Class 12 Physics · Chapter 2 NotesElectrostatic Potential and Capacitance
Revise Class 12 Physics Chapter 2 with clear notes on electrostatic potential, equipotential surfaces, conductors, capacitance, dielectrics and energy storage.
This chapter extends the ideas of work, energy and electric fields to electrostatic situations. It begins by showing that the Coulomb force is conservative, which allows us to define electrostatic potential energy and then electrostatic potential as work done per unit charge in bringing a positive test charge from infinity. You will learn how to calculate potential due to point charges, dipoles and systems of charges, and how equipotential surfaces relate to the electric field. The chapter then explains the behaviour of conductors in electrostatic equilibrium, including electrostatic shielding. Finally, it introduces capacitance, the parallel plate capacitor, the effect of dielectrics, combinations of capacitors, and the energy stored in a capacitor and in an electric field. These ideas underpin later topics in current electricity and alternating current, and are widely used in electronic circuits and energy storage devices.
What you'll learn
1Define electrostatic potential and potential difference, and relate them to work done by an external force.
2Calculate the potential due to a point charge, an electric dipole and a system of charges using the superposition principle.
3Describe equipotential surfaces and explain the relation between electric field and potential.
4Determine the potential energy of a system of charges and of a dipole in an external field.
5State and apply the electrostatic properties of conductors, including electrostatic shielding.
6Define capacitance and calculate the capacitance of a parallel plate capacitor with and without a dielectric.
7Find the equivalent capacitance of capacitors in series and in parallel.
8Calculate the energy stored in a capacitor and the energy density of an electric field.
Chapter at a glance
01Electric Potential and Potential Difference
02Potential Due to Point Charges and Systems
03Equipotential Surfaces and Relation to Electric Field
04Capacitance, Dielectrics, and Energy Storage
Detailed chapter notes
01
Electrostatic Potential and Potential Difference
The Coulomb force between two stationary charges is conservative, so the work done in moving a charge between two points does not depend on the path taken. If an external force just balances the electric force and moves a charge q slowly from point R to point P, the work done by the external force is stored as potential energy. The potential energy difference between P and R is U_P – U_R = W_RP. Dividing this work by the charge q gives the potential difference V_P – V_R, which is independent of q. Choosing the potential to be zero at infinity, the potential at a point is the work done in bringing a unit positive charge from infinity to that point without acceleration. Only potential differences are physically significant, and the zero of potential can be chosen conveniently.
Potential differenceV_P – V_R = W_RP / q
Potential at a pointV = W_∞P / q, with V = 0 at infinity
Work done by an external force in moving a charge q from R to P is q(V_P – V_R)
02
Potential Due to Point Charges and Systems of Charges
For a point charge Q at the origin, the potential at a distance r is V = Q / (4πε₀r). This result holds for any sign of Q; for a negative charge, the potential is negative. Because electrostatic potential obeys the superposition principle, the potential at a point due to a system of charges is the algebraic sum of the potentials due to individual charges: V = (1/4πε₀) Σ (q_i / r_i). For a continuous charge distribution, the sum becomes an integral. For a uniformly charged spherical shell of radius R and total charge q, the potential outside the shell is V = q / (4πε₀r) for r ≥ R, and inside the shell it is constant and equal to q / (4πε₀R). An electric dipole, consisting of charges +q and –q separated by a distance 2a, produces a potential V = (1/4πε₀) (p·r̂ / r²) for r much greater than a, where p is the dipole moment. This potential falls off as 1/r², faster than the 1/r fall-off for a point charge, and it depends on the angle between p and r.
Point chargeV = Q / (4πε₀r)
System of chargesV = (1/4πε₀) Σ (q_i / r_i)
Charged spherical shellV = q / (4πε₀R) inside and on the surface; V = q / (4πε₀r) outside
Dipole (r >> a)V = (1/4πε₀) (p cosθ / r²)
03
Equipotential Surfaces and Relation to Electric Field
An equipotential surface is a surface on which the electric potential has the same value at every point. For a single point charge, the equipotential surfaces are concentric spheres centred on the charge. For a uniform electric field, they are planes perpendicular to the field. In general, the electric field at any point is normal to the equipotential surface through that point, because any tangential component would do work in moving a charge along the surface, contradicting the definition of an equipotential surface. The electric field points in the direction of steepest decrease of potential, and its magnitude is given by the change in potential per unit displacement perpendicular to the equipotential surface: E = –dV/dl. This relation shows that the electric field is the negative gradient of the potential.
Equipotential surfaceV is constant on the surface
Electric field is always perpendicular to the equipotential surface
E = –dV/dl, where dl is measured normal to the equipotential surface
04
Potential Energy of a System of Charges and in an External Field
The potential energy of a system of two point charges q₁ and q₂ separated by a distance r₁₂ is U = (1/4πε₀) (q₁q₂ / r₁₂). For like charges, U is positive; for unlike charges, U is negative. For three charges, the total potential energy is the sum of the energies of all pairs: U = (1/4πε₀) (q₁q₂/r₁₂ + q₁q₃/r₁₃ + q₂q₃/r₂₃). The potential energy of a single charge q in an external potential V(r) is qV(r). For a system of two charges in an external field, the total potential energy includes both the interaction with the external field and the mutual interaction between the charges. The potential energy of a dipole p in a uniform external electric field E is U = –p·E = –pE cosθ, where θ is the angle between p and E. This energy is minimum when the dipole is aligned with the field and maximum when it is anti-aligned.
Two chargesU = (1/4πε₀) (q₁q₂ / r₁₂)
Charge in external fieldU = qV(r)
Dipole in uniform fieldU = –p·E
05
Electrostatics of Conductors and Electrostatic Shielding
In electrostatic equilibrium, the electric field inside a conductor is zero. Any net charge resides only on the outer surface of the conductor. At the surface of a charged conductor, the electric field is normal to the surface and its magnitude is E = σ/ε₀, where σ is the surface charge density. The electrostatic potential is constant throughout the volume of the conductor and has the same value on its surface. If a conductor has a cavity with no charges inside, the electric field inside the cavity is zero, regardless of external fields or the charge on the conductor. This phenomenon is called electrostatic shielding, and it is used to protect sensitive instruments from external electric influences. Charges placed inside a cavity do produce fields outside, so shielding does not work in the reverse direction.
Inside a conductorE = 0
At the surfaceE = σ/ε₀, normal to the surface
Potential is constant throughout the conductor
Cavity with no chargesE = 0 inside (electrostatic shielding)
06
Capacitance, Dielectrics and Energy Storage
A capacitor is a system of two conductors separated by an insulator. When the conductors carry charges +Q and –Q and have a potential difference V, the capacitance is defined as C = Q/V. Capacitance depends only on the geometry of the conductors and the nature of the dielectric between them. For a parallel plate capacitor with vacuum between the plates, C = ε₀A/d, where A is the area of each plate and d is the separation. When a dielectric of dielectric constant K is inserted, the induced surface charges reduce the net electric field and hence the potential difference, so the capacitance increases to C = K C₀. The product ε = Kε₀ is the permittivity of the medium. Capacitors can be combined in series, where 1/C = 1/C₁ + 1/C₂ + ..., and in parallel, where C = C₁ + C₂ + ... . The energy stored in a capacitor is U = (1/2) CV² = (1/2) QV = Q²/(2C). This energy can be thought of as stored in the electric field, with energy density u = (1/2) ε₀E².
CapacitanceC = Q/V
Parallel plate capacitor (vacuum)C = ε₀A/d
With dielectricC = K C₀, where K is the dielectric constant
The electric potential at a point is defined as the work done per unit charge in bringing a test charge from infinity to that point. What is the SI unit of electric potential?
AVolt (V)
BJoule (J)
CCoulomb (C)
DNewton (N)
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Answer: (A) Volt (V)
Electric potential is work done per unit charge, so its SI unit is Joule per Coulomb, which is defined as Volt (V).
Question 02
The electric potential at a point due to a point charge q at distance r is given by:
AV = kq/r²
BV = kq/r
CV = kq²/r
DV = kr/q
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Answer: (B) V = kq/r
The electric potential due to a point charge is V = kq/r, where k is Coulomb's constant. This is a fundamental formula from Coulomb's law applied to potential.
Question 03
What is an equipotential surface?
AA surface where all points have the same electric potential
BA surface where electric field is maximum
CA surface perpendicular to the direction of motion
DA surface where electric force is zero
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Answer: (A) A surface where all points have the same electric potential
An equipotential surface is defined as a surface on which all points have the same electric potential value.
Question 04
The capacitance of a parallel plate capacitor depends on which of the following factors?
AArea of plates and distance between them only
BCharge stored and voltage applied only
CMaterial between plates, plate area, and separation distance
DVoltage applied and time of charging
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Answer: (C) Material between plates, plate area, and separation distance
Capacitance C = ε₀εᵣA/d depends on the permittivity of the dielectric material (εᵣ), area of plates (A), and separation distance (d). Charge and voltage are related through capacitance but do not determine it.
Question 05
Two points A and B in an electric field have potentials 100 V and 50 V respectively. What is the potential difference between A and B?
A50 V
B150 V
C−50 V
D100 V
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Answer: (A) 50 V
Potential difference VAB = VA − VB = 100 − 50 = 50 V.
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Q1. Define electrostatic potential at a point. How is it related to potential energy of a charge q placed at that point?
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Model answer
Electrostatic potential at a point is the work done in bringing a unit positive test charge from infinity to that point without acceleration. For a charge q placed at a point where potential is V, the potential energy is U = qV.
Sample question3 marks
Q2. Define electrostatic potential at a point. Derive an expression for the potential due to a point charge Q at a distance r from it.
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Model answer
Electrostatic potential at a point is the work done by an external force in bringing a unit positive charge from infinity to that point without acceleration. For a point charge Q at origin, potential at distance r is V = (1/4πε₀) Q/r, derived by integrating work done against Coulomb force along radial path from infinity to r.
Sample question3 marks
Q3. Define an equipotential surface. Draw the equipotential surfaces for a uniform electric field along the x-axis.
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Model answer
An equipotential surface is a surface with a constant value of potential at all points on the surface. For a uniform electric field along the x-axis, the equipotential surfaces are planes perpendicular to the x-axis, i.e., planes parallel to the y-z plane.
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Q4. Define capacitance of a capacitor. On what factors does the capacitance of a parallel plate capacitor depend?
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Model answer
Capacitance C of a capacitor is defined as the ratio of the charge Q on one conductor to the potential difference V between the conductors: C = Q/V. For a parallel plate capacitor with vacuum between plates, capacitance depends on the area A of each plate, the separation d between them, and the permittivity of free space ε₀, as given by C = ε₀A/d.
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Q5. State the expression for electrostatic potential due to a point charge Q at a distance r. How does the potential vary with distance?
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Model answer
The potential due to a point charge Q at distance r is V = (1/4πε₀) Q/r. The potential varies inversely with distance (V ∝ 1/r). For a positive charge, potential is positive; for a negative charge, potential is negative.
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Electrostatic potential at a point is the work done by an external force in bringing a unit positive charge from infinity to that point without acceleration. It is a scalar quantity, measured in volts (V), and is given by V = W/q. Only potential differences are physically significant.
What is the difference between electric potential and electric potential energy?
Electric potential is the work done per unit charge in bringing a positive test charge from infinity to a point, and is independent of the test charge. Electric potential energy is the total work done in bringing a specific charge q to that point, and equals qV. Potential is a property of the field; potential energy depends on the charge placed in the field.
Why is the electric field inside a conductor zero?
In a conductor, free charges move until they reach electrostatic equilibrium. If an electric field existed inside, free charges would experience a force and drift, which is not a static situation. At equilibrium, the charges redistribute so that the net electric field inside the conductor is zero.
What is an equipotential surface?
An equipotential surface is a surface on which the electric potential is constant at every point. The electric field is always perpendicular to such a surface. For a point charge, equipotential surfaces are concentric spheres; for a uniform field, they are planes perpendicular to the field.
How does a dielectric affect the capacitance of a capacitor?
A dielectric becomes polarised in an electric field, producing induced charges that reduce the net electric field and hence the potential difference between the plates. Since C = Q/V, a lower V for the same Q means higher capacitance. The capacitance becomes C = K C₀, where K is the dielectric constant of the material.
What is the energy stored in a capacitor?
The energy stored in a capacitor of capacitance C with charge Q and potential difference V is U = (1/2) CV² = (1/2) QV = Q²/(2C). This energy is stored in the electric field between the plates, with energy density u = (1/2) ε₀E².