Electrostatics

Electrostatics – Study Notes (ITI Admission Test)

Electrostatics is the branch of physics that studies static electric charges and the forces, fields, and potentials they produce.
Key idea: All electric phenomena that involve stationary charges can be described by Coulomb’s law, electric field, potential energy, Gauss’s law, and capacitance.

1. Simple Explanation

2. Key Points (Bullet‑style Cheat Sheet)

2.1 Electric Charge

• Quantized in units of (e = 1.602×10^{-19}) C.
• Conservation: total charge in an isolated system is constant.
• Like charges repel, unlike charges attract.

2.2 Coulomb’s Law (Point Charges)

• (F = k \dfrac{|q_1q_2|}{r^2}), (k = 8.99×10^9 \text{ N·m}^2/\text{C}^2).
• Vector form: (\mathbf{F} = k \dfrac{q_1q_2}{r^3},\mathbf{r}).
• Force is central (along the line connecting charges).

2.3 Electric Field

• Defined as (\mathbf{E} = \dfrac{\mathbf{F}}{q_0}) (for a positive test charge (q_0)).
• For a point charge (q): (\mathbf{E} = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{r^2},\hat{r}).
• Field lines: originate from positive, terminate on negative, density ∝ field magnitude.

2.4 Electric Potential

• (V(\mathbf{r}) = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{r}) (reference at infinity).
• Potential difference (V_{AB} = V_B - V_A = -\int_A^B \mathbf{E}!\cdot d\mathbf{l}).
• For multiple charges, potentials add algebraically (superposition).

2.5 Work & Energy

• Work to move charge (q) through potential difference (V): (W = qV).
• Electric potential energy of two charges: (U = k \dfrac{q_1q_2}{r}).

2.6 Gauss’s Law

• (\displaystyle \oint_S \mathbf{E}!\cdot d\mathbf{A} = \dfrac{Q_{\text{enc}}}{\varepsilon_0}).
• Holds for any closed surface.
• Ideal for:
  • Infinite line charge
  • Infinite sheet of charge
  • Spherical charge distribution
  • Cylindrical charge distribution

2.7 Capacitance

• Parallel‑plate capacitor: (C = \varepsilon_0 \dfrac{A}{d}).
• Energy stored: (U = \dfrac{1}{2}CV^2 = \dfrac{Q^2}{2C}).
• Adding dielectric: (C' = \kappa C), where (\kappa) is dielectric constant.

2.8 Dielectrics

• Polarize under electric field; reduce field inside material.
• Relative permittivity (\kappa = \varepsilon_r = \dfrac{\varepsilon}{\varepsilon_0}).
• Energy density in dielectric: (u = \dfrac{1}{2}\varepsilon E^2).

3. Worked Example

Problem
Two point charges, (q_1 = +3,\mu\text{C}) and (q_2 = -5,\mu\text{C}), are separated by (r = 0.2,\text{m}). Calculate:

1. The magnitude and direction of the force on (q_1).
2. The electric potential at a point 0.1 m from (q_1) along the line joining the two charges.
3. The electric field at the same point.

Solution

1. Force on (q_1)

[ F = k \frac{|q_1 q_2|}{r^2} = (8.99\times10^9)\frac{(3\times10^{-6})(5\times10^{-6})}{(0.2)^2} = 8.99\times10^9 \times \frac{15\times10^{-12}}{0.04} \approx 3.37\times10^{3},\text{N} ]

• Direction: (q_1) is positive, (q_2) negative → attractive.
  Force on (q_1) points toward (q_2).

2. Potential at point P (0.1 m from (q_1))