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. P

    Electromagnetic induction in current loops

    would be really grateful if someone could help throw some light on this - its probably really simple but I am typing myself up in loops! I have a piece of wire which is placed in an external AC magnetic field which is varying sinusoidally. The wire is folded to make two rings of radii r1 and r2...
  2. Eclair_de_XII

    Is my short induction proof correct?

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  3. P

    Building speakers, induction and induced current

    i made a speaker from a cup, a coil of wire and a magnet. I hooked up both of the ends of the wire to a headphone jack and it works. Albeit very low sound, but it works. Can you point me in the direction of some equations that will show me the most influential factors on how to make it louder...
  4. K

    Is an induction coil used in a Gaussmeter?

    hi, i am newbie to this forum and physics i should say. i wanted to ask the physics guys whether induction coils are used in gaussmeters? thanks
  5. S

    I Hall Effect & Magnetic Induction Explained

    Can someone explain in detail if halls effect is magnetic induction?
  6. F

    Calculating Magnetic Flux Change in Moving Coil Experiment

    Homework Statement We have a coil with cross-sectional area of 2.4m^2. Magnetic flux density is of 0.29T Number of turns are 1. Initially the coil is parallel to magnetic field lines. (Coil's normal vector is perpendicular to field vector) Calculate the change in magnetic flux when coil is...
  7. B

    Question about Magnetic induction of circular loop

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  8. cnh1995

    Zero sequence braking of induction motor

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  9. X

    Why does induction in a loop depend on the contained flux?

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  10. M

    Proof by induction, ##(n)^{2} \le (2n)##.

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  11. G

    Some confusion of Faraday's law of induction.

    Hello. I have several confusions regarding Faraday's law of induction. EMF = \int_{}^{} {\vec E \cdot d\vec l} = - \frac{{d\Phi }}{{dt}} = - \frac{d}{{dt}}\int_{}^{} {\vec B \cdot d\vec S} . It means that If the magnetic flux Φ through the closed conducting loop changes in time, electric...
  12. UsableThought

    Help with algebraic deduction steps in a proof by induction

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  13. Tobi

    Faraday's law or Lorenzt force?

    A student in my physics class posted, in a group, a wrong answer to a question. The situation was: A plane has a wire extended between the tips of its wings and flies through a magnetic field, perpendicularly, while accelerating. The question was: What will be the induced current? His answer...
  14. Xubi

    What is the curl of the electric field produced by changing magnetic flux?

    Homework Statement My book says that the electric field produced by changing magnetic flux is nonconservative in nature but it doesn't give any explanation. Homework EquationsThe Attempt at a Solution
  15. Xubi

    Exploring Electromagnetic Induction: A Rotating Ring in a Magnetic Field

    Homework Statement A ring of mass m, radius R having charge q uniformly distributed over it and free to rotate about its own axis is placed in a region having a magnetic field B parallel to its axis. If the magnetic field is suddenly switched off, the angular velocity acquired by the ring...
  16. L

    Induced current in a metal ring

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  17. P

    Lorentz force on a charge due to a moving magnet

    Let's say I have a charge q which is viewed from its rest frame. So it's velocity v is 0. So the so-called magnetic component of its Lorentz force, which is q v x B, is 0. But I can have a magnet moving in this frame of reference. Let's say the velocity of this magnet according to the charge's...
  18. R

    Discrete Math - Strong Induction Question

    Homework Statement Prove by Strong Mathematical Induction[/B] Homework Equations N/A The Attempt at a Solution The steps to solving this problem are shown below. I understand all steps of the problem until the part where it says 44/49 becomes 49/49 since 44 < 49. Can someone please explain...
  19. M

    MHB Induction to find the determinant

    Hey! :o Suppose we have the matrix $A_n=(A_{ij})\in \mathbb{C}^{n\times n}$ with $a_{ij}=\left\{\begin{matrix} 1 , & i=j\\ -1 , & i=j-1\\ j^2, & i=j+1\\ 0 , & \text{ otherwise} \end{matrix}\right.$ for $1\leq i,j\leq n$. I want to find the determinant using induction. I have done the...
  20. A

    How Efficient Is Split-Return Coil Design for Quenching in Induction Heating?

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  21. P

    Exploring Tesla's Autos & Induction Motors

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  22. harini07

    Electrostatics: Charging a sphere by induction

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  23. G

    Thomson's jumping ring: Why net repulsion?

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  24. G

    Faraday's law for constant velocity, area and B-field

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  25. CynicusRex

    Induction: x+1/x is an int. Prove that x^n+1/x^n is an int...

    Homework Statement Algebra I.M. Gelfand, problem 134. You know that x+1/x is an integer. Prove that xn+1/xn is an integer for any n = 1, 2, 3, etc. Homework EquationsThe Attempt at a Solution I don't fully understand how the induction below proves anything. (Source) I did attempt the solution...
  26. B

    How Can Hydraulic Flow Be Controlled Between Two Pistons?

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  27. Clara Chung

    Question about induction cooker

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  28. M

    Induction Electric - Does Iron/Steel Work?

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  29. Asmaa Mohammad

    B The factors determining the induced EMF in a wire

    The induced emf in a straight wire is determined by the equation (emf=Blv sinθ) where θ is the angle between the direction of the motion and the lines of the magnetic field, and frequently, I see diagrams like these explaining the phenomenon: In those pictures the wire is perpendicular to the...
  30. A

    Practical Induction -- What Provides The Most Power?

    I'm having a hard time finding a starting point for this question. I'm looking for all the variables that can effect the amount generated by induction, and then hopefully ordering them from the most important to least important. Any references to a good starting point for someone who hasn't...
  31. Robertphysics

    Is Induction Heating More Efficient Than Resistive Heating for Water?

    I know for them for some time and I understand how induction heating works in general , where you have a coil through which a high frequency AC current is fed that creates a magnetic field which changes fast according to the frequency and this excites electrons in the metal introduced and heats...
  32. N

    Shield against induction of sounds from remote source of RF

    It has been demonstrated that it is possible to make humans hear sounds using waves. Abstract From Journal of Applied Physiology Is there a practical method for shielding? I mean using materials for simple attenuation, and/or altering characteristics of waves (frequency, phase and/or amplitude).
  33. hackhard

    Graphite electromagnetic induction

    since graphite conducts. if spiral shape is drawn with pencil on the wall (thick , dark lines) will a voltage difference appear between inner and outer end of the spiral when a varying magnetic field is switched on, normal to wall
  34. A

    Questions about batteries and magnets and EM induction

    Hello, I wanted to ask a few questions regarding magnets and batteries, and EM induction. First off, could I use batteries to replace a low voltage dc power supply, like say 2 V or so by connecting the ends of the battery in series to a circuit? As an extension, could I connect two batteries in...
  35. Davide Fabri

    Induction Heating Coil Design: Rectangular vs. Spiral Shape

    Goodmornig everybody, I'm working on the design of a power converter for high frequency induction heating. The induction coil should be a flat spiral (pancake) with a rectangular shape, but my customer would like to design it as shown in the following picture. I think it will not work because...
  36. Rod Alexei

    Direction of Electric field (EM induction)

    Homework Statement In which direction would the electric field in the wire point if the wire were pulled to the right? Homework Equations Right hand rule F=qvXB The Attempt at a Solution My answer is downward. I am confused whether I should take the moving charges as positive or negative...
  37. Clara Chung

    Question about magnetic induction.

    Homework Statement Homework EquationsThe Attempt at a Solution For cii), the answer is: Yes. When the rod moves down, it moves across the horizontal field, cutting field lines. A current will be induced to oppose the motion. The magnetic force due to the motion is vertically upward. But I...
  38. J

    What is the proof for n(n^4 - 1) = 10Q for some values Q and n being integers?

    For some values Q and n being integers, prove that n(n^4 - 1) = 10Q. So I've tried this with induction, but it gets pretty messy pretty quickly. So I can see that the LHS will be even no matter what, but I'm not sure where to go beyond this.
  39. Jhon Paul

    How Do You Calculate Frequency and Power for an Induction Heater?

    I would like to get a clue of how to arrive at the power and frequency for an Induction heater. I am planning to heat a graphite rod upto 1200 degC and also to provide a heat flux of 1 MW/m2. The graphite rod dimension will be for one case Dia 5 and length 100 mm and in the second case will be...
  40. mikemartinlfs

    Electromagnetic Induction (Long straight wire above circulua

    Homework Statement A circular loop of wire is placed next to a long straight wire. The current I in the long straight wire is increasing. What direction is the current that's induced in the circular loop? Homework Equations The Attempt at a Solution I'm aware that the answer is clockwise...
  41. MarkFL

    MHB How can we prove the recursive definition of $S$ by induction?

    I was recently sent the following question via PM (which we discourage here), so I thought I would start a thread and help here (and give the OP a link to this thread). Here's the question: (a) We can observe that we may write...
  42. Mzzed

    Voltages in an induced current

    I've been messing around with ampere's and faraday's laws as we have recently been applying them in undergrad level physics. I'm confused as to how voltage fits in with these laws when used for a solenoid inducing a current in a material placed inside the solenoid. I know that the induced...
  43. dormo715

    Reducing Radiation Heat from Induction Furnace on Lifting Magnet

    http://lookpic.com/O/i2/73/DYktKEGg.jpeg Hello Dear Friends We produce Lifting Electromagnet for Steel Scrap charging of Induction furnace, bottom plate of electromagnet is made of Stainless steel . my question : Can We use AL OR SILVER Spray for Spraying bottom plate that Radiation heat...
  44. R

    Calculating Poles in a 3-Phase Induction Motor

    how many poles does 3 phase, 60 hertz induction motor which runs at 715 r/min at no load and at 670 r/min at full load have and how is it calculated?
  45. H

    Resulting magnetic flux in the core of a transformer

    In a transformer, let's say we have: I1, I2 - currents through the primary and secondary winding V1, V2 - voltages N1, N2 - number of turns F1, F2 - magnetic fluxes through core, produced by the currents I1 and I2 (they are opposing...) R - the reluctance of the core We have V1/V2 = N1/N2...
  46. Asmaa Mohammad

    Using two electromagnets to study self-inductance

    Hi, I study physics and I know what the concept self-inductance means, and in one of my study books, a figure like this was used to explain the self inductance in a coil As shown in the figure, two electromagnets are connected together and the current is passing through them and would produce a...
  47. Jaroslav

    Prove divisibility, mathematical induction

    I'm still learning English, had to use dictionary and translator, so I'm sorry if its unclear, i will try to explain it more if needed. Homework Statement For n belonging to N when n is even and n > 3, prove that (4^(n-3) + 5^(n-3) + 9) is divisible by 9 Homework Equations 3. The Attempt at...
  48. dumbdumNotSmart

    Induced current of a coil on another coil

    Homework Statement We have two coils of copper wire, drawn below. One has a battery connected to it. The other one has a galvonometer connected. If the coil with the voltage is moved closer as shown in the picture, with considerable velocity, what direction will the current induced on the other...
  49. F

    MHB Question regarding pigeonhole method , number induction

    Hi i have few questions over here and need some expert/pro view with approaching the question. The proving one would be an issue to me as well. Hope people would be able to assist me here , because i am basically struggling with it :( thanks Questions : My Working : No idea how to...
  50. Z

    Engineering How Do You Calculate Current and Power Factor in a Synchronous Induction Motor?

    Homework Statement A star-connected, three-phase synchronous induction motor takes a current of 10 amps from a 415 volt supply at unity power factor when supplying a steady load. If the synchronous reactance is 5 ohms/phase and the resistance is negligible, draw the phasor diagram and...
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