Physics · Class 12 · JEE / NEET / AP Physics 2

Electrostatics: Coulomb's Law & Electric Potential

Electrostaticsstudies stationary electrical charges.Coulomb's Lawdictates the fundamental inverse-square electrostatic force, whileElectric Potential (VV)measures the work required to assemble charge distributions.

11. Visual Overview & Intuitive Discovery: Electrostatics: Coulomb's Law & Electric Potential

If you could remove just 1% of the electrons from a single grain of sand and place them 1 meter away from the remaining positive ions, the repulsive force between them would exceed the weight of the entire planet Earth! Electrostatic force is unimaginably powerful—103610^{36} times stronger than gravity. The only reason we don't experience this overwhelming force in daily life is the almost perfect balance of positive protons and negative electrons in ordinary matter.

Electrostatics: Coulomb Force & Dipole Potential: Coulomb's Law, Electric Field Vector, Electric Dipole, Electric Potential (V), Equipotential Contours Full size

Electrostatics: Coulomb Force & Dipole Potential

Detailed electrostatic field map showing Coulomb force vectors, electric dipole field geometry, and concentric equipotential contour surfaces.

  1. 1Coulomb's Law — F = k |q1 q2| / r² where k = 1/(4πε₀) ≈ 8.99 × 10⁹ N·m²/C²
  2. 2Electric Field Vector — E = F / q₀, radiating radially outward from +Q and inward toward -Q
  3. 3Electric Dipole — Two equal and opposite charges ±q separated by 2a, with dipole moment p = q · 2a directed from - to +
  4. 4Electric Potential (V) — Scalar field V = k q / r, representing potential energy per unit positive test charge
  5. 5Equipotential Contours — Surfaces of constant V; zero work is required to move a charge along an equipotential surface

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22. Physical Mechanism Flowchart: Electrostatics: Coulomb's Law & Electric Potential

Electrostatics: Coulomb's Law & Electric Potential Step-by-Step Flowchart

  1. 1

    Initial State

    Initial parameters and boundary conditions are established.

  2. 2

    Fundamental Force/Interaction

    Governing physical laws act continuously across the system.

  3. 3

    Energy & Vector Conservation

    Conservation principles restrict permitted trajectory states.

  4. 4

    Dynamic Equilibrium

    Final observable state satisfies equilibrium equations.

Follow the numbered pipeline above. In physics examinations, sketching this mental flowchart ensures you never miss intermediate steps in complex multi-concept numericals.

33. Rigorous Mathematical Proofs & Governing Equations

Derivation — Derivation of Governing Formula for Electrostatics: Coulomb's Law & Electric Potential

  1. Fnet=maF_{\text{net}} = m a

    Starting from Newton's foundational law

  2. ∫F dx=ΔK\int F \, dx = \Delta K

    Integrating over displacement

Coulomb Force & Electric Field Relations

F⃗=14πε0q1q2r2r^,E⃗=−∇⃗V=−dVdrr^\vec{F} = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r^2}\hat{r},\quad \vec{E} = -\vec{\nabla} V = -\frac{dV}{dr}\hat{r}
  • ε0\varepsilon_0 = Permittivity of free space (8.854 × 10⁻¹² C²/N·m²)
  • q1,q2q_1, q_2 = Magnitudes of interacting charges
  • V = Electric potential in Volts (J/C)
  • E⃗\vec{E} = Electric field in N/C or V/m

Electric Field & Potential of a Dipole ($p = 2aq$)

  • Axial Line (Distance r ≫ a)

    Electric Field Magnitude ($E$)
    2kpr3\frac{2kp}{r^3}
    Electric Potential ($V$)
    kpr2\frac{kp}{r^2}

  • Equatorial Line (Distance r ≫ a)

    Electric Field Magnitude ($E$)
    kpr3\frac{kp}{r^3}
    Electric Potential ($V$)
    00 (Potential is strictly zero everywhere on equator)

  • General Point (r,θ)(r, \theta)

    Electric Field Magnitude ($E$)
    kpr31+3cos⁡2θ\frac{kp}{r^3}\sqrt{1 + 3\cos^2\theta}
    Electric Potential ($V$)
    kpcos⁡θr2\frac{kp\cos\theta}{r^2}

44. Medical Application: Electrocardiograms (ECG) and Cardiac Dipoles

Every human heartbeat is fundamentally an electrostatic event. When pacemaker cells in the sinoatrial node fire, waves of sodium (Na+Na^+) and potassium (K+K^+) ions rush across cell membranes, creating a macroscopic traveling electric dipole vector p⃗\vec{p}. Hospital ECG electrodes placed on your chest and wrists detect the tiny millivolt potential differences (V=kpcos⁡θr2V = \frac{k p \cos\theta}{r^2}) produced by this cardiac dipole, plotting the famous P-QRS-T wave that reveals heart health.

Common mistake: Misconception: "If electric potential is zero at a point, the electric field must also be zero." Consider the exact midpoint between two equal and opposite charges (+q and -q). The electric potential is scalar: V=kqd+k(−q)d=0 VV = \frac{kq}{d} + \frac{k(-q)}{d} = 0\,\text{V}. However, both charges produce electric field vectors pointing in the exact same direction toward the negative charge, so the net electric field is non-zero (E=2kqd2≠0E = \frac{2kq}{d^2} \neq 0)!

55. Solved Standard & Advanced Exam Problems

Exam question · CBSE Class 12 & AP Physics Standard

Two point charges +4 μC and -1 μC are separated by a distance of 30 cm in air. At what point along the line joining them is the net electric potential zero?
  1. A.24 cm from +4 μC (between the charges)
  2. B.10 cm from +4 μC
  3. C.6 cm from -1 μC outside the charges
  4. D.Both 24 cm between them and 40 cm outside near -1 μC

Exam question · JEE Main & NEET

Four equal charges +q are placed at the four corners of a square of side a. What is the electric field at the center of the square?
  1. A.Zero
  2. B.4kq / a²
  3. C.2kq / a²
  4. D.8kq / a²

Remember

  • Coulomb force is an inverse-square force that obeys Newton's third law in vector form.
  • Electric field lines never cross each other because a single point cannot have two distinct field directions.
  • Work done moving a charge q between two points is W = q · ΔV and is path-independent in electrostatics.
  • The electric field inside an electrostatic conductor in equilibrium is always zero.

Check yourself

  1. Question 1

    What is the work done in moving a test charge of 5 μC along an equipotential surface of 50 V for a distance of 10 cm?

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