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.
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...
Three phase squirrel cage induction motor with given parameters (stator winding connected in delta)
nominal power: Pn=22.4kW
nominal stator voltage: Vsn=230V
nominal stator current: Isn=39.5A
nominal stator frequency: fsn=60Hz
nominal speed: nn=1168min−1
number of pole pairs: 3
stator...
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...
We know that both the interior and the surface of an electrostatically balanced conductor are equipotential. My question is if when we approach the loaded objects, the surface of the conductor will continue to be an equipotential. If not, then there could be a field line that left the region...
Hello,
I am studying the working principle of an induction motor. The attachment below (link) is a portion from Stephen J. Chapman's textbook (page 160-169). He proved the theory of the rotating magnetic field so well for a 2-pole induction motor from 160 to 164. However, on page 166, the...
If charging some smaller toy, phone, electric razor, toothbrush or the like with induction charging I assume that the speed (time it takes to charge an empty battery) will depend on
1. the battery to some extent, but for this sake let's omit that
2. the inductive charging it self
3. wall socket...
The study material says under 'Estimation of magnetising losses' :On no-load, the rotor current is very small and the slip almost zero. Therefore, R′2s is very high and virtually no current flows through the branch of the circuit containing R′2s. Consequently, the no-load current is virtually...
On examining Maxwell's third equation which is about time varying magnetic fields (Faraday's electromagnetic induction) we find that time varying magnetic fields produce loops of electric fields in space irrespective of whether a coil is present or not, if any coil is present then these loops of...
Hello everyone,
I have an old Hobart A200 mixer that I've rebuilt and the problem is the agitator shaft is spinning twice the rated speed when the motor comes up to speed. It has a repulsive induction motor. Speed 1 starts at about the rated 60 rpm but spins at about 120 when at running speed...
If we have a small dielectric sphere and a point charge, they will experience an attractive force due to electrostatic induction. (From the elongation/rotation of charges bound to individual atoms).
Likewise, if we have a small metallic sphere and a point charge, they will experience an...
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
I have gone through the principle of mathematical induction. I cannot understand why do we need to prove every statement for n=1. I mean why is it necessary? Why can't we start directly from n=k then n=k+1. For example see the below image. Thanks!
hello PF. I have this exercise to do but I don't know how to do it.
Let's assume that the core in the figure is made up of ferromagnetic permeability material
μ = 2000 μ0. Let a = 4 cm, b = 2 cm, c = 3 cm the geometric dimensions, S = 8 cm2 the section
transversal, N1 = 100, N2 = 300 the number...
Summary:: A planar condensator with the distance between the plates d, is moving with velocity V. The charge fidderence is U. Find the magnetic induction B, between the plates during the motion
I have solved this problem, but I am not sure about the solution, can anyone, please, tell me if I...
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...
A friend has a single phase, cap start induction motor with the following characteristics:
V 240/250
A 2.6
Hp 1/3
Rpm 1435 @ 50Hz
As for as I know it’s not fitted to a machine, but given the low hp rating I’d guess it’s for a drill press. So it would need only moderate to low starting torque...
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...
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}...
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...
this is a solution posted by my colleague, i have a problem in understanding how he got to conclude on equation (ii)
is this not supposed to be ##(k+2)!≥ 2^{k+1} (k+1)##
##(k+2)(k+1)!≥ 2^{k+1} (k+1)## ?...
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...
Hii Friends.
I was trying to model an Induction motor in ANSYS Maxwell 2D. The final output of Transient torque (negative torque) looks like shown below. The input passed is a sine wave current with 120 deg phase shift. Any Suggestion will be highly appreciated
b_{k} = b_{k - 1}/2 +b_{k-1}
b_{0} = 1
What would be the sequence for this expression, I calculated it to be 1, 2/3, 2/5, 2/7 ...
Is it right?
My explicit formula is b_{n} = 2/n+2
What would be the explicit formula in your view and how can that formula be proved by mathematical...
find the general rule and prove by induction
1 = 1
1 - 4 = -(1 + 2)
1 - 4 + 9 = 1 + 2 + 3
1 - 4 + 9 -16 = -(1 + 2 + 3 + 4)
I created this so far, but don't know if I am even going the correct direction
Faraday first demonstrated his law of induction by showing that when he created a current in one wire coil it created a second current in a second coil.
Another experiment showed the same effect. When he moved a magnet through a wire coil, it created a current in the coil.
My question is this...
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...
Consider the static field configuration shown in the image. There are three layers: 0 = vacuum, 1 = magneto-optic fluid and 2 = covering shell. Each of these layers have their own permittivity and permeability (ε_i,μ_i) (isotrope). A uniform electric field H_0 = H_0/sqrt(2) * (e_x + e_y) is...
So I was watching this video containing DIY experiments on electromagnetic induction .
At minute 4:45, the dude pretty much creates a transformer without using an iron core.
He runs 30-50 kHz AC in a coil (forming the primary circuit) and then brings another coil with its ends attached to a...
I will use examples of actual products:
1) 3 hp (2.2kW) 3 phase 2 pole AC Induction Motor
3 hp/2.2kW three-phase induction motor features 2 poles, 2840rpm rated speed, and 7.4Nm nominal torque, with a working voltage of 380V 50Hz. Cheap 3-phase asynchronous motor has outstanding performance in...
Summary:: prove that (n 0) + (n 1) + (n 2) + ... + (n n) = 2^n is true using mathematical induction.
note that (n n) is a falling factorial
Hello! I have trouble dealing with this problem:
Mod note: Thread moved from math technical section, so is missing the homework template.
Prove that (n...
In electromagnetic radiation, the electric field and the magnetic field mutually induce each other: but my impression is that it would be better to say that they are two aspects of the same wave, so that there is no time between them. However, an alternative would be that there would be that...
Hello;
I am trying to come up with an induction heater design for a machine that i am working on and am having some trouble planning it out.
Anybody have some good resources? I have scoured Google and there are tons on there. I am a bit unsure how to get started. I have an induction heater...
I'm setting up a Faraday/Lenz Law lab and was wondering if anyone had a suggestion on what gauge of wire I should use to get the best results. We don't have any wire here so I can't test it myself.
Thanks.
Hey! :o
1. Let $1\leq n\in \mathbb{N}$ and $\pi\in \text{Sym}(n)$. For $1\leq k\in \mathbb{N}$ we define $\pi^{-k}:=\left (\pi^n\right )^{-1}$.
Show for all $k,\ell\in \mathbb{Z}$ the equation $\pi^k\circ \pi^{\ell}=\pi^{k+\ell}$. 2. Let $1\leq n\in \mathbb{N}$. Show that $\pi^{n!}=\text{id}$...
I can create a fairly strong magnetic field by leading current through a copper coil. However, I cannot then with that magnetic field create a current (not a useful one at least) in an object made out of copper. I understand that this is because copper lacks of magnetic properties. For instance...
I think the initial assumptions would allow me to prove this without induction.
Suppose ##(x_n)## is a real sequence that is bounded above. Define $$ y_n = \sup\{x_j | j \geq n\}.$$
Let ##n \in \mathbb{N}##. Then for all ##j \in \mathbb{N}## such that ##j \geq n + 1 > n##
$$ x_{j} \leq y_n.$$...
As the summary says:
I have 2 induction heaters which are exhibiting different results when heating the same product. The 2 are different, having different sized coils, and I would like to work out the strength of the induced field in each of them (with a common item being heated) to establish...
Hello all,
In the attached picture there is an equation. I need to fill the general expression on the left hand side, and to prove by induction that the sum is equal to the expression in the right hand side.
I am not sure how to find the general expression. Can you kindly assist ?
Thank you !
Was curious if induction water heaters were a thing and if they had any benefits over a traditional electric resistance coil. On quora someone argues that a resistance coil is a near perfect conversion of electricity to heat, therefore no benefit to induction - is this correct? Is the...
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...
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...
$2^{n+2} < (n+1)!$ 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 $2^{k+3} < (k+2)!$