Revision: 11th Std >> Electrostatics MAH-MHT CET (PCM/PCB) Electrostatics

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.

The surface integral of the electric field intensity over a closed surface S is called the electric flux through that surface.

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

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

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

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 or dielectric constant.

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

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

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

“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.

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.

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

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 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.

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.

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.

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.

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

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

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

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

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

p = q × 2a

It is a vector quantity; its direction is from −q to +q.

λ = \[\frac {ΔQ}{Δl}\] C/m

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

σ = \[\frac {ΔQ}{ΔS}\] C/m²

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

ρ = \[\frac {ΔQ}{ΔV}\] C/m³

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

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

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.

Formula - Scalar Form:

Formula - 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​.

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 flux of the net electric field through a closed surface equals the net charge enclosed by the surface divided by ε_0​.

Formula - Gauss's Law:

Key Points of Gauss's Law:

• Applicable to any closed surface of regular or irregular shape.
• Only the enclosed charge contributes to the electric flux.
• The electric field at a point depends on the total charge distribution, both inside and outside the Gaussian surface.

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)`

• Thales (≈2500 years ago) observed that amber rubbed with wool attracts light objects like paper and straw.
• William Gilbert (1600) showed that many materials, such as glass, ebonite, and sulphur, also show this effect.
• This attractive property is produced by rubbing (friction); a material showing it is said to be electrified, and the process is called frictional electricity.
• An electrified material possesses electric charge and is therefore called a charged body.
• Electric charge is quantised (q = ±ne,  e = 1.6 × 10⁻¹⁹ C); there are two types of charges (positive and negative), as charges repel, unlike charges attract, and the SI unit of charge is coulomb (C).

• A charge creates an electric field around it, even if no other charge is present.
• The electric field does not depend on the test charge used to measure it (if the test charge is very small).
• The field of a positive charge points outward; the field of a negative charge points inward.
• The strength of the electric field decreases as the distance from the charge increases.
• At equal distances from a point charge, the electric field has the same magnitude (spherical symmetry).

• 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.