Important Formulas and Equations for UPSC Physics Optional Subject

Mechanics:
1. Newton's Laws of Motion:
- First Law: An object at rest will remain at rest, and an object in motion will continue moving with a constant velocity unless acted upon by an external force. (F = 0)
- Second Law: The force acting on an object is directly proportional to the rate of change of momentum of the object. (F = ma)
- Third Law: For every action, there is an equal and opposite reaction.

2. Work, Energy, and Power:
- Work (W) = Force (F) × Displacement (s) × cos(θ)
- Kinetic Energy (K.E.) = 0.5 × mass (m) × velocity (v)^2
- Potential Energy (P.E.) = mgh
- Power (P) = Work (W) / Time (t)

3. Conservation Laws:
- Conservation of Linear Momentum: Total momentum before an event is equal to the total momentum after the event. (m1v1 + m2v2 = m1v1' + m2v2')
- Conservation of Energy: Energy cannot be created or destroyed, only transformed from one form to another. (K.E. + P.E. = Constant)

Electromagnetism:
1. Coulomb's Law: The force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. (F = k(q1q2)/r^2)

2. Electric Field: The electric field at a point in space is defined as the force experienced by a unit positive test charge placed at that point. (E = F/q)

3. Ohm's Law: The current flowing through a conductor is directly proportional to the potential difference across it and inversely proportional to its resistance. (V = IR)

4. Magnetic Field: The magnetic field at a point in space is defined as the force experienced by a unit positive charge moving with a velocity at that point. (B = F/qv)

Thermodynamics:
1. First Law of Thermodynamics: The change in internal energy of a system is equal to the heat added to the system minus the work done by the system. (ΔU = Q - W)

2. Second Law of Thermodynamics: Heat cannot spontaneously flow from a colder body to a hotter body.

3. Carnot Cycle Efficiency: The maximum efficiency of a heat engine operating between two temperature reservoirs is given by the formula: Efficiency = (Th - Tc)/Th, where Th is the absolute temperature of the hot reservoir and Tc is the absolute temperature of the cold reservoir.

Modern Physics:
1. Einstein's Mass-Energy Equivalence: E = mc^2, where E is the energy, m is the mass, and c is the speed of light.

2. Planck's Quantum Theory: Energy is quantized and emitted or absorbed in discrete packets, known as quanta.

3. Bohr's Atomic Model: Electrons revolve around the nucleus in specific energy levels or orbits. The energy of an electron is quantized and can only take certain discrete values.

These are some of the important formulas and equations for the UPSC Physics optional subject. It is essential to understand