Revision: Electrostatics Physics HSC Science (General) 11th Standard Maharashtra State Board

The closed surface over which the surface integral of the electric field intensity (i.e. total electric flux) is considered in Gauss' Law is called a Gaussian surface.

The study of electricity/electric charges at rest is called electrostatics.

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]

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.

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 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 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 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.

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

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

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

“An electric line of force is an imaginary smooth curve drawn in an electric field along which a free, isolated positive charge moves. The tangent drawn at any point on the electric line of force gives the direction of the force acting on a positive charge placed at that point.”

OR

An imaginary curve drawn in such a way that the tangent at any given point on this curve gives the direction of the electric field is called an electric line of force.

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,

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

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).

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 midpoint of the line joining the two charges is called the centre 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 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).

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.

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 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.

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.

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.

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

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.

If a system contains n charges q₁, q₂, q₃,…, q_n then: 

Q = q₁​ + q_2​ + q₃ ​+ … + q_n​

Charge q on a body is always an integral multiple of electronic charge e:

q = ±ne

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

The dimensional formula of the electric field E is:

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

E = \[\frac{1}{4\pi\varepsilon_{0}}\frac{q}{r^{2}}\] newton / coulomb

where \[\frac{1}{4\pi\varepsilon_{0}}\] = 9.0 × 10⁹ newton meter² / coulomb².

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]

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

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

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

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

The total charge of a system is the algebraic sum of all individual charges present in the system. If a system contains n charges q₁, q₂, q₃, …, qₙ, then the total charge Q is given by:

This property states that charges add up like real numbers (scalars), taking into account their signs (positive or negative). This means a system with a +3C and a −3C charge has a net charge of zero.

The charge (q) on any body is always equal to an integral multiple of the elementary (electronic) charge (e). Mathematically:

where n is any positive integer (n = 1, 2, 3, …) and e is the charge of one electron (e = 1.6 × 10⁻¹⁹ C). This means charge cannot take arbitrary values — it only exists in discrete packets. You cannot have half or one-third of an electron's charge on a body.

For an isolated system, the net charge always remains constant. This law states that charge can neither be created nor destroyed; it can only be transferred from one body to another. For example, when a glass rod is rubbed with silk, the rod gains a positive charge and the silk gains an equal negative charge — the total charge of the system remains zero.

Like charges repel each other while unlike charges attract each other. This is a fundamental behaviour arising from the nature of electric charge. For example, two positively charged bodies brought near each other will experience a repulsive force, while a positively charged body brought near a negatively charged body will experience an attractive force.

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.

Statement

Coulomb’s law states that the electrostatic 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. The force acts along the line joining the two charges and is repulsive for like charges and attractive for unlike charges.

Explanation/Mathematical Form

Let two point charges q₁​ and q₂​ be placed at a distance r apart in vacuum (or air).

According to Coulomb’s law:

F ∝ q₁q₂

F ∝ \[\frac {1}{r^2}\] ​

Combining the above relations:

F = k\[\frac {q_1q_1}{r^2}\]​​

where
F = electrostatic force between the charges,
r = distance between the charges,
k = proportionality constant.

In vacuum (or air),

k = 9.0 × 10⁹ N m²C⁻²

Hence,

F = \[\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r^2}\]​​

where ε₀​ is the permittivity of free space, given by

ε₀ = 8.85 × 10⁻¹² C²N⁻¹m⁻²

If the charges are placed in a dielectric medium of permittivity ε,

F = \[\frac{1}{4\pi\varepsilon}\frac{q_1q_2}{r^2}\]​​

and since ε = Kε_0​,

F = \[\frac{1}{4\pi K\varepsilon_0}\frac{q_1q_2}{r^2}\]​​

where K is the dielectric constant of the medium.

Conclusion

Coulomb’s law quantitatively describes the force of attraction or repulsion between two point charges.
The force:

• depends on the magnitudes of charges,
• varies inversely as the square of the distance,
• acts along the line joining the charges, and
• decreases in a dielectric medium by a factor equal to its dielectric constant.

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.

"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

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)`

• Electric field lines originate from positive charges and terminate on negative charges (or at infinity).
• The tangent to a field line at any point gives the direction of the electric field; in a uniform field, the lines are parallel and straight.
• No two electric field lines intersect, as this would imply more than one direction of the electric field at a point.
• Electric field lines do not pass through a conductor, showing that the electric field inside a conductor is zero.
• The density of field lines indicates field strength—closer lines represent a stronger field, while wider spacing represents a weaker field; the lines are continuous and imaginary, though the field is real.

• 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)