induction Definition and Topics - 120 Discussions

Mathematical induction is a mathematical proof technique. It is essentially used to prove that a statement P(n) holds for every natural number n = 0, 1, 2, 3, . . . ; that is, the overall statement is a sequence of infinitely many cases P(0), P(1), P(2), P(3), . . . . Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder:

Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one (the step).
A proof by induction consists of two cases. The first, the base case (or basis), proves the statement for n = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that if the statement holds for any given case n = k, then it must also hold for the next case n = k + 1. These two steps establish that the statement holds for every natural number n. The base case does not necessarily begin with n = 0, but often with n = 1, and possibly with any fixed natural number n = N, establishing the truth of the statement for all natural numbers n ≥ N.
The method can be extended to prove statements about more general well-founded structures, such as trees; this generalization, known as structural induction, is used in mathematical logic and computer science. Mathematical induction in this extended sense is closely related to recursion. Mathematical induction is an inference rule used in formal proofs, and in some form is the foundation of all correctness proofs for computer programs.Although its name may suggest otherwise, mathematical induction should not be confused with inductive reasoning as used in philosophy (see Problem of induction). The mathematical method examines infinitely many cases to prove a general statement, but does so by a finite chain of deductive reasoning involving the variable n, which can take infinitely many values.

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  1. M

    Engineering Kirchoff's Law Problem about this Induction Generator circuit

    Hi, Please find attached below the Induction Generator circuit. I have the following parameters: Parameter Value Rs = Rr (Resistance on stator and rotor) 0.2 Ω Ls = Lr (inductance on stator and rotor) 5 mH = 0.005 H p (number of pairs of poles) 2 f (frequency) 60 Hz Vsl (stator line...
  2. M

    Engineering Induction motor calculations

    Given the AC Induction (asynchronous motor) in 3 phases: Parameter Value Pn (Nominal Power) = Pmechanical (output power at the shaft) 5 kW = 5000W uls (Voltage through the stator line) 220 V fstator (stator frequency) 50 Hz p (Number of pole pairs) 2 LFe (Iron loss) = Lmechanical...
  3. C

    Simple Induction Induction proof of Polynomial Division Theorem

    Theorem: Let ## f(x), g(x) \in \mathbb{F}[ x] ## by polynomials, s.t. the degree of ## g(x) ## is at least ## 1 ##. Then: there are polynomials ## q(x), r(x) \in \mathbb{F}[ x] ## s.t. 1. ## f(x)=q(x) \cdot g(x)+r(x) ## or 2. the degree of ## r(x) ## is less than the degree of ## g(x) ## Proof...
  4. J

    Metal Sphere and Charged Rod

    (A) incorrect, because opposite signs attract, and the sphere would've been drawn to the charged rod. (B) correct, according to the answer key, but if the charge of the sphere and the charge of the rod are the same, then wouldn't they repel each other? I'm confused as to why this is the correct...
  5. Jason-Li

    Engineering Mutual Inductance between two conductors

    Hello hope everyone at PF are keeping well! Looking at this problem but don't know if I have oversimplified it and my final answer doesn't seem right? So I drew the equivalent circuit: Then went onto calculate: I1 = 1/(R1+RL1) = 0.01A ω = 2πf = 2π*109 rads-1 so then with Vi being the inducted...
  6. Ale_Rodo

    "Barlow's wheel" used as an electrical generator

    In this thread, I hope to find some help in understanding one of the first application of Faraday's law of induction: the "Barlow's wheel". Basically the machine converts electrical power to mechanical, so as you can imagine, a battery, some conductor wires, a horseshoe magnet and a metal wheel...
  7. C

    Det of Triangular-like Matrix & getting stuck in Algorithmic Proof

    Find determinant of following matrix: ## A = \begin{pmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n-1} & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n-1} & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{n,1} & 0 & \cdots & 0 & 0 \end{pmatrix} ## Note: I tried to solve this question...
  8. M

    Help please solving this electromagnetism problem -- coil moving near a magnet

    A small coil is moved forward (without turning) between the poles of the electromagnet. Does an electric current develop in the coil? Explain the answer
  9. roybot

    Motion-powered GPS tracker to install permanently inside a bicycle frame

    Briefly, the problem; stolen bicycles are notoriously difficult to recover. Designing a GPS tracker to fit on one is challenging, because bicycles rarely have their own available power source. Whatever GPS unit is used must typically have its own battery as its sole source of power, which...
  10. C

    Electrostatic induction - how to quantify it with a theoretic integral or similar

    Assume that a certain charge distribution ##\rho## generates an electrical field ##E_{ext}## in the surrounding space. We also note the corresponding generated potential ##V_{ext}##. Assume furthermore that a conductor A, with a definite shape and volume, is placed in field ##E_{ext}##, and is...
  11. Mr_Allod

    Electromagnetic Induction of a Disk

    Hello I'm having trouble finding the right way to apply Faraday's law to this question. I've found the flux through the disc: ##\phi = \vec A \cdot \vec B = B_{0} \sin{\omega t} \left( \frac D 2 \right)^2 \pi ## and the EMF: ##\varepsilon = - \frac {d \phi} {dt} = -B_{0} \omega \cos{\omega t}...
  12. C

    Simple Induction Help with Lemma for proof of AM-GM inequality

    Summary:: x Hey, I'm learning calculus and had to prove the following Lemma which is used to prove AM-GM inequality, I had tried to prove it on my own and it is quite different from what is written in my lecture notes. I have a feeling that my proof of the Lemma is incorrect, but I just don't...
  13. C

    Simple Induction Direct Proofs regarding Induction

    Summary:: . When asked to prove by Induction, i'm asked to prove a statement of the form: Prove that for all natural numbers ##n##, ## P(n) ## Which means to prove: ## \forall n ( P(n) ) ## ( suppose the universe of discourse is all the natural numbers ) Then, I see people translating...
  14. K

    Eddy currents in Faraday's Law Experiment

    My explanation: A circular coil is connected to an AC supply at a frequency of 30-50 kHz (radio frequency). Therefore, an alternate current will be running through this “primary” coil, producing an alternating magnetic field. This magnetic field periodically decreases in strength, alternating...
  15. S

    Current Induced in one circuit by another circuit

  16. S

    Conservation of energy when placing two inductors next to each other

    This is more like a theoretical question of my own than actual homework. Say there is a circuit with a current source and an inductor. There is a current ##i(t)=at## going through the inductor. We now place a new circuit with an inductor and a resistor next to it. The current ##i(t)## causes a...
  17. S

    Find the speed of a copper loop falling in a magnetic field

    Hi all, so I had this problem and on the exam and I got a solution but I had an mass-term in there which wasn't given. I used Farraday's Law of Induction to get the Voltage induced. Then I used ##rho* \frac{A}{4a} ## for the resistance and divided the Voltage by that to get the current. I then...
  18. D

    50Hz 3 Phase Circuit - Twisting to reduce EMI?

    I have done the maths to work out the Induced Voltage on a communication cable, from a bunch of 3 phase circuits in a panel - with the minimum distance between them I calculated it to be about 1.5nV; but for a larger system where there could be larger loads on the cables I was thinking about...
  19. S

    Force on a copper loop entering into a magnetic Field B with speed v

    Hi, second problem in one evening, I'm sorry! But i'm also not quite sure if I did this one right. I had thought I need lenz's law but there is no current before entering the field so I just use the induced Voltage? My approach: ## V = \frac {B*A}{t} ## ## IR = \frac {B*A}{t} ## and ## A = v*t...
  20. P

    How to calculate the voltage induced in a coil by an AC magnetic field?

    So, as it says in the title, I am trying to calculate overall voltage induced in a coreless coil in the cases of it being stationary and moving in an alternating magnetic field. To go more into detail, I would like to create a mathematical model of a coil in an alternating magnetic field that...
  21. S

    Prove the decomposition of a graph w/ even edges produce a 2-path set

    For my base case I just used a graph with three vertices and 2 edges. Decomposing this would just give us the same graph, which has a path length of 2. The inductive step is where I'm having some trouble: One idea I have is that we take a graph G then inductively remove an edge to create two...
  22. S

    Proof by Induction: Arithmetic Sum

    Hi, I am self studying induction and came across the following problem. I am stuck on how to proceed (I need to use induction, I know there is a direct proof). My proof attempt is as follows: Let ## P (m) ## be the proposition that: $$ \sum_{i = m + 1}^{n} i = \frac{(n - m)(n + m + 1)}{2} $$...
  23. K

    Interesting Magnetic Field Induction Problem

    Homework Statement Suppose we have an isolated, long, narrow straight wire with low electrical resistivity. A constant current ##I## is sent through this wire. We know that if an electron is sent on a path which is perpendicular to the wire, towards it, with an initial speed ##v_0##, and the...
  24. atommo

    Stupid question relating to electric induction

    Hi, I've been interested in the science behind electrons/magnetism for quite a while. I've been learning quite a bit from various sources online. However there is one thing that's really nagging me. Magnetic fields result from moving electrons. That indicates that a permanent magnet has...
  25. I

    Maxwell's Law of Induction - How Does it Work?

    So, I was studying Maxwell's equations and I don't really understand the last one - Ampere's Law (with Maxwell's extra term added in). The bit I'm not able to understand is the term Maxwell added. How exactly does a changing electric field through a closed loop induce a magnetic field along that...
  26. Celso

    Induced EMF — generator coil area calculation

    Homework Statement What is the necessary area for a generator that produces an emf of ##\mathcal{E} = 150V## when it spins at a ratio of 60 revolutions per second, in a magnetic field of ##B = 0.5 T##? Homework Equations ##\oint_{c} E \cdot dl = \mathcal{E} = -\frac{d}{dt}\iint_{s} B \cdot dS...
  27. U

    Proof via mathematical induction

    Homework Statement Use mathematical induction to prove that (8n − 7n − 1) is divisible by 49 for any n ∈ N. Correction by mentor for better readability: ##49\,|\,(8^n-7n-1)## The Attempt at a Solution We can see that the base case is satisfied here: n = 1, 8^1-7*1-1 = 0 and 49 | 0 is true...
  28. mheruian

    Winding Direction Consideration (Proper Winding)

    Hello, I'm boy who suddenly been strucked by curiosity on how hand crank generator works and i watch and read some of how to create one, specially this one on youtube: and i want to create one but i don't have a 3D printer, but i saw some alternative but as i do/design my own coil based on...
  29. U

    Calculation of Change in Magnetic Flux Linkage Across a Wire

    Homework Statement A straight wire of length 0.20m moves at a steady speed of 3.0m/s at right angles to a magnetic filed of flux density 0.10T. Use Faraday's law to determine the e.m.f. induced across the ends of a wire. Homework Equations E= Nd Φ/dt but N=1 so E= dΦ/dt The Attempt at a...
  30. R

    Solar PV:What happens to the lost energy?inductive recovery?

    Hi there, I am new in here, thanks for any reply. I took same title of a previous closed 3D by pranj5. As understood main part of the energy absorbed by PV panels goes in reflection and heat. The heat origins mainly in electrons that are excited but not enough to jump or they jump but they...
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