Electric Charges and Fields

This chapter introduces electrostatics, the study of stationary electric charges. It begins with the fundamental properties of charge: it is quantized, conserved, and additive. You will learn to calculate the force between two point charges using Coulomb's Law, which is mathematically similar to Newton's Law of Gravitation but can be both attractive and repulsive.

The concept of force is then abstracted into the idea of an electric field, which describes the influence a charge exerts on the space around it. A significant part of the chapter involves calculating this field for various charge configurations, from single points to systems of charges, using the principle of superposition. This requires a strong command of vector addition. The chapter also introduces electric field lines as a way to visualize these fields.

Later, the chapter presents the electric dipole, a pair of equal and opposite charges, and analyzes its field and its behavior in an external uniform electric field. The most powerful tool introduced is Gauss's Law, which relates the electric flux through a closed surface to the net charge enclosed. Applying Gauss's Law to find the electric field for symmetric charge distributions (like infinite lines, sheets, and spheres) is a core skill. The chapter leans heavily on vector algebra and basic calculus from Class 11, and the application of Gauss's Law is where students often find the most difficulty.

Formulas for Electric Charges and Fields

Quantization of Charge: $q = ne$ n is an integer and e is the elementary charge (1.602 × 10⁻¹⁹ C).
Coulomb's Law: $F = \frac{1}{4\pi\epsilon_0} \frac{|q_1 q_2|}{r^2}$ Force between two point charges q₁ and q₂ separated by distance r.
Electric Field of a Point Charge: $\vec{E} = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \hat{r}$ Vector field at a distance r from a source charge q.
Electric Dipole Moment: $\vec{p} = q(2\vec{a})$ For charges +q and -q separated by vector distance 2a from negative to positive.
Torque on a Dipole in a Uniform Field: $\vec{\tau} = \vec{p} \times \vec{E}$ The magnitude is pE sin(θ), where θ is the angle between p and E.
Electric Field on the Axis of a Dipole: $E_{axial} \approx \frac{1}{4\pi\epsilon_0} \frac{2p}{r^3}$ For short dipoles, where r is the distance from the center.
Electric Field on the Equatorial Line of a Dipole: $E_{equatorial} \approx \frac{1}{4\pi\epsilon_0} \frac{p}{r^3}$ For short dipoles, where r is the distance from the center.
Electric Flux: $\Phi_E = \int \vec{E} \cdot d\vec{A}$ The surface integral of the electric field over a surface A.
Gauss's Law: $\oint \vec{E} \cdot d\vec{A} = \frac{q_{in}}{\epsilon_0}$ Relates net flux through a closed surface to the net charge enclosed.
Field of an Infinite Line Charge: $E = \frac{\lambda}{2\pi\epsilon_0 r}$ λ is the linear charge density; r is the perpendicular distance.
Field of an Infinite Plane Sheet of Charge: $E = \frac{\sigma}{2\epsilon_0}$ σ is the uniform surface charge density. The field is constant.