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

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

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

OR

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

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

Substances which allow electricity to pass through them easily are called conductors.

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

A process in which a charged object induces charge in an uncharged 'conductor' placed near it, without touching the conductor. This is called "charging by electrostatic induction”.

The charge induced on an uncharged conductor due to a nearby charged body is called an induced charge.

Charging by conduction is the process in which an uncharged conductor becomes charged by direct contact with a charged conductor due to the transfer of electrons.

The charge on the charged object which causes induction is called the inducing charge.

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

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

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 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.

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

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

“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 torque (couple) acting on an electric dipole placed in a uniform electric field, which tends to align the dipole along the field, is called the restoring couple.

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

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.

\[\overset{\rightarrow}{\operatorname*{\mathbf{E}}}=\frac{\overset{\rightarrow}{\operatorname*{\mathbf{F}}}}{q_{0}}=\frac{1}{4\pi\varepsilon_{0}}\int_{S}\frac{\sigma dS}{r_{21}^{2}}\overset{\wedge}{\operatorname*{\mathbf{r}_{21}}}\]

\[\overset{\rightarrow}{\operatorname*{\operatorname*{E}}}=\frac{\overset{\rightarrow}{\operatorname*{\operatorname*{F}}}}{q_{0}}=\frac{1}{4\pi\varepsilon_{0}}\int_{V}\frac{\rho dV}{r_{21}^{2}}\overset{\wedge}{\operatorname*{\operatorname*{r}_{21}}}\]

\[\overset{\rightarrow}{\operatorname*{\mathbf{E}}}=\frac{\overset{\rightarrow}{\operatorname*{\mathbf{F}}}}{q_{0}}=\frac{1}{4\pi\varepsilon_{0}}\int_{L}\frac{\lambda dl}{r_{21}^{2}}\overset{\wedge}{\operatorname*{\mathbf{r}_{21}}}\]

E = \[\frac{1}{4\pi\varepsilon_{0}}\frac{p}{r^{3}}\]

In vector notation:
\[\overrightarrow{\mathbf{E}}=-\frac{1}{4\pi\varepsilon_{0}}\frac{\overrightarrow{\mathbf{p}}}{r^{3}}\]

τ = p E sin θ

in vector form:
\[\vec τ\] = \[\vec p\] × \[\vec E\]

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.

• 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).

• Rubbing a glass rod with silk and an ebonite rod with cat-skin makes them electrically charged.
• A glass rod repels another glass rod, and an ebonite rod repels another ebonite rod when brought close.
• A rubbed glass rod attracts a rubbed ebonite rod, showing a different type of interaction.
• These experiments prove that electric charges are of two types.
• The charge on a glass rod is called positive, and the charge on an ebonite rod is called negative (named by Benjamin Franklin, 1750). Like charges repel, and unlike charges attract.

• Matter is made of atoms consisting of a positively charged nucleus (protons and neutrons) with negatively charged electrons revolving around it.
• In a neutral atom, the number of electrons equals the number of protons, so the atom as a whole is electrically neutral.
• Electrons are responsible for electrification; protons do not move because they are tightly bound in the nucleus.
• Loss of electrons makes a body positively charged, while gain of electrons makes a body negatively charged.
• When two different materials rub together, electrons are transferred from one to the other, producing frictional electrification.

• 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 eee (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.

• 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.