The Formula Marathon

Here is your one-stop shop for every mathematical tool you need for this chapter. Bookmark this page for quick revision!

Concept Formula Key Notes
Quantization of Charge Q=±neQ = \pm ne nn is an integer; e=1.6×10−19e = 1.6 \times 10^{-19} C
Coulomb's Law F=k∣q1q2∣r2F = k \frac{\lvert q_1 q_2 \rvert}{r^2} k=14πϵ0≈9×109k = \frac{1}{4\pi\epsilon_0} \approx 9 \times 10^9 Nm2^2/C2^2
Medium Effect Fmed=FvacKF_{med} = \frac{F_{vac}}{K} KK is the Dielectric Constant
Electric Field (Point Charge) E=14πϵ0qr2E = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} Directed away from ++ and towards −-
Dipole Moment p⃗=q×(2a⃗)\vec{p} = q \times (2\vec{a}) Direction: Negative →\to Positive
Axial Field (Short Dipole) Eaxial=14πϵ02pr3E_{axial} = \frac{1}{4\pi\epsilon_0} \frac{2p}{r^3} Parallel to p⃗\vec{p}
Equatorial Field (Short Dipole) Eequatorial=14πϵ0pr3E_{equatorial} = \frac{1}{4\pi\epsilon_0} \frac{p}{r^3} Opposite to p⃗\vec{p}
Torque on Dipole τ=pEsin⁡θ\tau = pE \sin \theta Max at 90∘90^{\circ}, Zero at 0∘0^{\circ} and 180∘180^{\circ}
Electric Flux Φ=E⃗⋅A⃗=EAcos⁡θ\Phi = \vec{E} \cdot \vec{A} = EA \cos \theta θ\theta is the angle with the Normal
Gauss's Law Φ=∮E⃗⋅da⃗=qinϵ0\Phi = \oint \vec{E} \cdot d\vec{a} = \frac{q_{in}}{\epsilon_0} Valid for any closed surface
Field: Infinite Wire E=λ2πϵ0rE = \frac{\lambda}{2\pi\epsilon_0 r} E∝1/rE \propto 1/r
Field: Infinite Sheet E=σ2ϵ0E = \frac{\sigma}{2\epsilon_0} Independent of distance rr
Field: Spherical Shell Eout=kQr2,Ein=0E_{out} = \frac{kQ}{r^2}, E_{in} = 0 Shell acts as point charge for r>Rr > R

Core Concepts Recap

  • Charge Properties: Remember the 'Big Three'—Additivity, Conservation, and Quantization.
  • Superposition Principle: Forces and fields are vectors. Net effect is the vector sum, not the scalar sum.
  • Conductors vs. Insulators: In conductors, charges reside only on the outer surface. In insulators, they stay where they are placed.
  • Electric Field Lines: Tangent gives E⃗\vec{E} direction; density gives E⃗\vec{E} magnitude. They never cross and never form loops.
  • Gauss's Law Strategy: Use it when there is high symmetry (Spherical, Cylindrical, or Planar). It simplifies complex integration into simple multiplication.

🎯 Exam Success Guide

For Board Exams:

  1. Derivations are Gold: Practice the Axial and Equatorial field derivations and the three Gauss's Law applications. They are almost guaranteed 3 or 5-mark questions.
  2. Units & Dimensions: Never forget to write 'N/C' for field or 'C m' for dipole moment. Losing half a mark for units is a common mistake.
  3. Diagrams: Always draw a clear diagram for Gauss's Law problems showing the Gaussian surface and the area vector.

For JEE Main & NEET:

  1. Short Dipole Approximation: Most competitive questions assume r≫ar \gg a. Use the 1/r31/r^3 formulas directly unless specified otherwise.
  2. Null Point Logic: Quickly find where E=0E=0 using the ratio of square roots: x=rq2/q1+1x = \frac{r}{\sqrt{q_2/q_1} + 1}.
  3. Flux Shortcuts: For charges at the corners or edges of cubes, use the 'Symmetry Extension' method (imagining multiple cubes to enclose the charge).
  4. Graph Identification: Be very comfortable with EE vs. rr graphs. Distinguish between the 'jump' in a shell's field vs. the 'linear rise' inside a solid non-conducting sphere.