Induction Definition and 999 Threads

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. Stephen Bulking

    Electromagnetic Induction exercise, is something wrong with these answers?

    1. |ε|=dΦ/dt B=μ0I×cos(0-cos180)/4πr =μ0I/2πr dΦ=BdS=μ0I×(vdtxdr)/2πr ΔΦ=(μ0I×(vdt)/2π)x∫dr/r=(μ0I×(vdt)/2π)xln(b/a) *the intergral goes from a to b |ε|=dΦ/dt=(μ0I×v×ln(b/a)/2π) i=ε/R=(μ0I×v×ln(b/a)/2πR) 2. dI/dt=-50 B=μ0NI/L ε=-dΦ/dt=-BdS/dt (*) =-NBS/dt (wait what?)...
  2. PainterGuy

    How Does Lenz's Law Apply to Induction Motors?

    Hi, The shown below, Figure 1, is an interaction of magnetic fields of stator and rotor of a squirrel cage induction motor. I believe the current is coming out of the squirrel cage bar on the left and entering on the right as is shown below, Figure 2. I have assumed that the bars highlighted in...
  3. TateTheL

    Electromagnetic Induction -- Insane Idea

    Would you be able to transfer a "current" through rings of conductive and paramagnetic material? As in, have a current, which causes a magnetic field to pop up in the ring, which fluxes, and causes the next ring to have a current? I am curious because this is a research project for FLL. (There...
  4. C

    Induction proof verification ##2^{n+2} < (n+1)## for all n ##\geq 6##

    $2^{n+2} < (n+1)!$ for all n $\geq 6$ Step 1: For n = 6, $256 < 5040$. We assume $2^{k+2} < (k+1)!$ Induction step: $2 * 2^{k+2} < 2*(k+1)!$ By noting $2*(k+1)! < (k+2)!$ Then $2^{k+3} < (k+2)!$
  5. Monoxdifly

    [ASK] Mathematical Induction: Prove 7^n-2^n is divisible by 5.

    Prove by mathematical induction that 7^n-2^n is divisible by 5.What I've done so far:For n = 1 7^1-2^1=7-2=5 (true that it is divisible by 5) For n = k 7^k-2^k=5a (assumed to be true that it is divisible by 5) For n = k + 1...
  6. I

    Electromagnetism EMF induction calculations

    A square conducting loop of side length a is in a non-uniform magnetic field. The loop occupies the first quadrant of the xy plane, i.e. the space between the origin (x, y) = (0,0) and the point (x,y) = (a, a). The magnetic field is in the +z direction. Develop an expression for the magnitude of...
  7. D

    I tried building an induction coil

    i tried to build an induction coil. the input voltage is 12.1 volts. the output voltage is 16.5 millivolts maximum. so the voltage in the primary coil is 12.1 volts. the voltage in the secondary coil is 16.5 millivolts maximum. the induction coil consist of a nail. thicker coils form the...
  8. M

    MHB Induction: Each square can be covered by L-stones

    Hey! :o A square with the side length $2^n$ length units (LU) is divided in sub-squares with the side length $1$. One of the sub-squares in the corners has been removed. All other sub-squares should now be covered completely and without overlapping with L-stones. An L-stone consists of three...
  9. K

    Efficiency of heating up aluminum vs steel by induction

    Summary: In a heating by induction experiment performed, the idea was to determine the efficiency of heating up a small steel cylinder, an aluminum cylinder and then compare the two efficiencies. The cylinder was surrounded by a metall coil that alternating current was going through...
  10. jisbon

    Showing that integral is an induction

    Hi all, Having this equation derived: ##\int _0^{\frac{\pi }{2}}\:sin^{n}x\:dx\:=\:\frac{n-1}{n}\int _0^{\frac{\pi }{2}}\:sin^{n-2}x\:dx## What I will do is simply substitue n with n+2, and I will get the following: ##\frac{2n}{2n+1}\int_{0}^{\pi /2}(sinx)^{2n-1}dx## What should I do from here?
  11. N

    Exploring Induction in Rotating Spheres: Is My Solution Valid?

    Let ##(r,\phi, \theta)## be the radial, polar and azimuthal coordinates respectively. As ##\vec{B}## is confined to ##xz## plane such that ##\theta = \alpha## I assumed ##\vec{B}## on the surface of shell to be ##\vec{B} = a\sin(\alpha) \hat x + \cos(\alpha) \hat z \tag{1}## Surface area...
  12. 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...
  13. Campeze

    Induction heating internal splines

    I want to surface heat internal splines with induction heating with an internal diameter coil. It is a cylinder, uniform shape, and the teeth are 1mm wide and 2mm deep. The inside diameter of the hollow section is 19mm and the length is 55cm. I was wondering when calculating power/resistance...
  14. F

    Calculating Current in Moving Rods: Exploring the Blv Rule and Faraday's Law

    Homework Statement: The rod is moving on a second rod with speed of v in the magnetic field perpendicular to rod. I know its resistance per meter. I am expected to omit megnetic field created by the current and resistance at the point where rods touch each other and calculate current. I...
  15. Mathew Joseph

    How to make the power factor of induction motors be a leading power factor?

    Hi guys, This month onwards our power distributor says that we only get powerfactor incentive when only we make the powerfactor leading and in between 0.95 and 1. Our industry fully consists of induction motor and compensative capacitors are provided but this month onwards we have to keep the...
  16. D

    Where Does the N in Faraday's Law of Induction Go?

    Hello Lets take an example: imagine a horizontal magnetic field, then a wire of length L. I push the wire with a force F through the field perpendicularly with respect to the magn. field for a distance of dS. EMF = work/charge --> F*dS/I*dt, where F = flux density*current*lenght of conductor...
  17. nagyn

    Induction stovetop design: am I using these equations correctly?

    I've been told an effective induction stovetop needs to deliver about 1000W of power. I have magnet wire that can tolerate at most about 0.2A of current, and was planning on using a 60Hz wall outlet as my source (obviously I'll need to bring down the outlet current quite a bit). So the energy...
  18. A

    Confused about Polarization and Induction - Static Electricity

    Hello. Recently we began taking static electricity and our teacher asked us to do a research paper on static electricity and charges in general to help us understand the material better. I've been stuck a bit on trying to understand what's the difference between polarization and induction and...
  19. Z

    Comp Sci Induction Proof for all strings: Can't understand the Question

    Hi, Can some body please explain me the following question: Use induction on ##n## to show that ##|t^n| = n |t| ## for all strings ##t## and all ##n## . Any idea how to that. I know we have a base case and an induction case but what would be the base case and what would be the induction case...
  20. gdritz

    Molten metal temperature vs Induction Coil Current

    Hi guys, I'm new here. I have been analyzing the behavior of cast iron melts in induction furnaces and realized that at the end of the melt the current increases slightly. The power remains the same. Physically, does anyone know the reason? I imagine to be related to the magnetic permeability...
  21. 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...
  22. C

    Engineering Understanding Windage/Friction Losses in Induction Motors

    So I've attached my graph and found that the windage/friction losses occur roughly around 230Watts. I just don't know how i further explain the magnetising current I0. I know I0 = input current on no load. So do i assumed from P = √3 x VL x IL x CosФ that CosФ=230/√3*220*6.8= 0.0849 or...
  23. 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...
  24. 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...
  25. bob012345

    Exploring the Role of the Vector Potential in Faraday's Law of Induction

    Suppose there is an almost infinitly long but narrow solenoid with an AC current surrounded by a much larger loop such that there is no magnetic field except in the solenoid. I had always thought it didn't matter what part of the outer loop the flux changed in, there would be an induced electric...
  26. T

    Induction heating flat spiral springs

    Summary: is it possible to heat a coiled spring If I had a spirally coiled spring like inserted pic, spiral coiled spring approx. 300mm dia and spring section +-12mm dia with +-1mm gauge wire, with a total length of spring approx. 4mtr-6mtr , is it possible to heat this with induction heating...
  27. 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...
  28. A

    AC Induction motor startup current

    Friend gave me this task he had back in school to calculate the necessary amperage for a fuse that is put in series with one of the phases for a 3 phase induction motor. I could calculate the working current of the motor from the rated kW and cos voltage/current angle, but I couldn't figure out...
  29. 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...
  30. S

    Prove using induction that if a graph has no odd cycles it is bipartite

    We learned in class how to do this problem the more traditional way, but we are required to reprove this using induction, which I'm not too sure how to do. My Attempt: For the base case I had three vertices connected by three edges, with the last one looping back to the first to create a cycle...
  31. S

    I Prove that a given graph contains a cycle using induction (Graph Theory)

    My Problem: Given a Graph G = (V,E), where the number of vertices is less than or equal to the number of edges, use induction to prove that the graph contains at least one cycle (the graph is not required to be completely connected). My attempt: For my base case, I used only one vertex with one...
  32. cnh1995

    Wrong answer key for this Induction Motor question?

    So this question was asked in GATE-2019 (EE). (GATE is the national Engineering post-graduate entrance exam in India and is conducted every year by one of the 7 top IITs (Indian Institute of Technology) and IISc Bangalore. This year, it was conducted by IIT Madras.) The answer I found in the...
  33. M

    Engineering Induction Motor's Speed at Maximum Torque

    Maximum Slip s=4/35 Synchronous Speed ns=50*60/p ns=3000 rev min-1 Speed at Maximum Torque Torque is maximum when R2=sX2 For maximum torque, X2=R2/s I'm not sure after this.
  34. themountain

    How can I calculate the magnetic field using electromagnetic induction?

    gotta be honest I am really struggling
  35. 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: $$ \sum_{i = m + 1}^{n} i = \frac{(n - m)(n + m + 1)}{2} $$...
  36. T

    Weak Induction implies Strong Induction

    Homework Statement [/B] Weak Induction: If (i) ##S(1)## holds, and (ii) for every ##k \geq 1(S(k) \Rightarrow S(k+1)##. Then ##\forall n \geq 1##, ##S(n)## holds. Strong Induction: If (i) ##S(k)## is true and (ii) ##\forall m\geq k [S(k) \land \cdots \land S(m)]\Rightarrow S(m+1)##. Then for...
  37. Rayanna

    How Does Coil Configuration Affect Solenoid Inductance?

    Two coils are made of copper wires of same length .In the first coil number of turns is 3n and radius is r . In the second coil number of turns is n and radius is 3r the ratio of self inductances of the coil is: I know that self inductance of a solenoid is μN2A/l ; where A = area of cross...
  38. D

    Mathematical Induction: Showing Sums of 3 Consecutive Integers

    On the outside rim of a circular disk the integers from 1 through 30 are painted in random order. Show that no matter what this order is, there must be three successive integers whose sum is at least 45.
  39. 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...
  40. N

    Challenging question about electromagnetic induction

    Homework Statement I have been stuck on this for weeks Homework Equations ∫E.dl = -dφ/dt The Attempt at a Solution Total EMF (V) = -dφ/dt (Where φ is the magnetic flux through the loop) ⇒V = -A(dB/dt) (Since Area remains constant) ⇒V = -Ax (x=dB/dt) ⇒V = -2xl^2 I do not know how to proceed.
  41. C

    MHB Proving General Associativity for Group by induction

    Dear Everyone, I am having some troubles with the problem. The problem states: Let $(G,\star)$ be a group with ${a}_{1},{a}_{2},\dots, {a}_{n}$ in $G$. Prove using induction that the value of ${a}_{1}\star {a}_{2} \star \dots \star {a}_{n}$ is independent of how the expression is bracketed...
  42. V

    Strong Induction: Proving Every Int n>1 is a Product of Primes

    Homework Statement Every positive integer n > 1 can be written as a product of primes. Proof: We prove the result by strong induction on n, where n≥2. Base Case: Note that 2 is prime, hence 2 = p1, where p1 is prime. Inductive Step: Let m ∈ ℤ with m ≥ 2 and assume for all integers k with 2...
  43. V

    Can the Horse Induction Proof Truly Confirm All Horses Are the Same Color?

    Homework Statement Let P(n) be the statement "In every set of n horses, all of the horses in the set have the same colour." Base Case: We must prove that P(1) is true. If our set only contains one horse, then all horses in the set have the same colour. Inductive Step: Let m ≥ 1 and assume P(m)...
  44. R

    Proof by Induction: Explaining Step 3 to 4 | Math Homework

    Homework Statement Attached are notes from class. Can someone please explain what happens to (-x(n+1)) in step 3 to step 4. Not sure why it goes away. Thanks! Homework EquationsThe Attempt at a Solution
  45. V

    Proofs By Induction: Proving P(n) for All n

    Homework Statement Suppose we want to prove using mathematical induction that for all positive integers n, 12+22+...+n2 = (n(n+1)(2n+1))/6. What do we need to prove in the inductive step of our proof? Homework EquationsThe Attempt at a Solution I am struggling to understand what this means...
  46. L

    Engineering Synchronous Induction Motor Phasor Diagram & Calcs Help

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  47. T

    Why Does a Voltmeter Show a Reading When an Iron Rod is Moved in Coil P?

    Homework Statement Two coils P and Q are placed close to one another. P is connected to a power supply whereas Q is only connected to a voltmeter and no supply (see attached diagram). The current in coil P is constant. An iron rod is inserted into coil P. a) Explain why during the time that...
  48. P

    Silicon not heating with induction heater?

    I’m using an inductive heater to try to melt some 99.85% polycrystalline silicon, as would occur in the Czochralski process, but the silicon workpiece is not melting let alone even getting hot. The silicon workpiece is about the size of a playing dice. It becomes conductive at 200 F and becomes...
  49. J

    Mathematical model of an induction motor

    A coil is modeled as Z = jwL or Z = sL or V = L dI/dt What are the equations for an induction motor? Single-phase one.
  50. Y

    Mathematical Induction 4+11+14+21+....+(5n+(-1)^n

    Dear all I am trying to prove by induction the following: I checked it for n=1, it is valid. Then I assume it is correct for some k, and wish to prove it for k+1, got stuck with the algebra. Can you kindly assist ? Thank you.
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