Important Derivations for CBSE Class 12 Physics: A Comprehensive Guide

The CBSE Class 12 Physics exam for the year 2017 has concluded, but the derivations covered in the syllabus can still be extremely useful for understanding the fundamental concepts and preparing for future exams.

Key Concepts and Derivations in CBSE Class 12 Physics

Mechanics: This branch of physics deals with the motion and forces between particles. Here are some important derivations:

Equations of Motion: The basic equations of motion describe the displacement, velocity, and acceleration of objects under various conditions. These equations are derived from the fundamental principles of calculus and are essential for understanding motion. Centripetal Force: The derivation for centripetal force is based on the centripetal acceleration formula, (a frac{v^2}{r}), where (a) is the centripetal acceleration, (v) is the velocity of the object, and (r) is the radius of the circular path. This derivation helps in understanding the force that keeps an object moving in a circular path. Work Done by Variable Force: Work done by a variable force is derived using the integral form, (W int_{x_1}^{x_2} F(x) , dx), where (F(x)) is the force as a function of position, and (dx) is the differential increment of distance. Potential Energy and Kinetic Energy: The derivation of these concepts is based on the work-energy theorem and conservation of energy principles. The potential energy is derived as the work done by a conservative force, while kinetic energy is derived as the energy associated with the motion of an object. Conservation of Mechanical Energy: The law of conservation of mechanical energy is derived from the principle that the sum of kinetic and potential energy is constant in a closed system, (E K U).

Electrostatics: This section deals with the electric charges and their interactions:

Coulomb's Law: The derivation of Coulomb's law is based on the inverse square law, (F k_e frac{q_1 q_2}{r^2}), where (F) is the force, (k_e) is Coulomb's constant, and (r) is the distance between the charges. Electric Field Intensity and Potential: The formula for the electric field intensity due to a point charge, (E k_e frac{q}{r^2}), and the electric potential due to a point charge, (V k_e frac{q}{r}), are derived using vector calculus and scalar calculus. Capacitance: The capacitance of a parallel plate capacitor is derived as (C frac{epsilon_0 A}{d}), where (A) is the area of the plates, (d) is the separation between the plates, and (epsilon_0) is the permittivity of free space.

Current Electricity: This section covers the flow of electric current through conductors:

Ohm's Law: Ohm's law is derived from the relationship between voltage, current, and resistance, (V IR). Resistance and Power: The formula for the resistance of a conductor, (R rho frac{l}{A}), and the power dissipated in a resistor, (P I^2R), are derived using the concepts of resistivity, length, and cross-sectional area of the conductor. EMF of a Cell: The formula for the EMF of a cell, (epsilon E - Ir), where (E) is the electromotive force, (I) is the current, and (r) is the internal resistance, is derived from the Kirchhoff voltage law. Internal Resistance: The internal resistance of a cell is derived using the concept of voltage drop across the internal resistance.

Magnetism: This branch of physics deals with magnetic fields and their effects on electric currents and moving charges:

Magnetic Field Due to a Current-Carrying Conductor: The formula for the magnetic field, (B frac{mu_0 I}{2pi r}), is derived using Ampere's circuital law. Force on a Moving Charge in a Magnetic Field: The force on a moving charge in a magnetic field is derived from the Lorentz force law, (F qv times B). Torque on a Current Loop: The torque on a current loop in a magnetic field is derived as (tau IAB sin theta), where (I) is the current, (A) is the area of the loop, (B) is the magnetic field, and (theta) is the angle between the loop and the magnetic field.

Electromagnetic Induction: This section covers the phenomena related to the generation of electric currents in response to changes in magnetic fields:

Faraday's Law of Electromagnetic Induction: Faraday's law is derived using the concept of changing magnetic flux through a loop, (epsilon -N frac{dPhi_B}{dt}). Lenz's Law: The law of electromagnetic induction is derived from the principle of conservation of energy and the opposing nature of induced currents. EMF Induced in a Coil: The EMF induced in a coil is derived as (epsilon -N frac{dPhi_B}{dt}).

Optics: This section deals with the behavior of light and imaging:

Lens Formula and Magnification: The lens formula, (frac{1}{f} frac{1}{v} frac{1}{u}), and the formula for magnification, (M frac{v}{u}), are derived using the principle of similar triangles. Refractive Index of a Prism: The refractive index, (n frac{sin i}{sin r}), is derived using Snell's law.

Modern Physics: This section covers recent developments in physics, including quantum mechanics and relativity:

De Broglie Wavelength: The de Broglie wavelength, (lambda frac{h}{p}), is derived using the relation between momentum and wavelength, where (h) is the Planck constant and (p) is the momentum of the particle. Energy of a Photon: The energy of a photon, (E hf), is derived using Planck's relation between energy and frequency. Half-Life of a Radioactive Substance: The half-life of a radioactive substance is derived using the exponential decay law, (N N_0 e^{-lambda t}), where (lambda) is the decay constant.

Conclusion

While the list provided here is not exhaustive, it covers the major derivations that are essential for understanding and excelling in CBSE Class 12 Physics. It is always recommended to refer to textbooks, previous year question papers, and practice problems to gain a comprehensive understanding of these concepts and their applications.