Revision: Electrostatics JEE Main Electrostatics

The closed surface over which the surface integral of the electric field intensity (i.e. total electric flux) is considered in Gauss' Law is called a Gaussian surface.

The study of electricity/electric charges at rest is called electrostatics.

The solids which have very small number of free electrons are called insulators. (e.g. Glass, Wood)

The solids which have a large number of free electrons are called conductors. (e.g. Iron, Aluminium)

The energy difference between the valence band and the conduction band is called forbidden energy gap.

The range of energies possessed by conduction electrons is called conduction band.

The range of energies possessed by valence electrons is called valence band.

The different energy levels with continuous energy variation are called energy bands.

The material with electrical conductivity between that of a conductor and an insulator, whose number of charge carriers can be controlled as per requirement, is called a semiconductor. (e.g. Silicon, Germanium)

A time-dependent combination of electric and magnetic fields that propagates through space and can transport energy is called an electromagnetic field.

Electric Field \[\vec E\] at a point is the electrostatic force \[\vec F\] experienced by a vanishingly small positive test charge q₀ placed at that point:

\[\vec E\] = \[\frac {\vec F}{q_0}\]

The charge per unit volume in a region of space, is called volume charge density.

OR

When charge is distributed over the volume of an object, it is called volume charge distribution.

The charge per unit area on a surface, is called surface charge density.

The charge per unit length along a line (such as a wire), is called linear charge density.

OR

When charge is distributed along a line, the charge distribution is called linear charge distribution.

A charge distribution in which charge is treated as continuously spread over a line, surface, or volume (ignoring microscopic discreteness), is called continuous charge distribution.

When charge is distributed along a line, the charge distribution is called linear charge distribution.

When charge is distributed over a surface, the charge distribution is called surface charge distribution.

When charge is distributed over the volume of an object, it is called volume charge distribution.

The phenomenon in which the electric field inside a cavity of a conductor is zero, irrespective of external charges or fields, is called electrostatic shielding.

A system consisting of two conductors having equal and opposite charges separated by an insulator or dielectric is called a capacitor.

The maximum electric field that a dielectric medium can withstand without breakdown (of its insulating property) is called its dielectric strength.

The ability of a conductor to store charge is called the capacity of conductor.

The ratio of the charge Q given to one of the conductors of a capacitor to the potential difference V between the conductors is called its capacitance, given by C = Q/V.

A capacitor that consists of two large, parallel, conducting plates separated by a small distance is called a parallel plate capacitor.

The ratio of the permittivity of a medium to the permittivity of vacuum.

K = ε / ε₀

OR

Dielectric constant is the factor by which the capacitance of a capacitor increases when a dielectric is completely inserted between its plates.

The product of vacuum permittivity and dielectric constant of the medium.

ε = ε₀K

For a system of n point charges q₁, q₂, q₃,…, qn, the total electric field at point P is:

E(r) = \[{\frac{1}{4\pi\varepsilon_0}\sum_{i=1}^n\frac{q_i}{r_{iP}^2}\hat{\mathbf{r}}_{iP}}\]

Symbol Reference

\[\vec τ\] = \[\vec p\] × \[\vec E\]

Magnitude: τ = pE sin θ

\[\vec{E}=\frac{1}{4\pi\varepsilon_0}\sum\frac{\rho\Delta V}{r^{\prime2}}\hat{r}^{\prime}\]

λ = \[\frac {ΔQ}{Δl}\] C/m

where ΔQ is the charge distributed over a small length Δl of the wire.

σ = \[\frac {ΔQ}{ΔS}\] C/m²

where ΔQ is the charge distributed over a small surface area ΔS.

ρ = \[\frac {ΔQ}{ΔV}\] C/m³

where ΔQ is the charge distributed over a small volume ΔV of the material.

\[V=\frac{Q}{4\pi\varepsilon_0r}\]

Potential due to System of Charges:

\[U=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r_{12}}\]

U = \[\frac{1}{4\pi\varepsilon_0}\cdot\frac{q_1q_2}{r_{12}}\]

V = \[\frac{1}{4\pi\varepsilon_0}\cdot\frac{q}{r}\]

Varies on spherical shell carrying charge q and radius R:

• Inside shell (r < R): V = \[\frac {1}{4πε_0}\] ⋅ \[\frac {q}{R}\]
• On surface (r = R): V = \[\frac {1}{4πε_0}\] ⋅ \[\frac {q}{R}\]
• Outside shell (r > R): V = \[\frac {1}{4πε_0}\] ⋅ \[\frac {q}{r}\]

\[V=\frac{1}{4\pi\varepsilon_{0}}\left[\frac{q_{1}}{r_{1}}+\frac{q_{2}}{r_{2}}+\frac{q_{2}}{r_{3}}+\frac{q_{4}}{r_{4}}+.........+\frac{q_{n}}{r_{n}}\right]\]

\[V=\frac{1}{4\pi\varepsilon_0}\sum_{i=1}^{i=n}\frac{q_i}{r_i}\]

\[\vec{E}=\frac{\sigma}{\varepsilon_0}\hat{n}\]

where
σ = surface charge density
\[\hat n\] = outward normal unit vector

Magnitude form:

E = \[\frac{\sigma}{\varepsilon_0}\]

C = 4πkε₀ · [\[\frac {ab}{(b − a)}\]]

C = \[\frac {2πkε₀ l}{2.303 log(b/a)}\]

C = Q/V

For two plates separated by distance d:

\[C=\frac{\varepsilon_0A}{d}\]

With a dielectric medium:

\[C=\frac{K\varepsilon_0A}{d}\]

"The electric field at any point due to a group of charges is the vector sum of the electric fields at that point due to each individual charge, calculated as if the other charges were not present."

• Each charge in the system contributes its own independent electric field at the point of interest.
• These individual fields are then added vectorially to give the total (resultant) field.

Gauss' Law states that the net electric flux through any closed surface is equal to `1/epsilon_0` times the net electric charge within that closed surface.

`oint  vec" E".d vec" s" = (q_(enclosed))/epsilon_o`

In the diagram, we have taken a  cylindrical gaussian surface of radius = r and length = l.
The net charge enclosed inside the gaussian surface `q_(enclosed) = lambdal`
By symmetry, we can say that the Electric field will be in radially outward direction.

According to gauss' law,

`oint  vec"E".d  vec"s" = q_(enclosed)/epsilon_o`

`int_1 vec"E" .d  vec"s" + int_2  vec"E" .d  vec"s" + int_3  vec"E". d  vec"s" = (lambdal)/epsilon_o`

`int_1  vec"E". d  vec"s"  &  int_3  vec"E". d  vec"s"  "are zero", "Since"  vec"E"  "is perpendicular to"  d  vec"s"`

`int_2  vec"E" . d  vec"s" = (lambdal)/epsilon_o`

`"at"  2,  vec"E" and d  vec"s"  "are in the same direction, we can write"`

`E.2pirl = (lambdal)/epsilon_o`

`E = lambda/(2piepsilon_o r)`

The flux of the net electric field through a closed surface equals the net charge enclosed by the surface divided by ε_0​.

Formula - Gauss's Law:

Key Points of Gauss's Law:

• Applicable to any closed surface of regular or irregular shape.
• Only the enclosed charge contributes to the electric flux.
• The electric field at a point depends on the total charge distribution, both inside and outside the Gaussian surface.

• Conductors → E_g = 0 - bands overlap, electrons flow freely.
• Semiconductors → E_g < 3 eV — small gap, conducts at room temperature.
• Insulators → E_g > 5 eV — large gap, no conduction.
• Ge = 0.72 eV, Si = 1.1 eV — both semiconductors.
• Metal conductivity decreases with temp. Semiconductor conductivity increases with temp. 

• The resultant field E is the vector sum of all individual fields.
• Each individual field E_i is calculated independently, as if no other charges exist.
• The unit vector \[\hat r_i\]P points from each charge q_i toward point P.
• The principle holds for any number of charges in any configuration.
• This is a direct application of the Superposition Principle to electric fields.

• \[\vec E\] = \[\vec F\]/q₀ — force per unit positive test charge
• Static case → Coulomb's Law is sufficient; field is a descriptive tool
• Accelerated charges → field becomes a real physical entity (EM waves)
• Time delay = d/c — information travels at the speed of light, not instantaneously
• An electric field carries and transports energy
• Field exists independently of whether any test charge is present
• Gravity is negligible for charged particles in typical electric fields