac{q_{1}}{4\pi\varepsilon_{0}}\sum_{i=2}^{n}\frac{q_{i}}{r_{1i}^{2}}\hat{\mathbf{r}}_{1i}\]
• The vector sum is obtained by the parallelogram law of addition of vectors.
• All of electrostatics is basically a consequence of Coulomb's law and the superposition principle.

"The electric field at any point due to a group of charges is the vector sum of the electric fields at that point due to each individual charge, calculated as if the other charges were not present."

• Each charge in the system contributes its own independent electric field at the point of interest.
• These individual fields are then added vectorially to give the total (resultant) field.

Gauss' Law states that the net electric flux through any closed surface is equal to `1/epsilon_0` times the net electric charge within that closed surface.

`oint  vec" E".d vec" s" = (q_(enclosed))/epsilon_o`

In the diagram, we have taken a  cylindrical gaussian surface of radius = r and length = l.
The net charge enclosed inside the gaussian surface `q_(enclosed) = lambdal`
By symmetry, we can say that the Electric field will be in radially outward direction.

According to gauss' law,

`oint  vec"E".d  vec"s" = q_(enclosed)/epsilon_o`

`int_1 vec"E" .d  vec"s" + int_2  vec"E" .d  vec"s" + int_3  vec"E". d  vec"s" = (lambdal)/epsilon_o`

`int_1  vec"E". d  vec"s"  &  int_3  vec"E". d  vec"s"  "are zero", "Since"  vec"E"  "is perpendicular to"  d  vec"s"`

`int_2  vec"E" . d  vec"s" = (lambdal)/epsilon_o`

`"at"  2,  vec"E" and d  vec"s"  "are in the same direction, we can write"`

`E.2pirl = (lambdal)/epsilon_o`

`E = lambda/(2piepsilon_o r)`

"The total electric flux through any closed surface is equal to \[\frac {1}{ε_0}\] times the net charge enclosed by that surface."

Three Forms of the Law

1. Verbal Form:
The net outward electric flux through a closed surface equals the net enclosed charge divided by ε₀.

2. Algebraic Form:

Φ_E = \[\frac {Q_enc}{ε_0}\]

3. Integral Form:

\[\oint\vec{E}\cdot d\vec{S}=\frac{Q_{\mathrm{enc}}}{\varepsilon_0}\]

Variable Legend

• The resultant field E is the vector sum of all individual fields.
• Each individual field E_i is calculated independently, as if no other charges exist.
• The unit vector \[\hat r_i\]P points from each charge q_i toward point P.
• The principle holds for any number of charges in any configuration.
• This is a direct application of the Superposition Principle to electric fields.

• \[\vec E\] = \[\vec F\]/q₀ — force per unit positive test charge
• Static case → Coulomb's Law is sufficient; field is a descriptive tool
• Accelerated charges → field becomes a real physical entity (EM waves)
• Time delay = d/c — information travels at the speed of light, not instantaneously
• An electric field carries and transports energy
• Field exists independently of whether any test charge is present
• Gravity is negligible for charged particles in typical electric fields

• Applicable to any closed surface, regardless of shape or size — sphere, cube, irregular shape
• Only enclosed charges contribute to the net flux; external charges do not
• The electric field E at the Gaussian surface is due to all charges (inside and outside), but the net flux depends only on enclosed charge​
• Gauss's Law is valid for both stationary and moving charges​
• It is one of Maxwell's four equations of electromagnetism​
• Gauss's Law can be derived from Coulomb's Law for static charges, and vice versa — both are equivalent​
• If net enclosed charge = 0, net flux = 0 (but E ≠ 0 necessarily)