Physics and Universe https://physicsanduniverse.com Wed, 08 Jul 2026 05:11:30 +0000 en-US hourly 1 https://wordpress.org/?v=6.8.5 52870843 Writing equations with Latex https://physicsanduniverse.com/writing-equations-with-latex/ https://physicsanduniverse.com/writing-equations-with-latex/#respond Mon, 13 Jul 2026 04:48:27 +0000 https://physicsanduniverse.com/?p=10867 LaTeX is a high-quality typesetting system; it includes features designed for the production of technical and scientific documentation. LaTeX is the de facto standard for the communication and publication of scientific documents. LaTeX is available as free software. Latex has been widely used in the field of physics and mathematics where writing equations is a must. Latex is also available for web and can be used easily. I am also using latex to write equations on this site and wordpress plugin called jetpack come with the latex facility. All you have to do is install Jetpack plugin for wordpress and activate the plugin. That’s it. You are now ready to write beautiful mathematical equations. The aim of this post is to give code of some of the most popular equation that comes up in Math and Physics.

Latex inputResult
E = mc^2E = mc^2
\nabla q \nabla p \ge \dfrac{\hbar}{2}\nabla q \nabla p \ge \dfrac{\hbar}{2}
F = \dfrac{1}{4\pi\epsilon_0} \dfrac{Q_1Q_2}{r^2}F = \dfrac{1}{4\pi\epsilon_0} \dfrac{Q_1Q_2}{r^2}
g’ = g \sqrt{1 – \dfrac{2 \omega^2 R cos^2 \phi}{g}}g' = g \sqrt{1 - \dfrac{2 \omega^2 R cos^2 \phi}{g}}
b^2+p^2=h^2b^2+p^2=h^2
x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
e^{i\pi}+1=0e^{i\pi}+1=0
F=G\frac{m_1m_2}{r^2}F=G\frac{m_1m_2}{r^2}
\nabla\cdot\mathbf{E}=\frac{\rho}{\varepsilon_0}\nabla\cdot\mathbf{E}=\frac{\rho}{\varepsilon_0}
E=h\nuE=h\nu
PV=nRTPV=nRT
e^{ix}=\cos x+i\sin xe^{ix}=\cos x+i\sin x
F(\omega)=\int_{-\infty}^{\infty}f(t)e^{-i\omega t}\,dtF(\omega)=\int_{-\infty}^{\infty}f(t)e^{-i\omega t}\,dt
\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
v=f\lambdav=f\lambda
i\hbar\frac{\partial\Psi}{\partial t}=\hat{H}\Psii\hbar\frac{\partial\Psi}{\partial t}=\hat{H}\Psi
\int_a^b f'(x)\,dx=f(b)-f(a)\int_a^b f'(x)\,dx=f(b)-f(a)
]]> https://physicsanduniverse.com/writing-equations-with-latex/feed/ 0 10867 De Morgan’s Theorem https://physicsanduniverse.com/de-morgans-theorem/ https://physicsanduniverse.com/de-morgans-theorem/#respond Mon, 06 Jul 2026 03:58:38 +0000 https://physicsanduniverse.com/?p=10850 The two De Morgan theorem can be stated as below

  1. The complement of sum of the variables is equal to the product of complement of variables. \overline{A+B} = \overline{A}.\overline{B}
  2. The complement of product of the variables is equal to the sum of the complement of variables. \overline{A.B} = \overline{A} + \overline{B}

To verify this theorem let’s construct a truth table

12345678910
\overline{A} \overline{B} \overline{A+B} \overline{A+B} \overline{A} \overline{B} \overline{A}.\overline{B} \overline{A.B} \overline{A.B} \overline{A} + \overline{B}
0001111011
0110100011
1010010011
1110000100

If we compare column 4 and 7, they are identical proving first law stating \overline{A+B} = \overline{A}.\overline{B}

If we compare column 9 and 10, they are identical proving second law stating \overline{A.B} = \overline{A} + \overline{B}

]]> https://physicsanduniverse.com/de-morgans-theorem/feed/ 0 10850 X-Ray https://physicsanduniverse.com/x-ray/ https://physicsanduniverse.com/x-ray/#respond Mon, 29 Jun 2026 06:36:02 +0000 https://physicsanduniverse.com/?p=10845 Production: X-ray is produced under the action of incident electron that excites an electron in inner orbit if target atom.

How is is different for light? Both are electromagnetic wave but X-ray has short wavelengths ranging from 0.001 Angstrom to 100 Angstrom. X rays are also more energetic than visible light and can penetrate through numbers of different substances which are generally opaque in visible spectrum.

X-ray as medical tool: X-ray is used in medicine due to its penetrating nature. It passes through flesh/skin but is block by bones, making it great tool for studying skeletal system. This is possible because of high absorption capacity of Calcium and Phosphorus present in bone.

Barium sulfate solution to check internal organs: Heavy atoms absorbs more x-rays so it becomes easier to detect obstacles like stone.

Reason to use Tungsten in Coolidge tube: 1. Its has high melting point and doesn’t melt when collided with high incident electron beam. 2. Its has high atomic number so X-ray produced is of high energy 3. The energy levels inside the target atom must consist of energy transition levels that lie in range of X-rays.

Hard X-ray are one with high penetration power and are with wavelength of 0.01 to 1.5 Angstrom.

Soft X-ray are one with little penetration power and with wavelength range of 1.5 to 100 Angstrom.

]]> https://physicsanduniverse.com/x-ray/feed/ 0 10845 3I Atlas – what’s going on? https://physicsanduniverse.com/3i-atlas-whats-going-on/ https://physicsanduniverse.com/3i-atlas-whats-going-on/#respond Tue, 25 Nov 2025 10:44:21 +0000 https://physicsanduniverse.com/?p=10627 3I Atlas just flew pas out sun and has revealed new details about its nature and makeup. In any comet’s life the most exciting time is when its approaches the sun and is at its shortest distance from Sun. We call this phase Perihelion. During this time comet gets to face full force of the sun and is continuously bombarded by radiation, heat and coronal mass ejections. Here is what has happened so far when 3I Atlas was closest to the Sun

  1. There was rapid change is brightness seen. The brightness increased some 400 times which is huge as most comets that pass by the sun only brighten up by 10 to 100 times.
  2. Light emitted by 3I Atlas was close to blue where normal expectation was it will be red due to coma of gas and dust.
  3.  3I Atlas started to experience non gravitational acceleration for the first time during its perihelion with the Sun. This is sort of expected as coma of the comet push around the object giving it non gravitational acceleration. The magnitude however is higher than expected.
  4. Its closest approach to the Sun was on October 29th 2025
  5. Data taken after closest approach shows seven distinct jets of gas going into space and are mostly pointing away from the sun. Couple of smaller jets are shooting directly into the sun which is not typical.
  6. Actual size of 3I Atlas is still debatable but its roughly estimated between 5km to 10km across and its more likely to be on the small side of the estimation range. This make 3I Atlas 20 times larger than Omua Mua.  
]]> https://physicsanduniverse.com/3i-atlas-whats-going-on/feed/ 0 10627 Oscillator https://physicsanduniverse.com/oscillator/ https://physicsanduniverse.com/oscillator/#respond Mon, 19 Sep 2022 02:51:42 +0000 https://physicsanduniverse.com/?p=9085 Oscillator is an electronic circuit that produces a repetitive electronic signal (sine wave or square wave). Some examples of signal generated by oscillators includes signal broadcast by radio and TV transmissions, clock signals and sound produced by electronic beeper and video games.

Phase shift oscillator

An oscillator circuit which uses RC network as a combination of resistors and capacitors for its frequency selective parts is called RC oscillator. Phase shift oscillator is an electronic oscillator which has inverting amplifier and a feedback filter which shifts the phase of the amplifier output by 180 degree at the oscillation frequency.

  • Oscillators converts DC signals into AC signal.
  • Oscillator generates pulse type output along with square, saw-tooth and sine wave.
]]> https://physicsanduniverse.com/oscillator/feed/ 0 9085 SI Units https://physicsanduniverse.com/si-units/ https://physicsanduniverse.com/si-units/#respond Sat, 31 Jul 2021 17:53:15 +0000 https://physicsanduniverse.com/?p=8984 General Conference on Weights and Measures in 1971 adopted seven quantities as fundamental base quantities. These seven base quantities are known as International System of units or SI in brief. There quantities listed below are seven fundamental quantities

QuantitySI UnitSymbol
Lengthmeterm
Masskilogramkg
Timeseconds
Electric currentampereA
TemperaturekelvinK
Amount of substancemolemole
Luminous intensitycandelacd

Two supplementary units are also added for Plane angle called radian (rad) and another unit for solid angle called steradian (Sr.)

Meter: One meter is defined as a distance between two lines engraved on a platinum-iradium bar kept at 0C at the International Bureau of Weights and measures at Paris. However a more accurate definition of meter is defined as distance traveled by light in vacuum in 1/299792458 second.

Kilogram: It is mass of cylinder made of platinum-iridium alloy kept at international Bureau of weight and measurements in Paris. With internationl agreement, mass of 1kg is assigned to it.

Second: One second is time taken to make 9,192,631,770 vibrations by a hyperfine transition in cesium-133 atom.

Ampere: It is current which when passed equally through two long parallel conductors kept one meter apart in vacuum, produces a force of 2 \times 10^{-7} newton/m on the wires.

Kelvin: It is 1/273.16 of the temperature of tripple point of water. The temperature of triple point of water is 273.16 kelvin.

Mole: It is the mass of a substance that contains 6.023 \times 10^{23} number of atoms or molecules. The number 6.023 \times 10^{23} is called Avogadro number. Example one mole of carbon-12 has a mass of 0.012kg exactly and contains 6.023 \times 10^{23} number of carbon atoms.

Candela: It is the luminous energy emitted per second by a source per unit solid angle. It has a radiant intensity of 1/683 watt per steradian.

Radian: It is the angle subtended at the center of a circle of radius r by an arc of length r of the same circle.

Solid Angle: It is defined as the angle subtended at a point by the surface area.

Steradian: It is the solid angle subtended at the center by unit area on the surface of a sphere of unit radius. Total solid angle subtended at the center by the surface area of a sphere of radius r in 4\pi steradian.

]]> https://physicsanduniverse.com/si-units/feed/ 0 8984 Calculating mass of planet https://physicsanduniverse.com/calculating-mass-of-planet/ https://physicsanduniverse.com/calculating-mass-of-planet/#respond Sun, 21 Jun 2020 13:13:26 +0000 https://physicsanduniverse.com/?p=8894 The mass of a planet can be determined by observing the time period of its satellite.

Let M be the mass of the planet and m be the mass of its satellite. For the satellite to be in the circular orbit, gravitational force is equal to centripetal force.

If r be the radius of the orbit of the satellite,

\dfrac{GMm}{r^2} = \dfrac{mv^2}{r}

M = \dfrac{v^2 r}{G}

If T is the period of revolution of the satellite,

v = \omega r = \dfrac{2 \pi r}{T}

Then,

M = \dfrac{4 \pi^2 r^2 r}{T^2 G} = \dfrac{4 \pi^2 r^3}{GT^2}

So, knowing the distance of the satellite from the planet and time period of revolution of the satellite, the mass of the planet can be determined.

]]> https://physicsanduniverse.com/calculating-mass-of-planet/feed/ 0 8894 Solar constant and temperature of the Sun https://physicsanduniverse.com/solar-constant-and-temperature-of-the-sun/ https://physicsanduniverse.com/solar-constant-and-temperature-of-the-sun/#respond Sat, 13 Jun 2020 12:47:46 +0000 https://physicsanduniverse.com/?p=8887 Solar constant is the amount of energy received from the sun per second per unit area by a perfect black body on the earth, the area being placed normal to the direction of radiation.

Let S be the solar constant r be the distance of the earth from the sun and R be the radius of the sun.

Total energy radiated by the sun per second is \sigma T^4 \times 4 \pi R^2

Here T is the temperature of the sun and \sigma is the Stefan’s constant.

Energy received per second per unit are on the surface of the earth is

= \dfrac{\sigma T^4 \times 4 \pi R^2}{4 \pi r^2} = \dfrac{\sigma T^4 R^2}{r^2}

According to the definition of the solar constant, this is the solar constant.

S = \dfrac{\sigma T^4 R^2}{r^2}

T^4 = \dfrac{Sr^2}{\sigma R^2}

T = \left(\dfrac{Sr^2}{\sigma R^2}\right)^\frac{1}{4}

Hence by knowing the value of S, the temperature of the sun can be estimated.

]]> https://physicsanduniverse.com/solar-constant-and-temperature-of-the-sun/feed/ 0 8887 Magnitude of star [Formula] https://physicsanduniverse.com/magnitude-of-star-formula/ https://physicsanduniverse.com/magnitude-of-star-formula/#respond Sat, 06 Jun 2020 12:46:59 +0000 https://physicsanduniverse.com/?p=8879 The magnitude of star is the measure of its brightness. When measured from the earth, two stars that differ by one magnitude have a brightness ratio of 2.512

\dfrac{\text{The brightness of star of nth magnitude}}{\text{The brightness of the star of (n+m)th magnitude}} = (2.512)^m

If two stars of magnitude m_1 and m_2 possess brightness l_1 and l_2 respectively, then

m_2 - m_1 = -2.5 log\dfrac{l_2}{l_1}

]]> https://physicsanduniverse.com/magnitude-of-star-formula/feed/ 0 8879 Motion down an inclined plane https://physicsanduniverse.com/motion-inclined-plane/ https://physicsanduniverse.com/motion-inclined-plane/#respond Sat, 23 Dec 2017 07:16:48 +0000 http://physicsanduniverse.com/?p=8642 Let P be a particle on a smooth plane inclined at an angle alpha to the horizon. If ABC represent a vertical section of the inclined planemotion down inclined plane through P, AB is the line of greatest slope and the angle ABC is alpha. The acceleration of P due to gravity is g and it is vertically downwards towards PQ. If PN be normal to the plane then angle QPN is alpha. The resolved parts of g are g cos\alpha along PN and g sin \alpha along PB.

Since the plane prevents any motion perpendicular to it, the particle moves down the plane along AB with acceleration g sin \alpha

If h be the height AC and l the length AB of the inclined plane sin \alpha = \dfrac{h}{l}

Let a particle slide down AB from rest and v be its velocity at B.

Then v^2 = 2gsin\alpha l = 2gh

Thus the velocity acquired is independent of alpha and is the same as the velocity acquired in falling through a height h.

If t be the time taken to slide down AB, l = \dfrac{1}{2}g sin\alpha t^2

t = \sqrt{\dfrac{2l}{g sin\alpha}}

If the particle is projected upwards along BA, its acceleration is -g sin\alpha

]]> https://physicsanduniverse.com/motion-inclined-plane/feed/ 0 8642 Motion under Gravity https://physicsanduniverse.com/motion-under-gravity/ https://physicsanduniverse.com/motion-under-gravity/#respond Sat, 16 Dec 2017 05:07:14 +0000 http://physicsanduniverse.com/?p=8629 When a body falls towards the earth we can notice that its velocity increases as it approaches earth. Hence the falling body is accelerating. Galileo performed several experiment and showed that falling bodies with different mass from same height reach the earth at the same time and hence they fall with constant acceleration. This acceleration is called acceleration due to gravity and is denoted by symbol g .

Experiments have shown that the value of g the same at the same place and varies slightly from place to place. A fairly correct value of g is 9.8 m/s^2 or 32 ft/sec^2 .

Vertically downward motion

A particle moving vertically downwards from height h goes downward with an acceleration g . If we take the downward directions as positive and replace s by h and a by g in equations of motion, we get

v = h + gt

h = ut + \dfrac{1}{2}gt^2

v^2 = u^2 + 2gh

Vertically upward motion

A particle thrown vertically upward moves with a retardation g . Its velocity slowly diminishes and becomes zero when the particle is at its maximum height. The body is then at rest for an instant and immediately begins to fall with acceleration g . We take upward direction as positive and replace g with -g then equations of motions becomes

v = u -gt

h = ut - \dfrac{1}{2} gt^2

v^2 = u^2 - 2gh

Greatest height and time of ascent when particle is thrown upward vertically

Let H be the maximum height attained in time T. Since velocity is 0 at the maximum height, 0 = u - gT and 0 = u^2 - 2gH

Time to reach the top: T = \dfrac{u}{g}

Maximum height reached: H = \dfrac{u^2}{2g}

Time of flight

When the particle returns to the earth, the vertical displacement is zero. If T’ is the time taken to come back, 0 = uT' - \dfrac{1}{2}gT'^2  = T'(u - \dfrac{1}{2} gT')

T' = 0

T' = \dfrac{2u}{g} this gives the total time of flight.

Since the time of ascent is \dfrac{u}{g} , time  of flight is twice the time of ascent.

So, time of ascent is equal to time of descent.

Time taken to reach a height of h

From h = ut - \dfrac{1}{2}gt^2

t = \dfrac{u \pm \sqrt{u^2-2gh}}{g}

When u^2 > 2gh , t has two real values which chows that particle reaches the height twice … once on its way up and once on its way down after it has reached the greatest height.

If h > \dfrac{u}{2g} or u^2 < 2gh , t becomes imaginary. It shows that the particle does not go higher than \dfrac{u^2}{2g}

Velocity at a height h

From v^2 = u^2 - 2gh we have

v = \pm \sqrt{u^2-2gh}

The positive value represents upward velocity and negative value represents downward velocity at the same height h. It shows that magnitude of the velocity is the same when the particle is at the same height whether it is ascending and descending.

]]> https://physicsanduniverse.com/motion-under-gravity/feed/ 0 8629