Category: Physics

Moment of inertia I is given by,

$$ I = \Sigma m_i r_i^2$$

The displacement Δx of a particle is given by the following formula,

Δx = x₂ – x₁

Where,
x₁ = Initial position of a particle.
x₂ = Final position of a particle.

The displacement vector of the particle is given by,

Δr = (x₂ – x₁ )i + (y₂ – y₁ )j + (z₂ – z₁ )k

Magnitude of Δ r is given by

| Δ r | = [ (x₂ – x₁ )² + (y₂ – y₁ )² + (z₂ – z₁ )² ]^1/2

Position vector is given by,
r = xi + yj + zk

Where,
i = unit vector along x direction.
j = unit vector along y direction.
k = unit vector along z direction.

First equation of uniform accelerated motion

v = u + at

Second equation of uniform accelerated motion

s = ut + ½ at²

Third equation of uniform accelerated motion

v² = u² + 2as

Where
s = Displacement.
v = Final velocity of the particle.
u = Initial velocity of the particle.
a = Acceleration of the particle

Formula of density is given by,
ρ = m/V
Where,
m = Mass.
V = Volume.

Formula of relative density ρ_r is given by,
ρ_r= ρ_s/ρ_w
Where,
ρ_s = Density of substance.
ρ_w = Density of water at 4^oC.

The formula of weight is given by,
W = mg
Where,
m = Mass of the object.
g = Acceleration due to gravity.

$$ \nabla . \vec{E} = \frac{\rho}{\epsilon_0}$$

$$ \nabla . \vec{B} = 0$$

$$ \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$$

$$ \nabla \times \vec{B} = \mu_0 \vec{J}$$

Laplace equation is given by,

∇² Φ = 0

Where,
Φ = Scalar function.
∇ = Laplacian operator.

$$ x’ = \frac{x-vt}{ \sqrt {1- \frac{v^2}{c^2}}}$$
$$ y’ = y$$
$$ z’ = z$$
$$t’ = \frac{t -\frac{vx}{c^2}}{ \sqrt {1- \frac{v^2}{c^2}}}$$

Displacement current is given by,

$$\vec{J_d} = \frac{\partial \vec{E}}{\partial t}$$

When a light source giving the line spectrum is placed in an uniform external magnetic field, the spectral line emited by the atoms of the source are split. This splitting of spectral line by a magnetic field is called zeeman effect.

At very low temperature, electrical resistivity of some metal and alloys drops suddenly to zero. This phenomenon is called superconductivity.

The venturimeter is a device to measure the speed of incompressible liquid and rate of flow of liquid through pipes. Working of venturimeter is based on Bernoulli’s theorem.

$$\Delta x.\Delta p_x \geq \frac{\hbar}{2}$$

Thermal expansion can be three types:

Linear expansion

$$\alpha_l = \frac{1}{\Delta T} \frac{\Delta l}{l}$$

Area expansion

$$\alpha_a = \frac{1}{\Delta T} \frac{\Delta A}{A}$$

Volume expansion

$$\alpha_v = \frac{1}{\Delta T} \frac{\Delta V}{V}$$

$$ H=kA \frac{T_C-T_D}{L}$$
Where,
k is a constant, called thermal conductivity. Si unit of thermal conductivity is Wm^-1k^-1.

Operators in quantum mechanics

Suppose a particle of mass m, constrained to move along the x-direction in the region of the potential V. Then Schrodinger equation for this particle is,

$$ i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} + V\psi$$

Where,
\(i\) is the square root of -1 and \(\hbar = \frac{h}{2\pi}\).
\(\psi\) is the wave function of the particle.
V is the specified potential.

What is a semiconductor?

Those material, which have resistivity or conductivity intermediate to metals or insulator, called semiconductor. The band gap of semiconductor is less than 3 eV. The band gap of Ge and Si respectively 1.1 eV and 0.7 eV.
Si, Ge, GaAs, CdTe etc are examples of semiconductors.
The band gap of Ge and Si respectively 1.1 eV and 0.7 eV.

Si, Ge and C are elemental semiconductor. Examples of compound semiconductors are following:
Inorganic: Cds, GaAs, Cdse, InP etc.
Organic: Anthracene, doped pthalocyanines etc.
Organic Polymers: Polypyrolle, Polyaniline, Polythiophene etc.

An intrinsic semiconductor is one which is made of the semiconductor material in its totally pure form and with no impurities or lattice defects.

If we added small amount ( parts per million, ppm ) of suitable impurities in pure semiconductor , Then this process is called dopping and suitable impurity is called dopent. Dopped intrinsic or pure semiconductor is called extrinsic semiconductor.

There are two types of extrinsic semiconductors-

(a) n-type semiconductor

The pentavalent impurities eg. Phosphorus, Arsenic, Antimony, Bismith etc are referred to as donor impurities. If pure semiconductor is dopped with donor impurities then this type of semiconductor is called n-type semiconductor.

(b) p-type semiconductor

Trivalent impurities eg. Boron, Aluminium, Indium and galium are referred as accepter impurities. If pure semiconductor is dopped with acceptor impurities then this type of semiconductor is called p-type semiconductor.

Newton’s laws of motion

Newton’s gives three law of motion:

1. Newton’s first law motion

A body continues in its state of rest or constant velocity along the same straight line, unless not disturbed by some external force. This is Newton’s first law motion.

2. Newton’s second law motion

Time-rate of change of momentum is proportional to the applied external force. This is Newton’s second law motion

3. Newton’s third law motion

To every action there is always equal and opposite reaction. This is Newton’s third law motion.

What is the errors of measurement in physics?

Errors of measurement = True value of a quantity – Measured value of a quantity

Suppose, the measured value of quantity be A_m and the error in measurement be ΔA. Then the true value of the quantity can be written as
A_t = A_m ± ΔA

Absolute error, Relative error and Percentage error

Absolute error

Suppose a physical quantity be measured n times and the measured values be \( a_1, a_2, a_3 — a_n \). The arithmetic mean of these values is given by,

$$ a_m = \frac{a_1 + a_2 + a_3 + … + a_n }{n}$$

The absolute errors in the individual measurement values are

$$ \Delta a_1 = a_m – a_1 $$

$$ \Delta a_2 = a_m – a_2 $$

$$ \Delta a_3 = a_m – a_3 $$

………….

$$ \Delta a_n = a_m – a_n $$

Mean absolute error

Mean absolute error \( \Delta a_m\) of a physical quantity is given by,

$$ \Delta a_m = \frac {|\Delta a_1| + |\Delta a_2| + |\Delta a_3| +…+ |\Delta a_n|}{n} $$

Relative error

Relative error \( R_e\) is given by,

$$ \ R_e = \frac{\Delta a_m}{a_m}$$

Percentage error

Percentage error \( p_e\) is given by,

$$ P_e = \frac{\Delta a_m}{a_m} \times 100 \% $$

Types of errors

1. Systematic errors
2. Random errors
3. Gross errors

For two particle system, the position vector of centre of mass of the two particle system is given by,

$$\vec{r} = \frac {m_1 \vec{r_1} + m_2 \vec{r_2}}{m_1+m_2}$$

For two particle system, velocity of centre of mass is given by,

$$\vec{v_c} = \frac {m_1 \vec{v_1} + m_2 \vec{v_2}}{m_1+m_2}$$

Whewe,

\(m_1\) and \(m_2\) are the masses of two particles.

\(\vec{r_1}\) and \(\vec{r_2}\) are the position vector of the particles \(m_1\) and \(m_2\) respectively.

\(\vec{v_1}\) and \(\vec{v_2}\) are velocities of the particles \(m_1\) and \(m_2\) respectively.

Truth table of AND Gate:

Truth table of NOT Gate.

Truth table of OR Gate.

Truth table of NAND Gate.

Truth table of NOR Gate.

Truth table of XOR Gate.