What is mutual induction

Sometimes programs rely on mutual recursion to do things. For example, here is an Agda program (taken from ∀ → odd m → even n → odd (m + n) e+e≡e zero en = en e+e≡e (suc om) en = suc (o+e≡o om en) o+e≡o (suc em) en = suc (e+e≡e em en) Another example is this (in Haskell) (from alternating :: [Int] -> Bool alternating l = (updown l) || (downup l) updown :: [Int] -> Bool updown [] = True updown [x] = True updown (x:y:ys) = (x Bool downup [] = True downup [x] = True downup (x:y:ys) = (x > y) && (updown (y:ys)) What are some other examples of mathematical proofs which are more naturally expressed with mutual induction? The most non-trivial example I can think of is the proof of Strong Normalization Theorem for Simply Typed Lambda Calculus (see page 42, section 6.2 In my opinion, the strong normalization theorem for simply typed $\lambda$-calculus is not a negligible result, at least in theoretical computer science (which is much closer to mathematics than software development or web programming). In general, proofs by mutual induction are quite common in theoretical computer science, which deals with discrete objects often defined by induction or mutual induction. In general, a proof by mutual induction of a statement $A$ consists in proving a stronger statement than $A$, usually a statement of the form $A \land B$ (i.e. $A$ and $B$), where $B$ can be seen as an "auxiliary statement". The advantage is that now one can use a stronger induction hypothesis than in a proof of th...

https://phys.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fphys.libretexts.org%2FBookshelves%2FUniversity_Physics%2FBook%253A_University_Physics_(OpenStax)%2FBook%253A_University_Physics_II_-_Thermodynamics_Electricity_and_Magnetism_(OpenStax)%2F14%253A_Inductance%2F14.0E%253A_14.E%253A_Inductance_(Exercise) \( \newcommand\) • • • • • • • • • • • • • • • • • • 1. Show that \(NΦ_m/I\) and \(ε/(dI/dt)\), which are both expressions for self-inductance, have the same units. 2. A 10-H inductor carries a current of 20 A. Describe how a 50-V emf can be induced across it. 3. The ignition circuit of an automobile is powered by a 12-V battery. How are we able to generate large voltages with this power source? 4. When the current through a large inductor is interrupted with a switch, an arc appears across the open terminals of the switch. Explain. 5. Does self-inductance depend on the value of the magnetic flux? Does it depend on the current through the wire? Correlate your answers with the equation \(NΦ_m=LI\). 6. Would the self-inductance of a 1.0 m long, tightly wound solenoid differ from the self-inductance per meter of an infinite, but otherwise identical, solenoid? 7. Discuss how you might determine the self-inductance per unit length of a long, straight wire. 8. The self-inductance of a coil is zero if there is no current passing through the windings. True or false? 9. How does the self-inductance per unit length near the center of a solenoid (away from the ends) com...

Mutual inductance occurs when a change in current in one coil induces voltage in a nearby coil through the interaction of their magnetic fields. Mutual Inductance Mutual inductance is a fundamental concept in electromagnetism, describing the interaction between two electrical circuits or coils. It occurs when a change in current in one coil, called the primary coil, induces a voltage in another nearby coil, known as the secondary coil, through the magnetic field generated by the primary coil. Understanding Mutual Inductance When a current flows through a coil, it creates a magnetic field around it. If another coil is placed within the vicinity of the first coil, the magnetic field generated by the first coil will interact with the second coil. This interaction induces a voltage in the second coil, which is the basis of mutual inductance. The amount of voltage induced in the second coil depends on the rate of change of the magnetic field and the number of turns in the coil. Factors Affecting Mutual Inductance Several factors influence the mutual inductance between two coils, including: • Distance between coils: As the distance between the coils increases, the mutual inductance decreases, as the magnetic field from the primary coil has less effect on the secondary coil. • Number of turns: The more turns in the coils, the greater the mutual inductance. • Coil orientation: The mutual inductance is maximized when the coils are aligned with their axes parallel to each other and ...

What is Mutual Inductance? Definition: The mutual inductance of two coils is defined as the emf induced due to the magnetic field in one coil opposes the change of current and voltage in another coil. That means the two coils are magnetically linked together due to the change in The current flowing in one coil induces the voltage in another coil due to the change in magnetic flux. The amount of magnetic flux linked with the two coils is directly proportional to the mutual inductance and current change. Mutual Inductance Theory Its theory is very simple and it can be understood by using two or more coils. It was described by an American scientist Joseph Henry in the 18th century. It is referred to as one of the properties of the coil or conductor used in the circuit. The property Oliver Heaviside introduced the term inductance in the year 1886. The property of mutual inductance is the working principle of many The main drawback of the mutual inductance is, leakage of the inductance of one coil can interrupt the operation of another coil utilizing electromagnetic induction. To reduce the leakage, electrical screening is required The positioning of two coils in the circuit decides the amount of mutual inductance that links with one to the other coil. Mutual Inductance Formula The formula of two coils is given as M= ( μ0.μr. N1. N2. A) / L N2= turns of coil 2 A= cross-sectional area in m 2 L = length of the coil in meters Unit of Mutual Inductance The unit of mutual inductance...

https://phys.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fphys.libretexts.org%2FBookshelves%2FUniversity_Physics%2FBook%253A_University_Physics_(OpenStax)%2FBook%253A_University_Physics_II_-_Thermodynamics_Electricity_and_Magnetism_(OpenStax)%2F14%253A_Inductance%2F14.03%253A_Self-Inductance_and_Inductors \( \newcommand\) • • • • • • • • • • • • • Learning Objectives By the end of this section, you will be able to: • Correlate the rate of change of current to the induced emf created by that current in the same circuit • Derive the self-inductance for a cylindrical solenoid • Derive the self-inductance for a rectangular toroid Mutual inductance arises when a current in one circuit produces a changing magnetic field that induces an emf in another circuit. But can the magnetic field affect the current in the original circuit that produced the field? The answer is yes, and this is the phenomenon called self-inductance. Inductors Figure \(\PageIndex\): The familiar security gate at an airport not only detects metals, but can also indicate their approximate height above the floor. (credit: “Alexbuirds”/Wikimedia Commons) Large induced voltages are found in camera flashes. Camera flashes use a battery, two inductors that function as a transformer, and a switching system or oscillator to induce large voltages. Recall from Example \(\PageIndex Wb.\] Significance The self-inductance and flux calculated in parts (a) and (b) are typical values for coils found in contemporar...

11 Magnetic Forces and Fields • Introduction • 11.1 Magnetism and Its Historical Discoveries • 11.2 Magnetic Fields and Lines • 11.3 Motion of a Charged Particle in a Magnetic Field • 11.4 Magnetic Force on a Current-Carrying Conductor • 11.5 Force and Torque on a Current Loop • 11.6 The Hall Effect • 11.7 Applications of Magnetic Forces and Fields • Learning Objectives By the end of this section, you will be able to: • Correlate two nearby circuits that carry time-varying currents with the emf induced in each circuit • Describe examples in which mutual inductance may or may not be desirable Inductance is the property of a device that tells us how effectively it induces an emf in another device. In other words, it is a physical quantity that expresses the effectiveness of a given device. When two circuits carrying time-varying currents are close to one another, the magnetic flux through each circuit varies because of the changing current I in the other circuit. Consequently, an emf is induced in each circuit by the changing current in the other. This type of emf is therefore called a mutually induced emf, and the phenomenon that occurs is known as mutual inductance ( M). As an example, let’s consider two tightly wound coils ( N 1 N 1 and N 2 N 2 turns and carry currents I 1 I 1 and I 2 , I 2 , respectively. The flux through a single turn of coil 2 produced by the magnetic field of the current in coil 1 is Φ 21 , Φ 21 , whereas the flux through a single turn of coil 1 d...

In the domain of electronics and electromagnetism, inductance is the crucial concept where it is the inclination of a conductor to contradict the current flow. The current flow generates a magnetic field across the conductor. The field strength is based on the current’s magnitude and goes as per current changes. The word inductance was coined by Oliver Heaviside in the year 1886. In the name of Heinrich Lenz, the symbol of inductance was provided. The story of electromagnetic induction is a component of electromagnetism initiated with the interpretations of electric charge, electric current, and magnetic attraction. By knowing the concord of these forces in nature and the scientific concept behind electromagnetism started in the 18 th era. The one prominent concept that arrived with this interpretation is Mutual What is Mutual Inductance? Mutual induction definition is stated that when two coils are brought in close with other than the magnetic field in any of these two coils direct to connect with the other coil. This directs to the development of voltage in the second coil. This scenario of one coil that impacts or modifies the voltage and current values in the other coil is called a mutual inductance property. Mutual inductance can impose few bad things also which are leakage and stray inductance. When these are released from one coil, they affect the functionality of another element through the process of electromagnetic induction. Because of this, some approach of ele...

11 Magnetic Forces and Fields • Introduction • 11.1 Magnetism and Its Historical Discoveries • 11.2 Magnetic Fields and Lines • 11.3 Motion of a Charged Particle in a Magnetic Field • 11.4 Magnetic Force on a Current-Carrying Conductor • 11.5 Force and Torque on a Current Loop • 11.6 The Hall Effect • 11.7 Applications of Magnetic Forces and Fields • Learning Objectives By the end of this section, you will be able to: • Correlate two nearby circuits that carry time-varying currents with the emf induced in each circuit • Describe examples in which mutual inductance may or may not be desirable Inductance is the property of a device that tells us how effectively it induces an emf in another device. In other words, it is a physical quantity that expresses the effectiveness of a given device. When two circuits carrying time-varying currents are close to one another, the magnetic flux through each circuit varies because of the changing current I in the other circuit. Consequently, an emf is induced in each circuit by the changing current in the other. This type of emf is therefore called a mutually induced emf, and the phenomenon that occurs is known as mutual inductance ( M). As an example, let’s consider two tightly wound coils ( N 1 N 1 and N 2 N 2 turns and carry currents I 1 I 1 and I 2 , I 2 , respectively. The flux through a single turn of coil 2 produced by the magnetic field of the current in coil 1 is Φ 21 , Φ 21 , whereas the flux through a single turn of coil 1 d...

https://phys.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fphys.libretexts.org%2FBookshelves%2FUniversity_Physics%2FBook%253A_University_Physics_(OpenStax)%2FBook%253A_University_Physics_II_-_Thermodynamics_Electricity_and_Magnetism_(OpenStax)%2F14%253A_Inductance%2F14.0E%253A_14.E%253A_Inductance_(Exercise) \( \newcommand\) • • • • • • • • • • • • • • • • • • 1. Show that \(NΦ_m/I\) and \(ε/(dI/dt)\), which are both expressions for self-inductance, have the same units. 2. A 10-H inductor carries a current of 20 A. Describe how a 50-V emf can be induced across it. 3. The ignition circuit of an automobile is powered by a 12-V battery. How are we able to generate large voltages with this power source? 4. When the current through a large inductor is interrupted with a switch, an arc appears across the open terminals of the switch. Explain. 5. Does self-inductance depend on the value of the magnetic flux? Does it depend on the current through the wire? Correlate your answers with the equation \(NΦ_m=LI\). 6. Would the self-inductance of a 1.0 m long, tightly wound solenoid differ from the self-inductance per meter of an infinite, but otherwise identical, solenoid? 7. Discuss how you might determine the self-inductance per unit length of a long, straight wire. 8. The self-inductance of a coil is zero if there is no current passing through the windings. True or false? 9. How does the self-inductance per unit length near the center of a solenoid (away from the ends) com...

In the domain of electronics and electromagnetism, inductance is the crucial concept where it is the inclination of a conductor to contradict the current flow. The current flow generates a magnetic field across the conductor. The field strength is based on the current’s magnitude and goes as per current changes. The word inductance was coined by Oliver Heaviside in the year 1886. In the name of Heinrich Lenz, the symbol of inductance was provided. The story of electromagnetic induction is a component of electromagnetism initiated with the interpretations of electric charge, electric current, and magnetic attraction. By knowing the concord of these forces in nature and the scientific concept behind electromagnetism started in the 18 th era. The one prominent concept that arrived with this interpretation is Mutual What is Mutual Inductance? Mutual induction definition is stated that when two coils are brought in close with other than the magnetic field in any of these two coils direct to connect with the other coil. This directs to the development of voltage in the second coil. This scenario of one coil that impacts or modifies the voltage and current values in the other coil is called a mutual inductance property. Mutual inductance can impose few bad things also which are leakage and stray inductance. When these are released from one coil, they affect the functionality of another element through the process of electromagnetic induction. Because of this, some approach of ele...