Class 12th Physics: Electrostatics Comprehensive Study Notes

Fundamentals of Electric Charge

• Definition of Electric Charge: Electric charge is an intrinsic physical property of matter that causes it to experience a force when placed in an electromagnetic field. Charges are classified into two types: positive and negative.

• Units of Charge:   - SI Unit: The Coulomb ($C$).   - CGS Unit: The statcoulomb (also known as the electrostatic unit of charge or esu).   - Conversion: $1\,C = 2.9979 \times 10^9\,\text{statcoulomb}$.

• Quantization of Charge:   - This principle states that the total charge of a body is always an integral multiple of a basic, elementary unit of charge, denoted as $e$.   - The formula is expressed as: $q = \pm ne$.   - Here, $n$ is an integer ($n = 1, 2, 3, \dots$) and $e$ is the charge of an electron/proton ($e \approx 1.6 \times 10^{-19}\,C$).   - Charge exists in discrete packets rather than continuous amounts.

• Calculation of Electrons in One Coulomb:   - Given $q = 1\,C$ and $e = 1.6 \times 10^{-19}\,C$.   - Using $n = \frac{q}{e}$:   - $n = \frac{1}{1.6 \times 10^{-19}} = 6.25 \times 10^{18}$.   - Thus, $6.25 \times 10^{18}$ electrons constitute one Coulomb of charge.

Dimensional Analysis in Electrostatics

• Electric Field ($E$):   - Defined as force per unit charge: $E = \frac{F}{q}$.   - Dimensional formula of Force ($F$): $[MLT^{-2}]$.   - Dimensional formula of Charge ($q = I \times t$): $[AT]$.   - Dimensional formula of Electric Field: $\frac{[MLT^{-2}]}{[AT]} = [MLT^{-3}A^{-1}]$.

• Electrical Permittivity ($\epsilon_0$):   - Derived from Coulomb's Law: $F = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2}$.   - Rearranging for $\epsilon_0$: $\epsilon_0 = \frac{q_1 q_2}{4\pi F r^2}$.   - Dimensional analysis: $\frac{[AT][AT]}{[MLT^{-2}][L^2]} = \frac{[A^2 T^2]}{[ML^3 T^{-2}]} = [M^{-1}L^{-3}T^4A^2]$.

Coulomb's Law and Electrostatic Force

• Vector Form of Coulomb's Law:   - Definition: The mathematical representation of the electrostatic force between two point charges that accounts for both magnitude and direction ($\mathbf{F} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2} \mathbf{\hat{r}}$).   - Importance:     - It proves that electrostatic forces obey Newton's Third Law ($\mathbf{F_{12}} = -\mathbf{F_{21}}$).     - It confirms that the electrostatic force is a central force, acting along the line joining the centers of the two charges.     - It provides the precise direction of force, showing attraction for unlike charges and repulsion for like charges.

• Effect of Changing Parameters:   - According to Coulomb's Law, $F \propto q_1 q_2$.   - If the magnitude of both charges is doubled ($q'_1 = 2q_1$ and $q'_2 = 2q_2$) while the distance ($r$) remains the same:   - $F' = k \frac{(2q_1)(2q_2)}{r^2} = 4 \times \left( k \frac{q_1 q_2}{r^2} \right) = 4F$.   - The force between the two charges is quadrupled.

• Dielectric Constant ($K$):   - Also known as relative permittivity ($\epsilon_r$), it is the ratio of the permittivity of a medium ($\epsilon$) to the permittivity of free space ($\epsilon_0$): $K = \frac{\epsilon}{\epsilon_0}$.   - It can also be defined as the ratio of the force between two charges in a vacuum ($F_{vac}$) to the force in a medium ($F_{med}$): $K = \frac{F_{vac}}{F_{med}}$.   - Value for Water: Water has a high dielectric constant, approximately $K = 80$ (or $81$) at room temperature.

Electric Field Theory

• Definition of Electric Field:   - The electric field at a point is the space around a charged particle or object within which a force would be exerted on other charged particles.   - Quantitatively, it is the force experienced by a unit positive test charge placed at that point: $\mathbf{E} = \lim_{q_0 \to 0} \frac{\mathbf{F}}{q_0}$.

• Significance of Electric Field:   - It provides a mechanism for charge-to-charge interaction without direct contact (action-at-a-distance).   - It describes the electrical environment of a charge independent of the test charge used to measure it.   - It allows for the calculation of the force on any charge placed in that space ($\mathbf{F} = q\mathbf{E}$).

• Electric Field Produced by a Point Charge:   - Consider a point charge $q$ at the origin. To find the field at distance $r$, place a test charge $q_0$.   - Force by Coulomb's Law: $F = \frac{1}{4\pi\epsilon_0} \frac{q q_0}{r^2}$.   - By definition, $E = \frac{F}{q_0}$.   - Substituting $F$: $E = \frac{\frac{1}{4\pi\epsilon_0} \frac{q q_0}{r^2}}{q_0} = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2}$.

Electric Dipoles

• Definition: An electric dipole is a system consisting of two equal and opposite point charges ($+q$ and $-q$) separated by a small, finite distance ($2a$).

• Electric Dipole Moment ($p$):   - A vector quantity representing the strength of the dipole.   - Magnitude: $p = q \times 2a$.   - Direction: From negative charge ($-q$) to positive charge ($+q$).

• Electric Field on the Axial Line:   - Let $P$ be a point on the axial line at distance $r$ from the center of the dipole.   - The resulting electric field is: $E_{axial} = \frac{1}{4\pi\epsilon_0} \frac{2pr}{(r^2 - a^2)^2}$.   - For a short dipole ($r \gg a$): $E_{axial} = \frac{1}{4\pi\epsilon_0} \frac{2p}{r^3}$.

• Electric Field on the Equatorial Line:   - Let $P$ be a point on the perpendicular bisector (equatorial line) at distance $r$ from the center.   - The resulting electric field is: $E_{equatorial} = \frac{1}{4\pi\epsilon_0} \frac{p}{(r^2 + a^2)^{3/2}}$.   - For a short dipole ($r \gg a$): $E_{equatorial} = \frac{1}{4\pi\epsilon_0} \frac{p}{r^3}$.

• Comparison:   - For a short dipole at the same distance $r$, the electric field on the axial line is twice the magnitude of the electric field on the equatorial line: $E_{axial} = 2 E_{equatorial}$.   - The direction of $E_{axial}$ is along the dipole moment $\mathbf{p}$, while the direction of $E_{equatorial}$ is opposite to $\mathbf{p}$.

Dipole in a Uniform Electric Field

• Torque Acceleration:   - When a dipole with moment $\mathbf{p}$ is placed in a uniform electric field $\mathbf{E}$ at an angle $\theta$, it experiences a torque.   - Each charge experiences a force $F = qE$. These forces are equal, opposite, and act along different lines of action, forming a couple.   - Torque Magnitude (\tau): $\tau = \text{Force} \times \text{Perpendicular distance}$.   - $\tau = (qE) \times (2a \sin(\theta)) = (q \times 2a) E \sin(\theta) = pE \sin(\theta)$.   - Vector Form: $\mathbf{\tau} = \mathbf{p} \times \mathbf{E}$.

• Discussion of Results:   - Maximum Torque: Occurs when $\theta = 90^{\circ}$. Here, $\tau_{max} = pE$.   - Stable Equilibrium: Occurs when $\theta = 0^{\circ}$. The torque is zero and the dipole aligns with the field.   - Unstable Equilibrium: Occurs when $\theta = 180^{\circ}$. The torque is zero, but any displacement will result in the dipole rotating back toward stable equilibrium.