Electrostatic Potential and Capacitance Class 12 Physics Notes

Electrostatics is one of the most important chapters in Class 12 Physics. After learning about Electric Charges and Fields, the next crucial topic is Electrostatic Potential and Capacitance. This chapter deals with concepts like electric potential, potential difference, energy in electric fields, capacitors, and their applications.

A strong understanding of these topics helps in solving numerical problems, conceptual questions, and applications in real-life situations like the working of capacitors in electronic circuits.

Table of Contents

Electrostatic Potential and Capacitance Class 12 Physics Notes | Detailed Explanation

Electrostatic Potential

What is Electrostatic Potential?

The electrostatic potential at a point is defined as the amount of work done per unit positive charge in bringing it from infinity to that point in the electric field without any acceleration.

Mathematically:

$\frac{W}{q}$

Where,

$V$ = Electrostatic potential
$W$ = Work done in bringing the charge
$q$ = Magnitude of charge

Electrostatic Potential due to a Point Charge

For a point charge $q$ at a distance $r$ :

$\frac{1}{4\pi \epsilon_0} \cdot \frac{q}{r}$

Unit: Volt (V)
Dimension: [ML²T⁻³A⁻¹]

Key Points:

Potential decreases as distance increases.
Potential is a scalar quantity.

Electrostatic Potential due to Multiple Point Charges

If $q1,q2,q3…qnq_1, q_2, q_3 \dots q_n$ are charges at distances $r1,r2,r3…rnr_1, r_2, r_3 \dots r_n$ , then

$\frac{1}{4\pi \epsilon_0} \left( \frac{q_1}{r_1} + \frac{q_2}{r_2} + \cdots + \frac{q_n}{r_n} \right)$

This is based on the principle of superposition.

Potential due to Electric Dipole

For an electric dipole with charges $+ q$ and $- q$ separated by distance $2 a$ :

$\frac{1}{4\pi \epsilon_0} \cdot \frac{p \cos \theta}{r^2}$

where $\times 2a$ is the dipole moment.

Equipotential Surfaces

Equipotential surfaces are surfaces where the potential remains constant.
No work is done in moving a charge along an equipotential surface.
Electric field is always perpendicular to the equipotential surface.

Relation between Electric Field and Potential

The electric field $E$ is the negative gradient of the potential:

$-\frac{dV}{dr}$

This shows that electric field points in the direction of maximum decrease of potential.

Electrostatic Potential Energy

The potential energy of a system of charges is the work done in assembling the charges from infinity to the given positions.

For two charges $q_1$ and $q_2$ separated by distance $r$ :

$\frac{1}{4\pi \epsilon_0} \cdot \frac{q_1 q_2}{r}$

Capacitance

Definition

Capacitance is the ability of a system to store charge per unit potential difference.

Mathematically:

$\frac{Q}{V}$

where

$C$ = Capacitance
$Q$ = Charge stored
$V$ = Potential difference

Unit: Farad (F)

Capacitance of an Isolated Spherical Conductor

For a sphere of radius $R$ :

$4\pi \epsilon_0 R$

Capacitance of a Parallel Plate Capacitor

For two plates of area $A$ separated by distance $d$ :

$\frac{\epsilon_0 A}{d}$

If a dielectric of constant $K$ is introduced:

$\frac{K \epsilon_0 A}{d}$

Combination of Capacitors

Series Combination

$1Ceq=1C1+1C2+⋯\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots$
Parallel Combination

$Ceq=C1+C2+⋯C_{eq} = C_1 + C_2 + \cdots$

Energy Stored in a Capacitor

The energy stored is given by:

$\frac{1}{2} C V^2$

Dielectrics and Polarization

A dielectric is a non-conducting material placed between the plates of a capacitor.
It increases the capacitance by reducing the electric field.

Applications of Capacitors

Used in tuning radios and TVs
Used in flash cameras
Used in electronic circuits for energy storage

Objective Questions

Q1. The SI unit of electric potential is:

(a) Joule (b) Volt (c) Coulomb (d) Newton
Answer: (b) Volt

Q2. Potential due to a point charge varies as:

(a) $1/ r$ (b) $1/r^2$ (c) $r$ (d) $r^2$
Answer: (a) $1/ r$

Q3. Equipotential surfaces are always:

(a) Parallel to electric field lines
(b) Perpendicular to electric field lines
(c) Inclined at any angle
(d) None of these
Answer: (b) Perpendicular to electric field lines

Q4. The work done in moving a charge along an equipotential surface is:

(a) Maximum (b) Minimum (c) Zero (d) Infinite
Answer: (c) Zero

Q5. The formula for potential due to a point charge is:

(a) $V = q / r$ (b) $1/(4\pi \epsilon_0) \cdot q/r$ (c) $V = q/r^2$ (d) $V = q^2/r$
Answer: (b) $\frac{1}{4\pi \epsilon_0} \cdot \frac{q}{r}$

Q6. Capacitance of a parallel plate capacitor with dielectric constant K is:

(a) $\epsilon_0 A/d$
(b) $\epsilon_0 A/d$
(c) $A/K\epsilon_0 d$
(d) $K^2 \epsilon_0 A/d$
Answer: (b) $\epsilon_0 A/d$

Q7. Energy stored in a capacitor is given by:

(a) $U = CV^2$ (b) $U = C / V$ (c) $U = 1/2 CV^2$ (d) $U = V^2/C$
Answer: (c) $\frac{1}{2}CV^2$

Q8. Capacitance depends on:

(a) Area of plates (b) Distance between plates (c) Dielectric medium (d) All of these
Answer: (d) All of these

Q9. For capacitors in series, the equivalent capacitance:

(a) Increases (b) Decreases (c) Remains same (d) Becomes infinite
Answer: (b) Decreases

Q10. For capacitors in parallel, the equivalent capacitance:

(a) Increases (b) Decreases (c) Remains same (d) None of these
Answer: (a) Increases

Q11. SI unit of capacitance is:

(a) Farad (b) Volt (c) Coulomb (d) Newton
Answer: (a) Farad

Q12. A capacitor is charged and then disconnected from the battery. On inserting a dielectric, the potential difference:

(a) Increases (b) Decreases (c) Remains same (d) Becomes zero
Answer: (b) Decreases

Q13. The electric field is related to potential as:

(a) $E = - d V / d r$ (b) $E = d V / d r$ (c) $\times dV$ (d) $E = 1/ V$
Answer: (a) $E = - d V / d r$

Q14. Potential at infinity is considered as:

(a) Zero (b) One (c) Infinite (d) Negative
Answer: (a) Zero

Q15. The potential difference between two points is the work done in moving:

(a) Unit negative charge
(b) Unit positive charge
(c) Any charge
(d) Both positive and negative charges
Answer: (b) Unit positive charge

Q16. The presence of a dielectric in a capacitor:

(a) Increases capacitance
(b) Decreases capacitance
(c) Has no effect
(d) Makes it zero
Answer: (a) Increases capacitance

Q17. Energy density in a capacitor is given by:

(a) $\frac{1}{2}\epsilon_0 E^2$
(b) $\frac{1}{2}CV^2$
(c) $U = CV^2$
(d) $U = V^2/C$
Answer: (a) $\frac{1}{2}\epsilon_0 E^2$

Q18. In a parallel plate capacitor, if the plate area is doubled, capacitance:

(a) Doubles (b) Halves (c) Remains same (d) Becomes zero
Answer: (a) Doubles

Q19. Work done in charging a capacitor is stored as:

(a) Magnetic energy (b) Thermal energy (c) Electrostatic energy (d) Chemical energy
Answer: (c) Electrostatic energy

Q20. A capacitor blocks:

(a) DC (b) AC (c) Both AC and DC (d) None
Answer: (a) DC

Short Answer Questions-Electrostatic Potential and Capacitance Class 12 Physics Notes

Q1. Define electrostatic potential.

Answer: Electrostatic potential at a point is the amount of work done in bringing a unit positive charge from infinity to that point in the electric field without acceleration. Its SI unit is Volt (V).

Q2. Write the formula for potential due to a point charge.

Answer: The potential at a distance $r$ from a point charge $q$ is given by $\frac{1}{4\pi \epsilon_0} \cdot \frac{q}{r}$ .

Q3. What are equipotential surfaces?

Answer: Equipotential surfaces are surfaces where the electric potential remains constant at all points. The electric field is always perpendicular to these surfaces, and no work is done in moving a charge along them.

Q4. State the relation between electric field and potential.

Answer: The electric field is the negative gradient of the potential, mathematically expressed as $-\frac{dV}{dr}$ .

Q5. What is a capacitor?

Answer: A capacitor is a device used to store electric charge and energy in the form of an electric field between two conductors separated by a dielectric medium.

Q6. What is the unit of capacitance?

Answer: The SI unit of capacitance is Farad (F), defined as the capacity to store one coulomb of charge at one volt potential difference.

Q7. Write the expression for energy stored in a capacitor.

Answer: The energy stored in a capacitor is given by $\frac{1}{2} C V^2$ , where $C$ is capacitance and $V$ is the potential difference.

Q8. Define dielectric constant.

Answer: The dielectric constant $K$ is the ratio of the capacitance with a dielectric material to the capacitance with vacuum between the plates: $\frac{C_{with}}{C_{without}}$ .

Q9. What is the effect of dielectric on capacitance?

Answer: The presence of a dielectric increases the capacitance because it reduces the effective electric field, allowing the capacitor to store more charge at the same voltage.

Q10. Write one application of capacitors.

Answer: Capacitors are widely used in electronic circuits for energy storage, filtering signals, tuning radios, and in devices like camera flashes.

Long Answer Questions-Electrostatic Potential and Capacitance Class 12 Physics Notes

Q1. Derive the expression for electrostatic potential due to a point charge.

Answer: Consider a point charge $q$ at point $O$ . The work done in bringing a small positive test charge $q_0$ from infinity to a point $P$ at a distance $r$ is

$\int_{\infty}^r \frac{q q_0}{4 \pi \epsilon_0 r^2} dr$

This gives the potential at point $P$ as

$\frac{W}{q_0} = \frac{1}{4 \pi \epsilon_0} \cdot \frac{q}{r}.$

Q2. Explain the concept of equipotential surfaces with examples.

Answer: Equipotential surfaces are imaginary surfaces where the potential is the same everywhere. For a point charge, equipotential surfaces are concentric spheres with the charge at the center. For a uniform electric field, they are planes perpendicular to the field lines. No work is done when moving a charge along these surfaces.

Q3. Derive the relation between electric field and potential.

Answer: The electric field is the rate of change of potential with distance. Considering a small displacement $d r$ in the field, the change in potential $d V$ is

$-\frac{dV}{dr}.$

The negative sign indicates that the electric field points in the direction of decreasing potential.

Q4. Explain the principle and working of a parallel plate capacitor.

Answer: A parallel plate capacitor consists of two large plates separated by a small distance. When connected to a battery, the plates acquire equal and opposite charges. The potential difference between the plates is directly proportional to the charge, and capacitance is given by $\frac{\epsilon_0 A}{d}$ , where $A$ is plate area and $d$ is distance between plates.

Q5. Derive the expression for capacitance of a parallel plate capacitor.

Answer: For plates of area $A$ separated by distance $d$ , the electric field is $\frac{\sigma}{\epsilon_0} = \frac{Q}{\epsilon_0 A}$ . The potential difference is $\frac{Qd}{\epsilon_0 A}$ . So, capacitance is $\frac{Q}{V} = \frac{\epsilon_0 A}{d}$ .

Q6. Explain the effect of dielectric on capacitance.

Answer: When a dielectric with constant $K$ is inserted, the electric field reduces by factor $K$ . This increases the charge stored at the same voltage, so the new capacitance becomes $C = K C_0$ , where $C_0$ is the original capacitance.

Q7. Derive the formula for energy stored in a capacitor.

Answer: The work done in charging a capacitor from 0 to $Q$ is

$\int_0^Q \frac{q}{C} dq = \frac{Q^2}{2C}.$

Since $Q = C V$ , energy stored becomes $\frac{1}{2}CV^2$ .

Q8. Explain series and parallel combination of capacitors.

Answer: In series, $1Ceq=1C1+1C2+…\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots$ , and voltage divides among capacitors. In parallel, $Ceq=C1+C2+…C_{eq} = C_1 + C_2 + \dots$ , and voltage remains same but charge divides.

Q9. Write the applications of capacitors in daily life.

Answer: Capacitors are used in tuning circuits of radios and TVs, as filters in power supplies, for storing energy in camera flashes, in motor starters, and in coupling and decoupling signals in electronics.

Q10. Explain energy density in an electric field.

Answer: The energy stored per unit volume in a parallel plate capacitor is given by $\frac{1}{2} \epsilon_0 E^2$ . This is called energy density, showing how much energy is stored in the electric field between the plates.

FAQs on Electrostatic Potential and Capacitance

Q1. What is the SI unit of potential?

The SI unit is Volt (V), where $\, V = 1 \, J/C$ .

Q2. What happens if we connect capacitors in series?

The equivalent capacitance decreases, and voltage divides among capacitors.

Q3. What is the role of a dielectric?

It increases capacitance by reducing the effective electric field.

Q4. Why is energy stored in capacitors?

Because work is done in charging the capacitor, which is stored as electrostatic energy.

Q5. Can electric field exist on an equipotential surface?

Yes, but it is always perpendicular to the surface.

Conclusion

The chapter Electrostatic Potential and Capacitance is fundamental in understanding how electric charges interact in terms of potential energy and how energy can be stored using capacitors. We explored key concepts such as electric potential, potential energy, equipotential surfaces, relation between electric field and potential, and the working principles of capacitors.

Capacitors play a crucial role in electronics and electrical engineering because of their ability to store energy efficiently and release it when required. The mathematical derivations, formulas, and applications discussed in this chapter form the foundation for solving complex problems in electrostatics and real-world circuits.

By mastering these concepts, students can confidently tackle both numerical and theoretical questions in board exams as well as competitive exams like JEE and NEET.

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Class 12 Physics: Electric Charges and Fields Notes | Complete Guide with FAQs