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Derivative of Product and Quotient The derivatives are widely used throughout Mathematics and the Sciences, and the Identities simplifying computations are invaluable tools. In this post, our intent is to derive the multiplication and division identities of derivatives using the very first principles. Derivative Suppose there be a continuous real function ` y = f(x) ` defined over an open interval `(a, b) \subseteq \mathbb{R}`. Then the derivative of the function `f(x)` at some point `c \in (a, b)` represented by, `f'(c) or \frac{d}{dx}f(x)` at `x = c` gives the rate of change of the function at the point `c` with respect to the independent variable. This idea however is still vague and we need a precise mathematical definition.   Right-hand & Left-hand Rate: In the neighbour of `c`, let us define an interval `I = [c - h, c + h]` for some ` h > 0`. Then we may define the right-hand average rate of change of the function in the interval as,  `\frac{f(c+h) - f(c)...

SHM Phasors This post is only an extension to the previous article on AC Phasors. The reader is advised to read the article on AC Phasors here  first, to gain an elementary idea on phasors. In this article, we consider phasors used in describing Simple Harmonic Motion. Simple Harmonic Motion: The motion governed by a force proportional to the displacement of the particle from the mean position and directed towards the mean position is called simple harmonic motion. i.e. for an ideal particle of mass `m`, we have, ` F = -kx` i.e.   ` m \frac{\text{d}^{2}x}{\text{dt}^{2}} = -kx ` Suppose we have the mass particle `m` attached to a spring that applies force according to the above equation. We stretch it from the mean position to some distance say `a` and then release it.  The differential equation of this motion is then, ` \frac{\text{d}^{2}x}{\text{dt}^{2}} = - \frac{k}{m}x` Since both `k` and `m` are positive quantities, we can let `\frac{k}{m} = w^{2}`, for some `w...

(Refresh the browser once more for the equations to load properly) Animating AC Phasors In this post, we try to get an intuitive feeling about special quantities called phasors, used to represent current and voltage in AC circuits. We will see the theory at work, learning about complex numbers, sinors, and how these are used to represent voltage and current. At the end, we'll be able to input our custom values in the animation! Sinusoids: Any quantity that varies in the form of a sine or a cosine function is called a sinusoid. When a sinusoidally time-varying voltage is applied across an electrical component we get a corresponding sinusoidally time-varying current which we call an Alternating Current (AC). The voltage applied is called the alternating voltage. In the general form if ` v(t) = V_{0} sin(ωt) `, then the current is in the form ` i(t) = I_{0} sin(ωt + φ) `, where ` φ ` is called the phase difference, created by the properties of the ele...

Magnetic Moment of A Rotating Charged Rod Magnetic Moment: In general for any closed loop magnetic moment is given by formula `\vec{μ}= ni \vec{A}` , where ` \vec{A}` represents the area vector of the loop determined as usual by right hand thumb rule. Now, the primary problem for us is like this: Prob. A rod with charge density ` ρ = ρ_{0}\frac{x}{L} ` is rotated about its one end along an axis perpendicular to its length. Then find the magnetic moment of the setup. For a Revolving Charged Particle: Before going into the problem, it is important to realise that we have been given a charged rod, and not a simple wire loop wherein a current flows. However as the rod shall rotate, the charges will tranverse through space thus creating an apparent current. In any circular path, when a charged particle say q revolves it appears to create and average current,  `<i> = qf = \frac{q}{T} = \frac{qω}{2π}` Thus in that case,  ` μ = (\frac{qw}{2π})(π r^{2}) = \frac{qωr^{2}}{2}` F...