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Syllabus

Overview: Electric Potential

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Estimated time: 24 minutes
CBSE: Class 12
CISCE: Class 12

Definition: Electric Potential at a Point

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.

CISCE: Class 12

Definition: Electron-volt

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.

CISCE: Class 12

Definition: Potential Gradient

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

CBSE: Class 12
CISCE: Class 12

Definition: Electric Potential Energy

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.

CBSE: Class 12
CISCE: Class 12

Definition: Equipotential Surface

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.

CBSE: Class 12
CISCE: Class 12

Definition: Electric Dipole

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.

CBSE: Class 12
CISCE: Class 12

Formula: Electric Potential

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)

CBSE: Class 12
CISCE: Class 12

Formula: Potential Difference between Two Points

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

OR

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

CISCE: Class 12

Formula: Electron-volt

1 electron-volt = 1.6 × 10-1 joule.

CBSE: Class 12
CISCE: Class 12

Formula: Electric Potential Due to a Point Charge

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

CBSE: Class 12
CISCE: Class 12

Formula: Electric Potential Energy of Two Point Charges

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

CBSE: Class 12
CISCE: Class 12

Formula: Dipole Potential on Axial Line

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.

CBSE: Class 12
CISCE: Class 12

Formula: Potential at any Point

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

CISCE: Class 12

Formula: Work Done in Rotating an Electric Dipole

W = pE (cos θ1 – cos θ2)

CISCE: Class 12

Key Points: Properties of Equipotential Surfaces

  • 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.
CISCE: Class 12

Key Points: Electric Potential Energy of an Electric Dipole at Electrostatic Field

  • The potential energy of an electric dipole in a uniform electric field is
    U = − p ⋅ E = − pE cos ⁡θ
  • When the dipole is parallel to the field (θ = 0∘),
    U = − pE
    This is the minimum potential energy (stable equilibrium).
  • When the dipole is perpendicular to the field (θ = 90∘),
    U = 0
  • When the dipole is anti-parallel to the field (θ = 180∘),
    U = +pE
    This corresponds to an unstable equilibrium.
  • Work done in rotating the dipole through an angle θ\thetaθ is W = pE(1 − cos⁡θ) and this work equals the increase in potential energy.
CBSE: Class 12

Definition: Electric Dipole Moment

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\[\vec a\])

CBSE: Class 12

Formula: Potential Due to a System of Charges

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:

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

CBSE: Class 12

Formula: Relation between Electric Field and Potential

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

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