Electrostatic Potential And Capacitance: Complete One-Shot Revision For CUET 2027

Electrostatic Potential and Capacitance is an important chapter in Class 12 Physics and CUET 2027 preparation. Understanding the key concepts and formulas can help you solve numerical questions and prepare for the exam with confidence.

This chapter covers electric potential, potential difference, the relation between electric field and potential, equipotential surfaces, electric dipoles, dielectrics, and capacitance. It also includes important topics such as capacitor combinations, energy stored in capacitors, and charge redistribution.

Electric Potential

Electric potential helps us understand the work required to bring a charge to a particular point in an electric field. It is measured relative to a reference point, usually infinity.

Definition And Units

When a charge is brought from infinity to a point P, work is done against the electric force. Electric potential at point P is defined as the work done per unit positive charge:

V = W / q0

Here, W is the work done and q0 is the test charge.

Electric potential is a scalar quantity.

• SI unit: joule per coulomb, or volt

• CGS unit: erg per coulomb

The electric potential around a charged object depends on the sign of its charge. A positive charge creates positive potential, whereas a negative charge creates negative potential. When two bodies are connected, charge can move between them due to their potential difference.

Relation Between Electric Field And Potential

Electric field and electric potential are related by:

E = -dV/dr

The differential form is:

dV = -E · dr

Therefore:

V = - integral of E · dr

The relationship between electric field and potential can be used in two ways:

• When the electric field is known, integration helps calculate the potential difference.

• When the potential function is known, differentiation gives the electric field.

• The negative sign indicates that electric potential falls along the direction of the electric field.

Therefore, the electric field is directed from a region of higher potential towards a region of lower potential.

Dimensional Formula Of Electric Potential

Electric potential is work divided by charge.

• Work has dimensions: ML2T-2

• Charge has dimensions: AT

Therefore, the dimensional formula of potential is:

[V] = ML2T-3A-1

Dimensional formulas can be asked from any Physics chapter and should be practised regularly.

Potential Difference

The potential difference between points A and B is equal to the external work required per unit positive charge to move it from A to B slowly.

VB - VA = WA to B / q0

Potential And Potential Difference

Electric potential describes the potential at a single point, while potential difference measures the change in potential between two points. Both are measured in volts, but they differ in their reference points and how they are calculated.

The relation between work and potential difference is:

W = q0 Delta V

In an electrostatic field, the potential difference between two points remains the same regardless of the path taken between them, as the field is conservative.

Potential Due To Charges

Electric potential can be calculated for a single point charge or a group of charges. Since electric potential is a scalar quantity, the total potential at a point is found by adding the contributions from all charges algebraically.

Point Charge

At a distance (r) from a point charge (q), the electric potential is calculated using the following formula:

V = kq/r

In a medium of dielectric constant K:

V = kq/(Kr)

Potential varies inversely with distance:

V is proportional to 1/r

Therefore, potential decreases as distance increases. The potential-distance graph is a rectangular hyperbola.

System Of Charges

When multiple charges are present, the net electric potential at a point is obtained by adding the potential contributions of all charges, including their positive or negative signs.

V = k(q1/r1 + q2/r2 + q3/r3 + ...)

Since potential is scalar:

• Positive charges contribute positive potential.

• Negative charges contribute negative potential.

• Potentials are added algebraically.

Potential Due To An Electric Dipole

The electric potential produced by a dipole at a point depends on its distance (r) from the dipole's centre and the angle (\theta) between the dipole moment and the position vector.

V = kp cos(theta)/r2

Here, p is the dipole moment.

Dipole moment is:

p = charge × separation between charges

The electric dipole moment is a vector quantity that points from the negative charge towards the positive charge.

At any point on the perpendicular bisector, or equatorial line, of a dipole:

V = 0

At any point on the perpendicular bisector of a dipole, the potentials due to its positive and negative charges cancel each other, making the net electric potential zero.

Potential Due To A Charged Conducting Sphere

For a charged conducting sphere of radius R:

Outside The Sphere

At distance r from the centre:

V = kQ/r

Potential decreases with distance outside the sphere.

At The Surface

V = kQ/R

Inside The Sphere

At every internal point, including the centre:

V = kQ/R

Therefore:

• Potential is constant inside the sphere.

• Surface potential and internal potential are equal.

• Potential decreases only outside the sphere.

For a charged conducting sphere, the electric potential remains constant throughout its interior. Therefore, the potential at the centre is equal to the potential at the surface.

Equipotential Surfaces

An equipotential surface is one where the electric potential has an identical value at every point.

Important properties are:

1. Electric field lines meet an equipotential surface at right angles, meaning the field is perpendicular to the surface at every point.

2. Moving a charge from one point to another along an equipotential surface requires no work because the potential difference between those points is zero.

3. Equipotential surfaces placed close together indicate a stronger electric field.

4. When equipotential surfaces are farther apart, the electric field is weaker.

5. Two distinct equipotential surfaces cannot cross each other because each point can have only one electric potential value.

Since the potential difference along an equipotential surface is zero:

W = q Delta V = 0

Electric Potential Energy

For two point charges q1 and q2 separated by distance r:

U = kq1q2/r

Like charges repel, while unlike charges attract.

For a dipole of moment p in a uniform electric field E:

U = -pE cos(theta)

In vector form:

U = -p dot E

The work done in rotating a dipole from theta1 to theta2 is:

W = pE[cos(theta1) - cos(theta2)]

For rotation from stable equilibrium:

W = pE(1 - cos(theta))

Stable equilibrium occurs when the dipole moment and electric field point in the same direction. This is an important examination concept.

Conductors, Insulators And Dielectrics

Materials behave differently in an electric field depending on how easily their charges can move. Conductors allow charges to move freely, while insulators restrict their movement. Dielectrics are insulating materials that become polarised when placed in an electric field.

Conductors And Insulators

Electrons in outer orbits experience weaker nuclear attraction and may become free after receiving energy. Inner-orbit electrons remain strongly bound.

In metals:

• Electrons are free charge carriers.

• Positive ions fixed in the lattice are bound charge carriers.

Dielectrics And Polarization

A dielectric is an insulating material in which positive and negative charges remain bound within molecules. When placed in an external electric field:

• Positive charges shift along the field.

• Negative charges shift opposite to the field.

• Opposite induced charges appear on the two surfaces.

• An internal field is produced opposite to the external field.

This alignment of charges is called polarisation.

Types Of Dielectrics

Dielectrics are insulating materials that become polarised when placed in an external electric field. They are classified into two types based on the positions of their positive and negative charge centres.

Polarization density is the induced dipole moment per unit volume:

Polarization density = induced dipole moment / volume

It is directly proportional to electric field:

P is proportional to E

Dielectric Strength

Dielectric strength is the maximum electric field a material can withstand before breakdown.

For air:

Dielectric strength = 3 × 10^6 V/m

Beyond this value, ionisation and dielectric breakdown may occur.

For a dielectric, the net internal electric field is:

E net = E/K

Capacitance

Capacitance is the ability to store charge. It is defined as:

C = Q/V

Therefore:

Q = CV

Capacitance depends on the arrangement and size of conductors and on the dielectric medium. It does not depend directly on charge or potential.

For an isolated spherical conductor:

C = 4 pi epsilon0 r

With a dielectric medium:

C = 4 pi epsilon0 K r

Combination Of Capacitors

Series Combination

1/Ceq = 1/C1 + 1/C2 + 1/C3

In series:

• Charge on each capacitor is the same.

• Potential difference is divided.

• Equivalent capacitance is less than the smallest individual capacitance.

Parallel Combination

Ceq = C1 + C2 + C3

In parallel:

• Potential difference across each capacitor is the same.

• Charge is divided.

• Equivalent capacitance is greater than the largest individual capacitance.

Energy Stored In A Capacitor

Energy stored in a capacitor is:

U = 1/2 CV2

Equivalent forms are:

U = 1/2 Q2/C

U = 1/2 QV

Energy density is:

u = 1/2 epsilon E2

Its SI unit is joule per cubic metre.

The battery supplies twice the energy finally stored:

Work done by battery = 2U

The remaining energy is lost during charging.

Charge Redistribution Between Conductors

When two conductors at different potentials are connected, charge flows from higher potential to lower potential until both reach a common potential.

The common potential is:

V common = (Q1 + Q2)/(C1 + C2)

Using initial potentials:

V common = (C1V1 + C2V2)/(C1 + C2)

After connection, the final charges satisfy:

Q1'/Q2' = C1/C2

The energy lost during redistribution is:

Energy loss = C1C2(V1 - V2)2 / [2(C1 + C2)]

This loss occurs because charge flows through the connecting wire.

Common Mistakes To Avoid While Calculating Electric Potential

Students often make mistakes while applying electric potential formulas in numerical questions. Keep these important points in mind to avoid errors during CUET 2027 preparation.

• Ignoring the sign of charge: Positive charges produce positive potential, while negative charges produce negative potential.

• Adding potentials incorrectly: Electric potential is a scalar quantity, so add individual potentials algebraically without resolving directions.

• Confusing potential and electric field: Potential due to a point charge varies as 1/r, whereas electric field varies as 1/r2.

• Forgetting the dielectric constant: In a medium with dielectric constant K, the potential due to a point charge becomes V=kq/(Kr).

• Using the wrong distance: Always measure the distance r from the charge to the point where potential is being calculated.

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