Revision: Class 12 >> Electrostatic Potential and Capacitance NEET (UG) Electrostatic Potential and Capacitance

"Potential difference is the work done to move a unit charge from one point to another in an electric field."

OR

The difference in electric potential between two points B and A, given by ΔV = V_B − V_A = \[\frac {W_AB}{q_0}\]​​, is called potential difference.

The work done against the electrostatic forces to achieve a certain configuration of charges in a given system is called electrostatic potential energy.

The work done by an external force in bringing a unit positive charge from infinity to that point is called electric potential at that point.

Electrostatic potential at a point is the work done by an external agent in bringing a unit positive test charge slowly from infinity to that point without acceleration.

The potential difference between two points P and R is the work done by an external force in moving a unit positive test charge from one point to the other.

The work done by an external agent in bringing a unit positive test charge slowly from infinity to a point in an electric field, against the electrostatic force, is called the electric potential at that point.

An electric dipole is a pair of equal and opposite charges separated by a small distance.

If the charges are separated by a distance 2a2a, then 2a2a is called the dipole length.

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

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

OR

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

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.

OR

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

The surface at which electric potential is the same at each point is called an equipotential surface.

The electrostatic force is a conservative force — the work done in moving a charge between two points is independent of the path and depends only on the initial and final positions. This is why the potential energy is well-defined.

The electrostatic potential energy of a system of charges is defined as the work done by an external agent in assembling the charges at their respective positions, bringing each charge from infinity, without any kinetic energy being imparted.

• Symbol: U
• SI Unit: Joule (J)
• Nature: Scalar quantity
• Reference: U = 0 when all charges are at infinity

A conductor in electrostatic equilibrium is an equipotential body, meaning all points on it are at the same electric potential.

Surface charge density is the charge per unit area on the surface of a conductor and is denoted by \[\sigma\].

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 condition in which charges in a conductor are at rest, and no further motion of charges occurs.

Non-conducting substances which cannot transmit electric charge through them are called dielectrics.

The molecule in which the centres of positive and negative charges are separated even when there is no external field, and which has a permanent dipole moment, is called a polar molecule. (e.g. HCl, H₂O, alcohol, NH₃)

The molecule in which the centres of positive and negative charges coincide and which has no permanent dipole moment in its normal state is called a non-polar molecule. (e.g. O₂, H₂, N₂, CO₂, benzene, methane)

A dielectric that has a permanent electric dipole moment even if the external electric field is absent is called a polar dielectric.

A dielectric in which every molecule has zero dipole moment in its normal state is called a non-polar dielectric.

Alignment of dipole moments (permanent or induced) in the direction of an applied electric field is called polarisation.

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 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.

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

The process by which the molecules of a dielectric develop induced dipole moments when placed in an external electric field. The induced dipole moments align opposite to the field, creating an opposing induced field E_P.

The maximum electric field a dielectric can withstand before it breaks down (becomes conducting). Measured in V/m. Example: Air ≈ 3 × 10⁶ V/m.

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 dielectric is a non-conducting (insulating) material in which charges are bound to their atoms/molecules and cannot move freely. When placed in an external electric field, the molecules of the dielectric get polarised — they develop induced dipole moments that partially oppose the external field.

The capacitance of a single capacitor that stores the same charge at the same voltage as the entire combination is called the equivalent capacitance of the combination.

The work done per unit charge in moving a charge from one plate of a capacitor to the other is called the potential difference between the plates.

Potential difference (V) between two points = Work done (W)/Charge (Q)
V = \[\frac {W}{Q}\]

The SI unit of electric potential difference is volt (V)

1 volt = \[\frac{1\mathrm{~joule}}{1\mathrm{~coulomb}}\] = 1 J C^-1

If the potential energies at points P and R are U_P and U_R​, then

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

If the work done in bringing charge q from infinity to point P is W, then

V_P ​= \[\frac {W​}{q}\]

\[V(r)=\frac{1}{4\pi\varepsilon_0K}\frac{q}{r}\]

• V(r) = electric potential at distance rr from the charge
• q = source charge
• ε₀ = permittivity of free space
• K = dielectric constant of medium
• Reference is taken such that V(∞) = 0.

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

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

Potential due to System of Charges:

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

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 = \[\frac {1}{4πε_0}\] ⋅ \[\frac {q}{r}\]

The electric dipole moment is:

\[\vec{p}=q(2a)\hat{p}\]

Its direction is from the negative charge to the positive charge.

\[V=\frac{1}{4\pi\varepsilon_{0}}\cdot\frac{p\cos\theta}{r^{2}}=\frac{1}{4\pi\varepsilon_{0}}\cdot\frac{\vec{p}\cdot\vec{r}}{r^{3}}(r>>a)\]

When a charge q₀​ is moved from point A to point B on the same equipotential surface:

Since V_A = V_B​ on the surface:

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

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

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

where

• σ = surface charge density
• \[\hat n\] = outward normal unit vector
• \[\varepsilon_0\] = permittivity of free space.

Magnitude form:

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

Vector form:

$$\vec{E} = \frac{\sigma}{\varepsilon_0}\hat{n}$$

Defined as dipole moment per unit volume:

\[P=\frac{\text{dipole moment}}{\mathrm{volume}}=np\]

C = Q/V

C = 4πkε₀ · [\[\frac {ab}{(b − a)}\]]

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

For two plates separated by distance d:

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

With a dielectric medium:

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

\[{C_P=C_1+C_2+C_3+\cdots}\]

For n identical capacitors of capacitance C each: C_P = nC

Physical Insight: Adding capacitors in parallel is like adding more storage tanks — the total storage capacity simply increases.

\[{\frac{1}{C_S}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}+\cdots}\]

For n identical capacitors of capacitance C each: C_S = \[\frac {C}{n}\]

For two capacitors in series, the voltage across each is:

\[V_1=\frac{C_2}{C_1+C_2}\cdot V\]

\[V_2=\frac{C_1}{C_1+C_2}\cdot V\]

Physical Insight: The smaller the capacitor, the larger the voltage drop across it in a series combination. This is why identical series capacitors share voltage equally.

W = \[\frac {1}{2}\]qV

OR

U = \[\frac {Q^2}{2C}\] ​= \[\frac {1​}{2}\]QV = \[\frac {1}{2}\]​CV²

SI unit: Joule (J)

• Electric potential at a point is the work done per unit positive test charge in bringing it slowly from infinity to that point, against the electric field.
• For a point charge q in air/vacuum:
  V(r) = \[\frac{1}{4\pi\varepsilon_0}\frac{q}{r}\]
• In a medium of dielectric constant K:
  V(r) = \[\frac{1}{4\pi\varepsilon_0K}\frac{q}{r}\]
• Positive charge produces positive potential; negative charge produces negative potential.
• Potential due to a point charge is spherically symmetric and depends only on distance r.
• Distance dependence:
  F ∝ 1/r², E ∝ 1/r², V ∝ 1/r.
• The potential at infinity is taken as zero; only potential differences are physically significant.
• The electrostatic field is conservative, so the work done in moving a charge between two points is path independent.

Capacitors in Series:

Equivalent capacitance: \[\frac{1}{C_s}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}+\cdots\]

• Same voltage (V) across all capacitors
• Charge divides
• The equivalent capacitance is greater than the largest capacitor

Capacitors in Parallel:

\[C_p=C_1+C_2+C_3+\cdots\]

• Same voltage (V) across all capacitors
• Charge divides
• The equivalent capacitance is greater than the largest capacitor