induction

Examples of induction in the following topics:

• Inductance

  • Specifically in the case of electronics, inductance is the property of a conductor by which a change in current in the conductor creates a voltage in both the conductor itself (self-inductance) and any nearby conductors (mutual inductance).
  • Self-inductance, the effect of Faraday's law of induction of a device on itself, also exists.
  • where L is the self-inductance of the device.
  • Units of self-inductance are henries (H) just as for mutual inductance.
  • The inductance L is usually a given quantity.

• Inductance

  • The answer is yes, and that physical quantity is called inductance.
  • Mutual inductance is the effect of Faraday's law of induction for one device upon another, such as the primary coil in transmitting energy to the secondary in a transformer.
  • The larger the mutual inductance M, the more effective the coupling.
  • Self-inductance, the effect of Faraday's law of induction of a device on itself, also exists.
  • where L is the self-inductance of the device.

• Logic

  • Francis Bacon (1561-1626) is credited with formalizing inductive reasoning.
  • "Bacon did for inductive logic what Aristotle did for the theory of the syllogism.
  • Statistical inference is an application of the inductive method.
  • While inductive methods are useful, there are pitfalls to avoid.
  • Abduction is similar to induction.

• Faraday's Law of Induction and Lenz' Law

  • This relationship is known as Faraday's law of induction.
  • The minus sign in Faraday's law of induction is very important.
  • As the change begins, the law says induction opposes and, thus, slows the change.
  • This is one aspect of Lenz's law—induction opposes any change in flux.
  • Express the Faraday’s law of induction in a form of equation

• Proof by Mathematical Induction

  • Proving an infinite sequence of statements is necessary for proof by induction, a rigorous form of deductive reasoning.
  • The assumption in the inductive step that the statement holds for some $n$, is called the induction hypothesis (or inductive hypothesis).
  • To perform the inductive step, one assumes the induction hypothesis and then uses this assumption to prove the statement for $n+1$.
  • This completes the induction step.
  • Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes.

• Sequences of Mathematical Statements

  • Sequences of statements are logical, ordered groups of statements that are important for mathematical induction.
  • Sequences of statements are necessary for mathematical induction.
  • Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true for all natural numbers.
  • For example, in the context of mathematical induction, a sequence of statements usually involves an algebraic statement into which you can substitute any natural number $(0, 1, 2, 3, ...)$ and the statement should hold true.
  • This concept will be expanded on in the following module, which introduces proof by mathematical induction.

• Different Lines of Reasoning

  • Apply two different lines of reasoning—inductive and deductive—to consciously make sense of observations and reason with the audience.
  • One important aspect of inductive reasoning is associative reasoning: seeing or noticing similarity among the different events or objects that you observe.
  • Here is a statistical syllogism to illustrate inductive reasoning:
  • The conclusion of an inductive argument follows with some degree of probability.
  • In order to engage in inductive reasoning, we must observe, see similarities, and make associationsbetween conceptual entities.

• Changing Magnetic Flux Produces an Electric Field

  • Faraday's law of induction states that changing magnetic field produces an electric field: $\varepsilon = -\frac{\partial \Phi_B}{\partial t}$.
  • We have studied Faraday's law of induction in previous atoms.
  • In a nutshell, the law states that changing magnetic field $(\frac{d \Phi_B}{dt})$ produces an electric field $(\varepsilon)$, Faraday's law of induction is expressed as $\varepsilon = -\frac{\partial \Phi_B}{\partial t}$, where $\varepsilon$ is induced EMF and $\Phi_B$ is magnetic flux.
  • Therefore, we get an alternative form of the Faraday's law of induction: $\nabla \times \vec E = - \frac{\partial \vec B}{\partial t}$.This is also called a differential form of the Faraday's law.
  • Faraday's experiment showing induction between coils of wire: The liquid battery (right) provides a current which flows through the small coil (A), creating a magnetic field.

• RL Circuits

  • Recall that induction is the process in which an emf is induced by changing magnetic flux.
  • Mutual inductance is the effect of Faraday's law of induction for one device upon another, while self-inductance is the the effect of Faraday's law of induction of a device on itself.
  • An inductor is a device or circuit component that exhibits self-inductance.
  • The characteristic time $\tau$ depends on only two factors, the inductance L and the resistance R.
  • The greater the inductance L, the greater it is, which makes sense since a large inductance is very effective in opposing change.

• Reasoning and Inference

  • Scientists use inductive reasoning to create theories and hypotheses.
  • An example of inductive reasoning is, "The sun has risen every morning so far; therefore, the sun rises every morning."
  • A faulty example of inductive reasoning is, "I saw two brown cats; therefore, the cats in this neighborhood are brown."
  • As you can see, inductive reasoning can lead to erroneous conclusions.
  • Can you distinguish between his deductive (general to specific) and inductive (specific to general) reasoning?