Ampere's circuital law

 Ampere's circuital law states that the line integral of a magnetic field around any closed loop is equal to the permeability of free space multiplied by the net electric current enclosed by that loop.

While the Biot-Savart Law gives the magnetic field at a point due to a current element using vector calculus, Ampere’s law provides a simpler alternative when symmetry is present. For instance, in the case of a long straight wire, a toroid, or a solenoid, the law allows for a direct and often less mathematically intensive calculation of the magnetic field. The law states that the line integral of the magnetic field around any closed loop is proportional to the net current enclosed by that loop. However, it is important to note that Ampere’s law is strictly valid for steady currents, meaning the current must remain constant with time. The complications that arise in time-varying situations lead to the introduction of the concept of displacement current, which extends the law to dynamic fields, a topic explored in later chapters.

We draw a circular loop with radius ' r' passing through point  'P', with the current-carrying wire lying on its axis. This loop is called an Amperian loop.

The Amperian loop can be of any shape, but for practical calculations, it is often chosen with symmetry to simplify the mathematics.

It is important to choose the direction of the current and the direction in which you traverse the loop properly; this is determined by the right-hand rule. If you curl the fingers of your right hand in the direction you traverse the loop, then your right thumb points in the direction in which the current is considered positive.

Pythagorous theorem for sides of a triangle of any angle theta

 Expansion of the Pythagorous theorem for a triangle with any angle other than 90 degree....also called as Law of Cosines.

Law of Cosines...gives general relation between the three sides of any triangle ( it is also true for right angle triangles).

For small x, the expression (1+x)^n approximates to 1+nx.

 

What are cis trans isomers

 Cis and trans isomers are [stereoisomers] (molecules with the same formula and bonds, but different 3D shapes). In a cis isomer, the same or high-priority groups sit on the same side of a rigid structure (like a double bond or ring). In a trans isomer, those groups sit on opposite sides.

Is there some rotation too when a ball is slipping?

 

A solid sphere (A) of mass 5m and a spherical shell (B) of mass m, both having same radius, are placed on a rough surface. When a force of same magnitude is applied tangentially at the highest points of A and B, they start rolling without slipping with an acceleration of a and b, respectively. The ratio of a and b is?

 

The double-slit experiment to illustrates wave interference pattern, and diffraction.

 

The double-slit experiment primarily illustrates wave interference, though diffraction plays a critical role as the waves bend around each individual slit to overlap.

 When light or matter passes through two close slits, diffraction spreads the waves out, and they interfere to create an alternating pattern of bright and dark bands on a screen.

How the Two Concepts Work Together

Diffraction: Waves bend and spread out as they pass through each narrow slit.
Interference: The spread-out waves from both slits cross paths and overlap.
Constructive Interference: Waves meet in sync and make bright bands.
Destructive Interference: Waves meet out of sync and cancel out to make dark bands.








m = number of maximum
D= distance between screen and slits
d = seperation between the two slits
lambda = wavelength
y = is the gap between the maximum and the center



Here, the Green lines show the points of Constructive interference, hence a maximum i.e. a bright spot while the orange line shows the points where destructive interferance takes place and hence points of mimnimum i.e. dark spots.

If two bodies of mass 10kg and 7 kg hang on the two sides of a single pully, then what is the speed of the bodies and what is the net force acting on the axis of the pully?

 

What is 'Molecularity' in chemical kinetics? why it cannot be fractional? and how to calculate it from the balanced equation of a reaction?

 What is 'Molecularity' in chemical kinetics? 

 why it cannot be fractional?

how to calculate it from the balanced equation of a reaction?

In determinants matrix A and its inverse is A^-1, then A = 1/(A^-1)?

 

What are natural magnets and why do they have magnetic properties?

 Natural magnets are permanent magnets found in the earth's crust, such as lodestone (a magnetized form of the iron oxide mineral magnetite). They form naturally when deposits are exposed to intense magnetic fields—like the Earth's magnetic field over long periods or powerful natural events like lightning strikes. 

Why They Have Magnetic Properties

Electron Spin: Every atom has electrons that spin and create tiny magnetic fields.
Unpaired Electrons: In most materials, these magnetic fields cancel each other out. In magnetic materials like iron, many electrons point in the same direction.
Magnetic Domains: Atoms group into tiny regions called domains.
Alignment: In a natural magnet, these domains point in one uniform direction, creating a strong overall magnetic force.

Electrons drift in a conductor from lower to higher potential, does this mean that all free electrons are moving in the same direction?

 

find value of a for which f(x) = x^2 -2ax + b f(x) is an increasing function for x>0

 

Determinant of a skew symmetric matrix

 

If A is a square matrix of order 3 such that det(A)=9, then det(9A^-1) is?

 

If A is a 3x3 matrix , B is a 3x1 matric and C is a 1x3 matric then only AB and BC are defined?

 The statement is false because, in addition to AB and BC, the matrix products AC and  CB are also perfectly well-defined. 

Conclusion

Apart from AB and BC, the products CB and CA are also defined. 

How can we build a chatting app like whatsapp that works on phone text messaging

To build a chat app like WhatsApp that runs on phone text messaging, you need to use an SMS Gateway API, set up a backend server, and create a mobile user interface. [1, 2]

Core Architecture & Tools

SMS Gateway API: Use services like Twilio or Vonage to send and receive real phone text messages (SMS) programmatically.
Backend Server: Build a server using Node.js or Erlang to process incoming and outgoing text logs.
Database: Store user phone numbers, message history, and profile data using databases like PostgreSQL, Firebase Firestore, or MongoDB.
Mobile Frontend: Develop cross-platform mobile screens using Flutter or React Native so the app works on both Android and iOS. [1, 2, 3, 4, 5]

Development Steps

User Verification: Implement phone number login by sending a random verification code via SMS.
Sync Contacts: Request permission to read the phone’s address book and match numbers registered in your database.
Message Routing: Connect your mobile app to the backend so that when a user hits send, the app triggers an API request to dispatch an SMS or sync through your server data pipeline