Notes-Part-1-Class-11-Science-Physics-Chapter-10-Electrostatics-Maharashtra Board

Topics to be Learn : Part-1 Introduction Electric Charges Basic Properties of Electric Charge Coulomb’s Law Principle of Superposition Topics to be Learn : Part-2 Electric Field Electric Flux Gauss' Law Electric Dipole Continuous Charge Distribution

Introduction :

Electrostatics : The study of electric charges at rest (called static charges), the forces between them, and the electric field and potential associated with them is known as electrostatics.

Electrification by friction :

• Many materials may be charged by rubbing them together. An uncharged body is electrically neutral because it has equal amounts of positive and negative charge.
• When two bodies are rubbed against one other, friction transfers some of the loosely bound electrons from one substance to the other; the one that obtains electrons is said to be negatively charged. The second body that loses electrons has a negative charge deficit and is considered to be positively charged.
• Materials can be arranged in an ordered list such that when any two of these are rubbed against each other, the one ahead in the list becomes positively charged while the other becomes negatively charged.
• Examples, (1) fur (2) glass (3) silk (4) human body (5) metals (6) ebonite (7) sealing wax (8) amber (9) resin (10) sulphur. Thus, glass becomes positively charged when rubbed with silk but negatively charged when rubbed with fur.
• That the charges on the two bodies being rubbed are of two different kinds was discovered by Charles Dufay. The terminology, positive and negative (or plus and minus) is due to Benjamin Franklin. Dufay also showed that like charges repel while unlike charges attract each other.

Examples of electrification by friction :

• When we wear or remove our woollen sweater or a synthetic dress on a dry day, electric charges are produced by friction between wool or the fabric and our body. This produces crackling sound and the sweater or dress appears to stick to our body.
• Electric charges develop on clouds due to friction. This results in thunder and lightning under suitable conditions.
• When a glass rod is rubbed with silk, both get charged due to friction.

Electric Charges :

Electrostatic induction :

Electrostatic induction is the process by which an uncharged conductor is brought close to a charged body such that the charge opposing the body's is formed on the nearer side of the conductor and the charge similar to the body's is developed on the far side of the conductor.

Explanation :

• When a charged body is brought near an uncharged conductor, charges are separated to the two ends of the conductor Fig. (a). The charges at the end nearer to the charged body are called bound charges while those at the other end are called free charges.
• When the conductor is grounded, i.e., connected to the earth with a metal wire, electrons flow from the earth to that end and neutralize the free positive charge there. The bound negative charge remains on the conductor, Fig. (b),
• When the grounding is removed, the negative bound charges remain on the conductor, Fig. (c).
• Now, if the charged body is removed, the negative charge on the conductor is distributed, Fig. (d).

Basic Properties of Electric Charge

Additive Nature of Charge:

Electric charge, like mass, is additive. To compare the additive property of charge with that of mass.

(i) Mass is always positive, while electric charge on a particle is positive or negative. For example, the proton is positively charged and the electron is negatively charged.

(ii) Ignoring binding energy, the total mass of a body is the sum of the masses of its constituents, and is always positive. The total or net charge on a body is the algebraic sum of the charges on its constituent particles. The net charge can be positive, negative or zero.

• It is positive if the number of protons exceeds the number of electrons
• It is negative if the number of protons is less than the number of electrons
• It is zero if the number of protons equals the number of electrons.

(iii) Electric charge, like mass, is a scalar quantity.

Quantization of Charge:

• The smallest magnitude e of electric charge observable as an isolated charge is that of the charge on an electron or a proton, e ≈6 x 10^—19 C.
• The natural unit of negative charge is that on an electron, and a proton has an equal amount of positive charge.
• A macroscopic body that has equal numbers of electrons and protons is electrically neutral. If the body has an excess of electrons, it is negatively charged while deficit of electrons results in a positive charge.
• Thus, electrically charged large scale matter always possesses a charge that is an integral multiple of the natural unit e, i.e., q = ± ne, where n is an integer. Hence, charge is said to be quantized.

Know This : Quarks are the more basic building blocks of protons and neutrons. Quarks are of six types, three having fractional charge (— 1/3)e and the other three of (+2/3)e. The electric charges of the antiquarks (which combine to form antiparticles) have opposite sign. A proton or a neutron consists of a combination of three quarks. Even in the quark model, charge is quantized, only that the step size of the charge decreases from e to e/3. Quarks are never directly observed or found in isolation. It is possible to observe the discrete nature of charge in very sensitive devices such as a single electron transistor (SET).

 Conservation of Charge:

Law or principle of conservation of charge : In any closed or isolated system, the algebraic sum of all the electric charges is constant, i.e., the net charge cannot be created or destroyed.

Forces between Charges:

When charged objects are brought close to one another without touching, they feel force. This force can be attractive or repulsive. Like charges repel each other and unlike charges attract each other.

Coulomb’s Law

Coulomb's law: The electrostatic force of attraction or repulsion between two static point charges is directly proportional to the product of the magnitudes of the charges, inversely proportional to the square of their separation and acts along the line joining them.

Know This : The Coulomb force is a conservative force. ∴ W = Δ KE The law was first published in 1785 by Charles Augustin de Coulomb (1736 - 1806) French physicist. It was found independently by Henry Cavendish (1731 — 1810) British chemist and physicist

Relative Permittivity or Dielectric Constant :

When the charges are kept in a material medium, such as water, mica or parafined paper, the medium affects the force between the charges. The force between the two charges placed in a medium may be written as,

F_med = \(\frac{1}{4πε}(\frac{q_1q_2}{r^2})\)       …..(1)

where ε is called the absolute permittivity of the medium. The force between the same two charges placed in free space or vacuum at distance r is given by,

F_vac = \(\frac{1}{4πε_0}(\frac{q_1q_2}{r^2})\)     …..(2)

Dividing Eq. (1) by (2)

\(\frac{F_{med}}{F_{vac}}\) = \(\frac{\frac{1}{4πε}(\frac{q_1q_2}{r^2})}{\frac{1}{4πε_0}(\frac{q_1q_2}{r^2})}\) = \(\frac{ε}{ε_0}\)

The ratio ε/ε₀ is the relative permittivity or dielectric constant of the medium and is denoted by ε_r or K.

K or ε_r = \(\frac{ε}{ε_0}\) = \(\frac{F_{med}}{F_{vac}}\)

• The relative permittivity (ε_r) ( ≡ dielectric constant) of a medium is the ratio of the absolute permittivity of the medium (ε) to the absolute permittivity of free space (≡ vacuum (ε₀)).
• ε_r is the ratio of the force between two point charges placed 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. ε_r is a dimensionless quantity.
• ε_r is also called specific inductive capacity.

 SI unit and dimensions of absolute permittivity :

(1) In the usual notation, F = \(\frac{1}{4πε}(\frac{q_1q_2}{r^2})\)

∴ ε = \(\frac{q_1q_2}{4πFr^2}\)

The SI unit of charge is the coulomb, that of force is the newton, and that of distance is the metre.

Hence the SI unit of absolute permittivity ε is \(\frac{coulomb^2}{newton.metre^2}\) usually written as C²/N.m².

(2) We have [ε] = \(\frac{[q_1q_2]}{[F][r^2]}\)  as 4π is a pure number.

Now, [q] = [IT], [F] =[MLT^—2] and [r] = [L]

∴ [ε] = \(\frac{[I^2T^2]}{[MLT^{-2}][L^2]}\) = [M^—1L^—3T⁴I²]

Thus, the dimensions of ε are —1 in mass, —3 in length, 4 in time and 2 in current I.

Definition of Unit Charge from the Coulomb’s Law:: 

• The SI unit of charge is the coulomb (C).
• Using Coulomb's law, it is defined as follows : One coulomb is the amount of charge, which placed at a distance of one metre from another charge of the same magnitude in vacuum, experiences a force of 9.0 x 10⁹

Know This : Comparison of gravitational and electrostatic forces: Similarities : Both forces obey inverse square law : F ∝ 1/r2 Both are central forces : act along the line joining the two objects. Differences : Gravitational force between two objects is always attractive while electrostatic force between two charges can be either attractive or repulsive depending on the nature of charges. Gravitational force is about 36 orders of magnitude weaker than the electrostatic force.

Principle of Superposition :

The electrostatic force between two point charges is given by Coulomb's law, irrespective of the presence of other similar point charges in the vicinity.

Principle of superposition of forces : The resultant, i.e., the net force on a point charge due to its interaction with other point charges in its vicinity the vector sum of the forces simultaneously exerted on it by the individual charges at a given instant.

Consider, for example, two point charges q₁ and q₂ in free space exerting forces simultaneously on a third point charge q₃ (see Fig.).

Then the forces \(\vec{F}_{31}\) (≡ \(\vec{F}\) _{on 3 by 1}) and \(\vec{F}_{32}\) ((≡ \(\vec{F}\) _{on 3 by 2}) are

\(\vec{F}_{31}\) = \(\frac{1}{4πε_0}.\frac{q_1q_3}{r_{13}^2}\,\hat{r}_{13}\)

and

\(\vec{F}_{32}\) = \(\frac{1}{4πε_0}.\frac{q_2q_3}{r_{23}^2}\,\hat{r}_{23}\)

where r₁₃ is the distance between q₁ and q₃, r₂₃ is that between q₂ and q₃,

\(\hat{r}_{13}\) is a unit vector directed from q₁ to q₃, \(\hat{r}_{23}\) is a unit vector directed from q₂ to q₃.

The resultant force on q₃ is \(\vec{F}_3\)

\(\vec{F}_3 = \vec{F}_{31} + \vec{F}_{32}\)

= \(\frac{1}{4πε_0}[\frac{q_1q_3}{r_{13}^2}\hat{r}_{13}+\frac{q_2q_3}{r_{23}^2}\hat{r}_{23}]\)

This can be extended to any number of point charges.

In general, if there are N charges,

\(\vec{F}_1 = \vec{F}_{12} + \vec{F}_{13}+\vec{F}_{14}+ ....+\vec{F}_{1N}\) is the resultant force on q₁.

and is the resultant force on q₁.

and

\(\vec{F}_i = \sum_{j=1}^{N}\vec{F}_{ij}\)  .... j ≠ i

 = \(\frac{1}{4πε_0} \sum_{j=1}^{N}\frac{q_iq_j}{r_{ij}^2}\hat{r}_{ij}\)                    

is the resultant force on q_i