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Maharashtra State BoardHSC Science (Computer Science) 12th Standard Board Exam
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पाठ्यक्रम

Overview: Electrostatics

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Topics

Estimated time: 48 minutes
Maharashtra State Board: Class 12

Law: Gauss' Law

The total electric flux through a closed surface is equal to \[\frac {1}{ε_0}\] times the total charge enclosed within the surface.

Mathematical Form:

It is one of Maxwell’s equations and is widely used to calculate electric fields for symmetrical charge distributions (spherical, cylindrical, planar symmetry).

Maharashtra State Board: Class 12

Formula: Electric Field of a Charged Spherical Shell

\[E=\frac{\sigma R^2}{\varepsilon_0r^2}\]

Case (i): Electric Field on the Surface of the Shell

\[E=\frac{q}{4\pi\varepsilon_0R^2}=\frac{\sigma}{\varepsilon_0}\]

Case (ii): Electric Field Inside a Uniformly Charged Spherical Shell

E = 0

Maharashtra State Board: Class 12

Formula: Electric Field of a Charged Wire

\[E=\frac{\lambda}{2\pi\varepsilon_0r}\]

Where:

  • λ = linear charge density
  • r = distance from the wire
  • ε0​ = permittivity of free space
Maharashtra State Board: Class 12

Formula: Electric Field due to an Infinite Plane Sheet

\[E=\frac{\sigma}{2\varepsilon_0}\]

Where:

  • σ = surface charge density
  • ε0​ = permittivity of free space
Maharashtra State Board: Class 12

Definition: Potential Difference

Potential difference between two points is the work done per unit charge in moving a charge between them.

Maharashtra State Board: Class 12

Formula: Potential Energy

\[U\left(r\right)=\left(\frac{1}{4\pi\epsilon_{0}}\right)\left(\frac{q_{1}q_{2}}{r}\right)\]

SI unit = joule (J)
l eV = 1.6 × 10-19 J
1 meV = 1.6 × 10-22 J
1 kev = 1.6 × 10-16 J

Maharashtra State Board: Class 12

Formula: Potential Difference

\[V_2-V_1=\frac{U_2-U_1}{q}=\frac{W}{q}\]

Maharashtra State Board: Class 12

Formula: Electric Field–Potential Relation

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

Maharashtra State Board: Class 12

Formula: Electric Potential due to a Point Charge

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

Potential Energy of Two Point Charges:

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

Maharashtra State Board: Class 12

Formula: Electric Potential due to an Electric Dipole

General Expression:
V = \[\frac{1}{4\pi\varepsilon_0}\frac{p\cos\theta}{r^2}\]

Vector Form of Dipole Potential:
V = \[\frac{1}{4\pi\varepsilon_0}\frac{\vec{p}\cdot\vec{r}}{r^3}\]

Maharashtra State Board: Class 12

Formula: Electric Potential due to a System of Charges

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

For continuous distribution:

V = \[\frac{1}{4\pi\varepsilon_0}\int\frac{dq}{r}\]

Maharashtra State Board: Class 12

Definition: Equipotential Surfaces

An equipotential surface is a surface on which the electric potential is the same at every point.
No work is done in moving a charge along an equipotential surface.

Maharashtra State Board: Class 12

Key Points: Equipotential Surfaces

  • Equipotential surfaces for a point charge are concentric spheres, and for a line charge, they are cylindrical in shape.
  • Electric field is always perpendicular (normal) to an equipotential surface at every point.
  • No work is done in moving a charge along an equipotential surface, and such surfaces never intersect each other.
Maharashtra State Board: Class 12

Definition: Electrostatic Energy of Point Charges

Electrostatic potential energy of a system of point charges is defined as the total amount of work done to assemble the system of charges by bringing them from infinity to their present locations.

Maharashtra State Board: Class 12

Formula: Potential Energy of a Dipole in Uniform Field

Maharashtra State Board: Class 12

Key Points: Potential Energy of Charges and Dipoles

  • For two charges, only the second charge requires work to assemble the system.
  • For many charges, total energy is the sum of all pairwise interaction energies.
  • In an external field, a charge has potential energy depending on its position.
  • For charges in an external field, the total energy includes mutual energy and external-field energy.
  • A dipole in a uniform field has minimum energy when aligned with the field and maximum energy when opposite to it.
Maharashtra State Board: Class 12

Key Points: Conductors, Insulators and Charges

  • Conductors have free electrons; insulators do not.
  • Inside a conductor, the electric field is zero, and the potential is constant.
  • An excess charge on a conductor remains on its surface.
  • The electric field outside a conductor is perpendicular to the surface.
  • Free charges can move; bound charges remain fixed to atoms.
Maharashtra State Board: Class 12

Definition: Dielectric

A dielectric is an insulating material that can be polarised when placed in an external electric field.

Maharashtra State Board: Class 1

Definition: Polarization

Polarization is the process in which positive and negative charges inside a dielectric are slightly displaced in opposite directions under the influence of an external electric field, producing a dipole moment.

Maharashtra State Board: Class 12

Definition: Polar Molecule

A polar molecule is a molecule in which the centre of positive charge does not coincide with the centre of negative charge, resulting in a permanent dipole moment.

Maharashtra State Board: Class 12

Definition: Polar Dielectric

A polar dielectric is a dielectric material made up of polar molecules having permanent dipole moments.

Maharashtra State Board: Class 12

Definition: Non-Polar Molecule

A non-polar molecule is a molecule in which the centre of positive charge coincides with the centre of negative charge, resulting in zero dipole moment in the normal state.

Maharashtra State Board: Class 12

Definition: Non-Polar Dielectric

A non-polar dielectric is a dielectric material made up of non-polar molecules that do not possess permanent dipole moments.

Maharashtra State Board: Class 12

Key Points: Polarization of Polar Dielectrics

  • In a non-polar dielectric, an external electric field induces dipoles by slightly shifting the charges.
  • In a polar dielectric, permanent dipoles align with the applied electric field.
  • Polarisation is the dipole moment per unit volume and increases with the applied field.
  • Polarisation produces induced surface charges that create an opposing internal field.
  • The net electric field inside a dielectric is reduced, and very strong fields can cause dielectric breakdown.
Maharashtra State Board: Class 12

Definition: Capacitance

Capacitance is defined as the ratio of charge to potential difference.

C = \[\frac {Q}{V}\]

Maharashtra State Board: Class 12

Definition: One Farad

A capacitor has a capacitance of 1 farad if a charge of 1 coulomb produces a potential difference of 1 volt across it.

1 F = 1 C/V

Maharashtra State Board: Class 12

Key Points: Capacitors: Principle and Combinations

  • A capacitor stores electric charge and electrical energy.
  • Capacitance is given by C = \[\frac {Q}{V}\] and depends on plate size, distance, and dielectric.
  • In series: the same charge; the voltage drops.
    \[\frac{1}{C_{eq}}=\frac{1}{C_1}+\frac{1}{C_2}+\cdots\]
  • In parallel: the voltage is the same; the charge divides.
    Ceq​ = C1 ​+ C2​ + ⋯
  • A series is used for high voltage; a parallel is used for large capacitance.
Maharashtra State Board: Class 12

Formula: Capacitance of a Parallel Plate Capacitor

C = \[=\frac{Q}{V}=\frac{Q}{\left(\frac{Qd}{A\varepsilon_{0}}\right)}=\frac{A\varepsilon_{0}}{d}\]

Maharashtra State Board: Class 12

Formula: Electric Field in a Dielectric-Filled Capacitor

E = \[\frac {Q}{Aε_{0}k}\] or Q = Akε0E

Maharashtra State Board: Class 12

Formula: Capacitance with Partial Dielectric Filling

\[C=\frac{\varepsilon_0A}{d-t+\frac{t}{k}}\]

Maharashtra State Board: Class 12

Definition: Displacement Current

The current due to the time rate of change of electric field in a dielectric (or in space), even in the absence of free charge flow.

Maharashtra State Board: Class 12

Formula: Displacement Curren

In a Dielectric:

\[i_d=Ak\varepsilon_0\frac{dE}{dt}\]

In Vacuum/Air:

\[i_d=A\varepsilon_0\frac{dE}{dt}\]

Maharashtra State Board: Class 12

Definition: Energy Stored in a Capacitor

The work done in charging a capacitor is stored as electrostatic potential energy in the electric field between its plates.

Maharashtra State Board: Class 12

Formula: Electrostatic Energy Stored in a Capacitor

U = \[\frac {Q^2}{2C}\]

Using Q = CV: 
U = \[\frac {1}{2}\]CV2
U = \[\frac {1}{2}\]QV

Maharashtra State Board: Class 12

Definition: Van de Graaff Generator

A Van de Graaff generator is a device that produces very high electric potentials (on the order of 107 volts) by accumulating charge on a hollow metallic conductor.

Maharashtra State Board: Class 12

Key Points: Von de Graaff Generator

  • It produces very high voltage (about 107 V) by collecting charge on a hollow metal dome.
  • It works on corona discharge and the property that the charge stays on the outer surface of a conductor.
  • A moving insulating belt carries charge to the dome, thereby continuously increasing its potential.
  • It is used to accelerate charged particles for nuclear experiments and other applications.
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