Physics · Electromagnetism

Electrostatics formulas for JEE

Every Electrostatics formula you need for JEE, grouped by concept.

FormulasRevision notes
48 formulas5 concepts
01Electric Charge and Coulomb's Law02Electric Field03Gauss's Law and Flux04Electric Potential05Capacitors and Dielectrics
01

Electric Charge and Coulomb's Law

6 formulas

Coulomb's Law

F21=14πϵ0q1q2r212r^21\mathbf{F}_{21} = \frac{1}{4\pi\epsilon_0}\frac{q_1 q_2}{r_{21}^2} \hat{\mathbf{r}}_{21}

The mutual electrostatic force between two stationary point charges.

applies whenValid for point charges at rest in a vacuum.
electrostaticsforcecoulomb

Superposition Principle for Electric Force

F1=q14πϵ0∑i=2nqir1i2r^1i\mathbf{F}_1 = \frac{q_1}{4\pi\epsilon_0} \sum_{i=2}^{n} \frac{q_i}{r_{1i}^2} \hat{\mathbf{r}}_{1i}

The total force on a given charge is the vector sum of the individual forces exerted on it by all other charges.

applies whenValid for a system of discrete point charges.
electrostaticsforcesuperposition

Linear Charge Density

λ=ΔQΔl\lambda = \frac{\Delta Q}{\Delta l}

Charge per unit length for a 1D line charge distribution.

applies whenMacroscopic continuous approximation.
electrostaticscharge_densitylinear

Quantization of Charge

q=neq = ne

Electric charge on a body is always an integral multiple of the basic unit of charge e.

applies whenApplicable for discrete charges; at macroscopic levels, quantization can often be ignored.
electrostaticschargequantization

Volume Charge Density

ρ=ΔQΔV\rho = \frac{\Delta Q}{\Delta V}

Charge per unit volume for a 3D volume charge distribution.

applies whenMacroscopic continuous approximation.
electrostaticscharge_densityvolume

Surface Charge Density

σ=ΔQΔS\sigma = \frac{\Delta Q}{\Delta S}

Charge per unit area for a 2D surface charge distribution.

applies whenMacroscopic continuous approximation.
electrostaticscharge_densitysurface
02

Electric Field

9 formulas

Electric Field of Dipole on Axis

Eaxis=14πϵ02pr3\mathbf{E}_{axis} = \frac{1}{4\pi\epsilon_0} \frac{2\mathbf{p}}{r^3}

The electric field at a point on the axis of an electric dipole at a distance r from its center.

applies whenValid for distances much larger than the dipole separation (r≫ar \gg a). Direction is parallel to the dipole moment.
electrostaticselectric_fielddipole

Electric Field of Dipole on Equatorial Plane

Eeq=−14πϵ0pr3\mathbf{E}_{eq} = -\frac{1}{4\pi\epsilon_0} \frac{\mathbf{p}}{r^3}

The electric field at a point on the equatorial plane of an electric dipole at a distance r from its center.

applies whenValid for distances much larger than the dipole separation (r≫ar \gg a). Direction is anti-parallel to the dipole moment.
electrostaticselectric_fielddipole

Electric Dipole Moment

p=q(2a)\mathbf{p} = q(2\mathbf{a})

Dipole moment of a pair of equal and opposite point charges separated by a distance 2a.

applies whenVector directed from the negative charge to the positive charge.
electrostaticsdipole

Definition of Electric Field

E=lim⁡q→0Fq\mathbf{E} = \lim_{q \to 0} \frac{\mathbf{F}}{q}

Electric field is the electrostatic force experienced by a vanishingly small positive test charge q.

applies whenThe test charge must be infinitesimally small so it does not disturb the source charge configuration.
electrostaticselectric_fielddefinition

Electric Field of a Point Charge

E=14πϵ0Qr2r^\mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{Q}{r^2}\hat{\mathbf{r}}

The electric field produced by a point charge Q at a distance r.

applies whenValid for stationary point charges in a vacuum.
electrostaticselectric_field

Superposition Principle for Electric Field

E(r)=14πϵ0∑i=1nqiriP2r^iP\mathbf{E}(\mathbf{r}) = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^{n} \frac{q_i}{r_{iP}^2} \hat{\mathbf{r}}_{iP}

The electric field at a point due to a system of charges is the vector sum of the electric fields produced by individual charges at that point.

applies whenValid for a system of discrete point charges.
electrostaticselectric_fieldsuperposition

Electric Field at the Surface of a Charged Conductor

E=σϵ0n^\mathbf{E} = \frac{\sigma}{\epsilon_0} \hat{\mathbf{n}}

The electric field just outside the surface of a charged conductor.

applies whenStatic situation; σ\sigma is the local surface charge density and n^\hat{\mathbf{n}} is the outward normal.
electrostaticsconductorelectric_field

Force on a Charge in an Electric Field

F=qE\mathbf{F} = q\mathbf{E}

The electrostatic force experienced by a charge q placed in an electric field E.

applies whenE is the external electric field at the exact location of charge q.
electrostaticsforceelectric_field

Torque on a Dipole

τ=p×E\boldsymbol{\tau} = \mathbf{p} \times \mathbf{E}

Torque experienced by an electric dipole tending to align it with an external electric field.

applies whenAssumes a uniform external electric field.
electrostaticsdipoletorque
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03

Gauss's Law and Flux

6 formulas

Electric Field of an Infinite Plane

E=σ2ϵ0n^\mathbf{E} = \frac{\sigma}{2\epsilon_0} \hat{\mathbf{n}}

Electric field produced by an infinite plane sheet of uniform surface charge density.

applies whenIndependent of distance from the plane.
electrostaticsgaussplane

Electric Field of an Infinite Wire

E=λ2πϵ0rn^\mathbf{E} = \frac{\lambda}{2\pi\epsilon_0 r} \hat{\mathbf{n}}

Electric field radially outward (or inward) from an infinitely long straight wire.

applies whenAssumes an infinitely long wire with uniform linear charge density.
electrostaticsgausswire

Electric Flux

Φ=∫E⋅dS\Phi = \int \mathbf{E} \cdot d\mathbf{S}

Electric flux representing the number of electric field lines crossing a small planar area element.

applies whenArea element must be treated as a vector directed along its normal.
electrostaticsflux

Gauss's Law

∮E⋅dS=qencϵ0\oint \mathbf{E} \cdot d\mathbf{S} = \frac{q_{enc}}{\epsilon_0}

The net electric flux through any closed Gaussian surface is equal to the total enclosed charge divided by the permittivity of free space.

applies whenValid for any closed surface regardless of shape or size.
electrostaticsgauss

Electric Field Inside a Spherical Shell

E=0\mathbf{E} = \mathbf{0}

Electric field at any point inside a uniformly charged thin spherical shell.

applies whenFor r<Rr < R, where R is the radius of the shell.
electrostaticsgaussspherical_shell

Electric Field Outside a Spherical Shell

E=14πϵ0qr2r^\mathbf{E} = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \hat{\mathbf{r}}

Electric field at a distance r outside a uniformly charged thin spherical shell.

applies whenFor r≥Rr \ge R, where R is the radius of the shell.
electrostaticsgaussspherical_shell
04

Electric Potential

14 formulas

Potential Energy of a Charge in an External Field

U=qV(r)U = qV(\mathbf{r})

Potential energy of a single charge q placed at a position r in an external electric potential V.

applies whenThe external field is not produced by the given charge q.
electrostaticspotential_energyexternal_field

Potential Energy of Two Charges in an External Field

U=q1V(r1)+q2V(r2)+14πϵ0q1q2r12U = q_1 V(\mathbf{r}_1) + q_2 V(\mathbf{r}_2) + \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_{12}}

Total potential energy of a system of two interacting charges in an external electric field.

applies whenCombines the energy of individual charges in the external field with their mutual interaction energy.
electrostaticspotential_energysystem

Potential Energy of a Three-Charge System

U=14πϵ0(q1q2r12+q1q3r13+q2q3r23)U = \frac{1}{4\pi\epsilon_0} \left( \frac{q_1 q_2}{r_{12}} + \frac{q_1 q_3}{r_{13}} + \frac{q_2 q_3}{r_{23}} \right)

The total work done in assembling a system of three point charges at given locations from infinity.

applies whenCharges are brought in from infinite separation without acceleration.
electrostaticspotential_energysuperposition

Potential Energy of a Two-Charge System

U=14πϵ0q1q2r12U = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_{12}}

The electrostatic potential energy of a system of two charges separated by distance r12.

applies whenPotential energy is taken to be zero when charges are infinitely far apart.
electrostaticspotential_energytwo_charges

Electric Potential Difference

ΔV=VP−VR=WRPq\Delta V = V_P - V_R = \frac{W_{RP}}{q}

Potential difference between two points is the work done by an external force in moving a unit positive charge from one point to the other.

applies whenExternal force must move the charge infinitesimally slowly.
electrostaticspotentialwork

Potential due to an Electric Dipole

V=14πϵ0p⋅r^r2V = \frac{1}{4\pi\epsilon_0} \frac{\mathbf{p} \cdot \mathbf{\hat{r}}}{r^2}

Electrostatic potential at a point having position vector r relative to the center of the dipole.

applies whenApproximation valid for distances much larger than the size of the dipole (r≫ar \gg a).
electrostaticspotentialdipole

Potential Energy of a Dipole in External Field

U=−p⋅EU = -\mathbf{p} \cdot \mathbf{E}

The potential energy stored by a dipole placed in a uniform external electric field.

applies whenAssumes zero potential energy when the dipole is perpendicular to the electric field.
electrostaticsenergydipole

Relation between Electric Field and Potential

∣E∣=−δVδl|\mathbf{E}| = -\frac{\delta V}{\delta l}

The magnitude of the electric field is given by the change in the magnitude of potential per unit displacement normal to the equipotential surface.

applies whenThe field is in the direction of the steepest decrease of potential.
electrostaticspotentialgradient

Potential due to a Point Charge

V(r)=14πϵ0QrV(r) = \frac{1}{4\pi\epsilon_0}\frac{Q}{r}

The electrostatic potential at a point due to a point charge Q placed at the origin.

applies whenPotential is taken to be zero at infinity.
electrostaticspotentialpoint_charge

Potential Inside a Spherical Shell

V=14πϵ0qRV = \frac{1}{4\pi\epsilon_0} \frac{q}{R}

Electrostatic potential inside a uniformly charged thin spherical shell.

applies whenFor r≤Rr \le R. The potential is constant everywhere inside and equals the surface potential.
electrostaticspotentialspherical_shell

Potential Outside a Spherical Shell

V=14πϵ0qrV = \frac{1}{4\pi\epsilon_0}\frac{q}{r}

Electrostatic potential at a distance r outside a uniformly charged thin spherical shell.

applies whenFor r≥Rr \ge R, where R is the radius. The shell behaves as a point charge.
electrostaticspotentialspherical_shell

Potential due to a System of Charges

V=14πϵ0∑i=1nqiriV = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^{n} \frac{q_i}{r_i}

The electrostatic potential at a point due to a system of point charges is the algebraic sum of the potentials due to individual charges.

applies whenValid for discrete point charges applying the superposition principle.
electrostaticspotentialsuperposition

Work Done in Rotating a Dipole

W=pE(cos⁡θ0−cos⁡θ1)W = pE(\cos\theta_0 - \cos\theta_1)

Work done by an external torque in rotating an electric dipole from angle θ0\theta_0 to θ1\theta_1 in a uniform electric field.

applies whenAssuming uniform external electric field E. The rotation is performed at an infinitesimal angular speed.
electrostaticsdipolework

Work and Potential Energy Difference

ΔU=UP−UR=−WRP\Delta U = U_P - U_R = -W_{RP}

The change in electrostatic potential energy between two points is the negative of the work done by the electrostatic force.

applies whenValid for conservative electrostatic fields.
electrostaticspotential_energywork
05

Capacitors and Dielectrics

13 formulas

Capacitance

C=QVC = \frac{Q}{V}

The ratio of the magnitude of charge on either conductor to the potential difference between them.

applies whenDepends only on the geometrical configuration of the conductors and the intervening medium.
electrostaticscapacitance

Capacitance with Dielectric

C=Kϵ0AdC = \frac{K \epsilon_0 A}{d}

Capacitance of a parallel plate capacitor fully occupied by a dielectric material.

applies whenDielectric constant K must be greater than 1.
electrostaticscapacitancedielectric

Electric Field inside a Parallel Plate Capacitor

E=σϵ0=Qϵ0AE = \frac{\sigma}{\epsilon_0} = \frac{Q}{\epsilon_0 A}

The uniform electric field localized between the two plates of a parallel plate capacitor.

applies whenAssumes vacuum between plates and ignores fringing fields at the edges (valid when d2≪Ad^2 \ll A).
electrostaticscapacitanceelectric_field

Energy Stored in a Capacitor

U=12CV2=Q22C=12QVU = \frac{1}{2} C V^2 = \frac{Q^2}{2C} = \frac{1}{2} Q V

The total electrostatic potential energy stored when assembling charges onto the plates of a capacitor.

applies whenEnergy can be viewed as residing in the electric field between the plates.
electrostaticscapacitanceenergy

Capacitance of a Parallel Plate Capacitor

C=ϵ0AdC = \frac{\epsilon_0 A}{d}

Capacitance of a parallel plate capacitor in vacuum.

applies whenThe plate separation dd must be much smaller than the linear dimensions of the plates (d2≪Ad^2 \ll A).
electrostaticscapacitanceparallel_plate

Capacitors in Parallel

Ceq=C1+C2+⋯+CnC_{eq} = C_1 + C_2 + \dots + C_n

Equivalent capacitance of multiple capacitors connected in parallel.

applies whenThe potential difference across each capacitor in the parallel combination is identical.
electrostaticscapacitanceparallel

Potential Difference across Parallel Plate Capacitor

V=Ed=Qdϵ0AV = Ed = \frac{Qd}{\epsilon_0 A}

The potential difference across the plates of a parallel plate capacitor with a uniform internal electric field.

applies whenAssumes uniform electric field between the plates.
electrostaticscapacitancepotential_difference

Capacitors in Series

1Ceq=1C1+1C2+⋯+1Cn\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots + \frac{1}{C_n}

Equivalent capacitance of multiple capacitors connected in series.

applies whenThe charge on each capacitor in the series sequence remains identical.
electrostaticscapacitanceseries

Dielectric Constant

K=ϵϵ0=CC0K = \frac{\epsilon}{\epsilon_0} = \frac{C}{C_0}

The factor by which the capacitance increases from its vacuum value when a dielectric is inserted fully between the plates.

applies whenDielectric must completely fill the space between the capacitor plates.
electrostaticsdielectricconstant

Induced Surface Charge Density in Dielectric

σ−σp=σK\sigma - \sigma_p = \frac{\sigma}{K}

The relation between free surface charge density (σ\sigma) on capacitor plates and the induced bound charge density (σp\sigma_p) on the dielectric surface.

applies whenValid for linear isotropic dielectrics fully occupying the space between the plates.
electrostaticscapacitancedielectriccharge_density

Energy Density of an Electric Field

u=12ϵ0E2u = \frac{1}{2} \epsilon_0 E^2

The electrostatic energy stored per unit volume in a region containing an electric field.

applies whenGeneral relationship holding true for the electric field of any charge configuration in vacuum.
electrostaticsenergy_density

Permittivity of a Medium

ϵ=ϵ0K\epsilon = \epsilon_0 K

The absolute permittivity of a dielectric medium calculated using the permittivity of free space and the dielectric constant K.

applies whenKK is the dimensionless dielectric constant (where K>1K > 1).
electrostaticspermittivitydielectric

Polarisation of a Dielectric

P=χeϵ0E\mathbf{P} = \chi_e \epsilon_0 \mathbf{E}

The induced dipole moment per unit volume in a dielectric material under an external electric field.

applies whenValid for linear isotropic dielectrics.
electrostaticsdielectricpolarisation
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