Revision: Electrostatics >> Electric Charges and Fields Physics Science (English Medium) Class 12 CBSE

An electric charge which can be considered to exist at a single point is called a point charge.

The basic property of matter due to which it experiences electric force and shows attraction or repulsion, is called electric charge.

OR

The fundamental property of subatomic particles that gives rise to the phenomenon of experiencing force in the presence of electric and magnetic fields is called electric charge.

A unit positive charge used to test the strength of electric fields is called a test charge.

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)

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

A field whose magnitude and direction are not the same at all points is called a non-uniform electric field.

A field whose magnitude and direction is the same at all points is called a uniform electric field.

The region in which the charge experiences an electric force is the electric field around the charge.

The space surrounding an electric charge q in which another charge q₀ experiences a (electrostatic) force of attraction or repulsion, is called the electric field of the charge q.

OR

Electric field due to a charge Q at a point in space may be defined as the force that a unit positive charge would experience if placed at that point.

OR

The region surrounding an electric charge or a group of charges in which another charge experiences a force is called an electric field.

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

“An electric line of force is an imaginary smooth curve drawn in an electric field along which a free, isolated positive charge moves. The tangent drawn at any point on the electric line of force gives the direction of the force acting on a positive charge placed at that point.”

OR

An imaginary curve drawn in such a way that the tangent at any given point on this curve gives the direction of the electric field is called an electric line of force.

A measure of electric field through a surface, given by the number of electric lines of force per unit area enclosing the electric lines of force, is called electric flux.

An electric dipole is a pair of equal and opposite point-charges placed at a short distance apart.

OR

A system formed by two equal and opposite point charges placed at a small distance apart is called an electric dipole.

“The line joining the two charges, pointing from the negative charge to the positive charge. This is known as the ‘direction of dipole axis’.”

The electric dipole moment is defined as the product of the magnitude of one of the charges and the distance between the two equal and opposite charges.

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

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.

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.

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.

“1 radian is the angle which an arc of length equal to the radius of a circle subtends at the centre of the circle.”

The arc of a circle subtends an angle at the centre of the circle. This angle is called a 'plane angle'.

In an electric field, the ratio of electric flux through a surface to the area A of the surface is called the 'electric flux density' at the location of the surface.

Mathematical Definition:

Electric flux density = \[\frac {Φ_E}{A}\]

For a plane surface normal to the electric field:

Electric flux density = \[\frac {E A}{A}\] = E

The electric flux linked with a surface in an electric field may be defined as the surface integral of the normal component of the electric field over that surface.

The electric flux is a measure of the number of lines of force passing through some surface held in the electric field. It is denoted by Ф_E.

OR

The electric flux through a surface is the dot product of the electric field and the area vector of the surface, is called electric flux.

“1 steradian is the solid angle subtended by a part of the surface of a sphere at the centre of the sphere, when the area of the part is equal to the square of the radius of the sphere.”

Conductors are those through which electric charge can easily flow. Metals, human body, earth, mercury and electrolytes are conductors of electricity.

OR

Substances which offer high resistance to the passage of electricity and do not allow electricity to pass through them easily, are called insulators.

OR

The material through which electric charge can flow easily is called a conductor.

Substances whose resistance to the movement of charges is intermediate between conductors and insulators, are called semiconductors.

Those substances in which electric charge cannot flow are called ‘insulators' (or dielectrics). Glass, hard-rubber, plastics and dry wood are insulators. Insulators have practically no free electrons.

OR

The material in which electrons are tightly bound to the nucleus and thus not available for conductance is called an insulator.

\[\vec{E}=\frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2}\hat{r}\]

E = \[\frac {\text {Number of electric lines of force}}{\text {Area enclosing the electric lines of force}}\]

p = q × 2a

It is a vector quantity; its direction is from −q to +q.

\[\vec τ\] = \[\vec p\] × \[\vec E\]

Magnitude: τ = pE sin θ

λ = \[\frac {ΔQ}{Δl}\] C/m

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

ρ = \[\frac {ΔQ}{ΔV}\] C/m³

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

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

σ = \[\frac {ΔQ}{ΔS}\] C/m²

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

Ф_Е = E A cos θ

• If the plane surface is normal to the electric field (θ = 0):
  ФЕ = ЕА cos 0 = ЕА
• If the plane surface is parallel to the electric field (θ = 90°):
  Or = E A cos 90° = 0
• For field lines entering the plane surface normally (θ = 180°):
  ФЕ = ЕА cos 180° = -EA

\[\Phi_{E}=\int_{A}\vec{\mathbf{E}}\cdot d\vec{\mathbf{A}}\]

where,
∫_A = is the (surface) integral over the entire surface
Φ_E =  is positive when lines leave the surface, negative when they enter.

OR

Electric flux through a small area element:

dΦ = E ⋅ dS = E dS cos θ

Total electric flux through a surface:

Φ = \[\int_S\vec{E}\cdot d\vec{S}\]

E = \[\frac{1}{4\pi\varepsilon_{0}}\frac{p}{r^{3}}\]

In vector notation:
\[\overrightarrow{\mathbf{E}}=-\frac{1}{4\pi\varepsilon_{0}}\frac{\overrightarrow{\mathbf{p}}}{r^{3}}\]

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.

Statement

Gauss’s theorem in electrostatics states that the total electric flux through any closed surface (called a Gaussian surface) is equal to \[\frac {1}{ε_0}\] times the net charge enclosed by the surface, irrespective of the shape and size of the surface.

Mathematical Form

Φ_E = \[\oint\vec{E}\cdot d\vec{A}=\frac{q_{\mathrm{enc}}}{\varepsilon_0}\]​​

where

• \[\vec E\] = electric field intensity
• d\[\vec{A}\] = outward normal area element
• q_enc = net charge enclosed
• ε_0​ = permittivity of free space

Proof (Outline)

Consider a point charge +q placed inside a closed surface.
The electric field at a point on the surface is

E = \[\frac{1}{4\pi\varepsilon_0}\frac{q}{r^2}\]​

The flux through a small area element dA is

dΦ_E = \[\vec{E}\cdot d\vec{A}=EdA\cos\theta\]

Since dAcos⁡θ = r²dΩ,

dΦ_E = \[\frac{q}{4\pi\varepsilon_0}d\Omega\]

Integrating over the entire closed surface,

Φ_E = \[\frac{q}{4\pi\varepsilon_0}\int d\Omega\]

But the total solid angle subtended by a closed surface is 4π,

Φ_E = \[\frac  {q}{ε_0}\]​

Hence proved.

Key Note

Gauss’s law is most useful for symmetric charge distributions (spherical, cylindrical, planar) and is valid for all inverse-square law fields.

• Thales (≈2500 years ago) observed that amber rubbed with wool attracts light objects like paper and straw.
• William Gilbert (1600) showed that many materials, such as glass, ebonite, and sulphur, also show this effect.
• This attractive property is produced by rubbing (friction); a material showing it is said to be electrified, and the process is called frictional electricity.
• An electrified material possesses electric charge and is therefore called a charged body.
• Electric charge is quantised (q = ±ne,  e = 1.6 × 10⁻¹⁹ C); there are two types of charges (positive and negative), as charges repel, unlike charges attract, and the SI unit of charge is coulomb (C).

• 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. 

• A charge creates an electric field around it, even if no other charge is present.
• The electric field does not depend on the test charge used to measure it (if the test charge is very small).
• The field of a positive charge points outward; the field of a negative charge points inward.
• The strength of the electric field decreases as the distance from the charge increases.
• At equal distances from a point charge, the electric field has the same magnitude (spherical symmetry).

• The electric field due to many charges is the force on a unit test charge at that point.
• The total electric field is the vector sum of fields due to each charge (superposition principle).
• The electric field depends on the positions of the charges and changes from point to point in space.

• An electric field describes the electrical effect of a system of charges and does not depend on the test charge used to measure it.
• It is a vector quantity defined at every point in space and can vary from point to point.
• In changing situations, electromagnetic fields travel at the speed of light and can carry energy from one place to another.

• Electric field lines originate from positive charges and terminate on negative charges (or at infinity).
• The tangent to a field line at any point gives the direction of the electric field; in a uniform field, the lines are parallel and straight.
• No two electric field lines intersect, as this would imply more than one direction of the electric field at a point.
• Electric field lines do not pass through a conductor, showing that the electric field inside a conductor is zero.
• The density of field lines indicates field strength—closer lines represent a stronger field, while wider spacing represents a weaker field; the lines are continuous and imaginary, though the field is real.

• SI unit of Electric flux = N·m²·C⁻¹ or V·m (since E = N / C = V/m).
• Dimensions of electric flux: [Φ_E] = [E] [A] = [ML³T⁻³A⁻¹]

• Point Charge:
  Using a spherical Gaussian surface, the electric field due to a point charge is
  E = \[\frac{1}{4\pi\varepsilon_0}\frac{q}{r^2}\]​
  which directly leads to Coulomb’s law.
• Infinite Line of Charge:
  For a uniformly charged infinite wire with linear charge density λ,
  E = \[\frac{\lambda}{2\pi\varepsilon_0r}\]​
  The field is radial and varies inversely with distance r.
• Infinite Plane Sheet of Charge:
  For a sheet with surface charge density σ,
  E = \[\frac{\sigma}{2\varepsilon_0}\]​
  The field is independent of distance from the sheet.
• Two Parallel Charged Sheets:
  The electric field is uniform between the sheets and zero outside when the sheets carry equal and opposite charges.
• Charged Conductor:
  The electric field inside a conductor is zero, and just outside the surface,
  E = \[\frac{\sigma}{\varepsilon_0}\]​
  where σ is surface charge density.
• Uniformly Charged Spherical Shell / Conducting Sphere:
  Outside the shell: behaves like a point charge at the centre
  Inside the shell: the electric field is zero
• Uniformly Charged Non-conducting Sphere:
  Outside: E ∝ \[\frac {1}{r^2}\]​
  Inside: E ∝ r, increasing linearly from centre to surface

• A Gaussian surface is an imaginary closed surface used to calculate the electric flux of a vector field.
• It must be a closed surface (e.g., a sphere, cylinder, or cube); open surfaces such as discs or squares are not valid.
• The shape of the Gaussian surface should match the symmetry of the charge distribution so that the electric field is uniform or normal to the surface.
• The surface must not pass through any discrete charge, though it may pass through a continuous charge distribution.
• Electric flux through a Gaussian surface depends only on the charges enclosed, even though the electric field on the surface is due to both internal and external charges.

• Gauss’ theorem establishes a connection between the electric flux through a closed surface and the charge enclosed, and is especially useful for highly symmetric charge distributions.
• An area can be treated as a vector quantity, with both magnitude (area) and direction (the outward normal to the surface).
• A solid angle is the three-dimensional analogue of a plane angle and describes how a surface appears from a point.
• The total solid angle subtended at a point by a closed surface, irrespective of its shape, is always the same.
• The solid angle subtended by an area element depends on its orientation and position relative to the point, being maximum when the area faces the point directly.