Revision: Electrostatics >> Electrostatic Potential and Capacitance Physics Science (English Medium) Class 12 CBSE

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.

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

A capacitor is connected across the terminals of a d.c. battery.

The energy stored on a capacitor is equal to the work done by the battery.

The work required to transport a small amount of charge (dQ) from the negative to positive plates of a capacitor is equal to V dQ, where V represents the voltage across the capacitor.

dU = V dQ

= `Q/C dQ`

∴ Energy stored (U) = ∫V dQ

= `1/C int Q dQ`

= `1/2 Q^2/C`

= `1/2 CV^2`    ...(i)

Energy density is defined as the total energy per unit volume of the capacitor.

For a parallel plate capacitor,

C = `(A epsilon_0)/d`

Putting in eqn. (i),

U = `1/2 (A epsilon_0)/d V^2`

= `epsilon_0/2 Ad(V/d)^2`

= `epsilon_0/2 Ad E^2`    ...[Putting `V/d` = E]

A × d = Volume of space between plates

So, energy is stored per unit volume.

 Potential difference: The potential difference between two points may be defined as the work done in moving a unit positive charge from one point to the other.

Electric potential is a measure of work done on the unit's positive charge to bring it to that point against all electrical forces. It is represented as ‘V’.

The potential at a point is defined as the amount of work done per unit charge in bringing a positive test charge from infinity to that point.

The potential difference (p.d.) between two points is equal to the work done per unit charge in moving a positive test charge from one point to the other.

OR

The work done per unit positive charge in moving a charge from one point to another in an electric field, is called potential difference between those two points.

The rate of change of potential with distance in the electric field is called the 'potential gradient'.

The electric potential energy of a system of charges is the work that has been done in bringing those charges from infinity to near each other to form the system.

OR

The total work done by an external agency in assembling the charges from infinity to their specified positions (without acceleration), is called the electrostatic potential energy of the system.

1 electron-volt is the work done in taking one electron from one point to the other, when the potential difference between these points is 1 volt.

OR

1 electron-volt is the (kinetic) energy which an electron acquires when accelerated through a potential difference of 1 volt.

Any surface over which the electric potential is same everywhere is called an equipotential surface.

OR

A surface on which the electric potential has the same value at every point, is called an equipotential surface.

An electric dipole is a pair of equal and opposite point charges, placed at a small distance. Its moment, known as electric dipole moment.

OR

A system of two equal and opposite charges separated by a small distance, is called an electric dipole.

The product of the magnitude of one charge and the separation vector directed from negative to positive charge, is called the electric dipole moment.

\[\vec p\]​ = q × (2\[\vec a\])

The work done by an external agent in carrying a unit positive test charge from infinity to a point in the electric field is called the electric potential at that point.

OR

The work done in bringing a unit positive charge (without acceleration) from infinity to a given point in an electric field, is called electrostatic potential at that point.

The capacitance of a capacitor is defined as the ratio of the charge given to a plate of the capacitor to the potential difference produced between the plates.

The capacitance of a conductor is defined as the ratio of the charge given to the rise in the potential of the conductor.

Mathematical definition: C = \[\frac {Q}{V}\]

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

A capacitor is a pair of two conductors of any shape which are close to each other and have equal and opposite charges. These conductors are called the 'plates' of the capacitor.

OR

A system of two conductors separated by an insulator, is called a capacitor.

To sum up, an electric field produces in a dielectric (non-polar or polar) a net dipole moment in the direction of the field. This phenomenon is known as 'dielectric polarisation' or 'electric polarisation of matter'.

“The total amount of work in charging the capacitor is stored up in the capacitor in the form of electric potential energy.”

The dielectric constant (or specific inductive capacity) of a material is the ratio of the capacitance of a given capacitor completely filled with that material to the capacitance of the same capacitor in vacuum.

OR

A non-conducting substance that has no (or negligible) free charge carriers and can be polarised in an external electric field, is called a dielectric.

A ‘polar' molecule is one in which the centre of gravity of the positive charges (protons) is separated from the centre of gravity of the negative charges (electrons) by a finite distance.

OR

A molecule in which the centres of positive and negative charges are separated, giving it a permanent dipole moment, is called a polar molecule.

The Molecules in which the centres of positive and negative charges coincide and so the molecules have zero electric dipole moment. Such molecules are called ‘non-polar' molecules.

OR

A molecule in which the centres of positive and negative charges coincide and has no permanent dipole moment, is called a non-polar molecule.

Dielectric strength is defined as the maximum value of the electric field that it can tolerate without its electric breakdown.

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

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

Magnitude form:

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

V = \[\frac {W}{Q}\]

or

W = QV

W = pE (cos θ₁ – cos θ₂)

Potential due to a continuous charge distribution:

\[V=\frac{1}{4\pi\varepsilon_0}\int\frac{\rho dV}{r}\]

Potential outside a uniformly charged spherical shell:

\[V=\frac{1}{4\pi\varepsilon_0}\frac{q}{r}\quad(r\geq R)\]

Potential inside a uniformly charged spherical shell:

\[V=\frac{1}{4\pi\varepsilon_0}\frac{q}{R}\quad(r<R)\]

E = -\[\frac {dV}{dl}\]

V = \[\frac {W}{q_0}\]

• Dimensions: [V] = [ML²T⁻³A⁻¹]
• SI unit is volt (V), where 1 V =1 J C⁻¹

^OR

\[V=\frac{W_{\infty\to P}}{q}\]

Unit: 1 volt=1 joule per coulomb (J/C)

\[V_A-V_B=\frac{W}{q_0}\]

OR

\[V_P-V_R=\frac{U_P-U_R}{q}\]

1 electron-volt = 1.6 × 10^-1 joule.

V = \[\frac{1}{4\pi\varepsilon_0}\frac{P}{r^2-l^2}\]

Far-field, r ≫ 2l: V = \[\frac{1}{4\pi\varepsilon_{0}}\frac{p}{r^{2}}\] volt.

V = \[\frac{1}{4\pi\varepsilon_{0}}\frac{p\cos\theta}{r^{2}}\] volt.

U = \[\frac{1}{4\pi\varepsilon_{0}}\frac{q_{1}q_{2}}{r}joule\]

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

C = 4 π ε_0​ a farad

U = \[\frac {1}{2}\] C V²

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

If there is vacuum (or air) between the plates, then K = 1

C₀ = \[\frac {ε_0 A}{d}\] farad

\[U=\frac{1}{2}\frac{Q^{2}}{C}=\frac{1}{2}CV^{2}joule.\]

• Zero Work: No work is done in moving a charge along an equipotential surface because the potential difference is zero.
• Relation with Electric Field: The electric field is always perpendicular to an equipotential surface; there is no electric field component along the surface.
• Spacing and Field Strength: Equipotential surfaces are closer where the electric field is strong and farther apart where the field is weak.
• Non-intersection: Equipotential surfaces never intersect, since that would imply two directions of the electric field at one point, which is impossible.

• Series combination: All capacitors connected in series carry the same charge Q, while the total potential difference is the sum of individual potential differences.
• Equivalent capacitance in series is given by​
  \[\begin{aligned}
  \frac{1}{C}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}
  \end{aligned}\]
  and is less than the smallest individual capacitance.
• In a series combination, the potential difference across each capacitor is inversely proportional to its capacitance, and the capacitor with the least capacitance has the highest voltage.
• Parallel combination: All capacitors connected in parallel have the same potential difference, while the charge distributes according to capacitance.
• Equivalent capacitance in parallel is given by​
  C = C₁ ​+ C₂​ + C₃​
  and is greater than any individual capacitance.

• An electric field produces dipoles in non-polar dielectrics and aligns them in polar dielectrics, resulting in a net dipole moment along the field.
• Polarisation causes bound charges to appear only on the surfaces of the dielectric slab; the interior remains electrically neutral.
• The polarisation charges create an electric field opposite to the applied field, reducing the field inside the dielectric.
• When a dielectric is inserted in an isolated capacitor, the electric field and potential difference decrease, while capacitance increases.
• A dielectric can withstand the electric field only up to a certain limit, beyond which electrical breakdown occurs.

• Capacitance is directly proportional to the area of the plates
  C ∝ A
  Increasing the effective overlapping area increases capacitance.
• Capacitance is inversely proportional to the distance between the plates
  C ∝ \[\frac {1}{d}\]​
  Reducing the separation between plates increases capacitance.
• Capacitance depends on the medium between the plates
  It increases when a dielectric is introduced and is directly proportional to the dielectric constant K:
  C ∝ K

• In metals, electric current is due to the drift of free electrons; positive ions remain fixed in the lattice and do not move.
• Valence electrons in the outermost orbit are loosely bound and can become free (conduction) electrons, especially at room temperature.
• When an external electric field is applied to a conductor, free electrons acquire a drift velocity opposite to the field, producing current.
• The electrical conductivity of a solid depends on the number of free electrons available for conduction.
• In dielectrics, an applied electric field causes electric polarisation; charges appear on the surface, but no charge flows through the material.