# 14.1 Mutual inductance

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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 ( [link] ). Coils 1 and 2 have ${N}_{1}$ and ${N}_{2}$ turns and carry currents ${I}_{1}$ and ${I}_{2},$ respectively. The flux through a single turn of coil 2 produced by the magnetic field of the current in coil 1 is ${\text{Φ}}_{21},$ whereas the flux through a single turn of coil 1 due to the magnetic field of ${I}_{2}$ is ${\text{Φ}}_{12}.$

The mutual inductance ${M}_{21}$ of coil 2 with respect to coil 1 is the ratio of the flux through the ${N}_{2}$ turns of coil 2 produced by the magnetic field of the current in coil 1, divided by that current, that is,

${M}_{21}=\frac{{N}_{2}{\text{Φ}}_{21}}{{I}_{1}}.$

Similarly, the mutual inductance of coil 1 with respect to coil 2 is

${M}_{12}=\frac{{N}_{1}{\text{Φ}}_{12}}{{I}_{2}}.$

Like capacitance, mutual inductance is a geometric quantity. It depends on the shapes and relative positions of the two coils, and it is independent of the currents in the coils. The SI unit for mutual inductance M is called the henry (H)    in honor of Joseph Henry (1799–1878), an American scientist who discovered induced emf independently of Faraday. Thus, we have $1\phantom{\rule{0.2em}{0ex}}\text{H}=1\phantom{\rule{0.2em}{0ex}}\text{V}·\text{s/A}$ . From [link] and [link] , we can show that ${M}_{21}={M}_{12},$ so we usually drop the subscripts associated with mutual inductance and write

$M=\frac{{N}_{2}{\text{Φ}}_{21}}{{I}_{1}}=\frac{{N}_{1}{\text{Φ}}_{12}}{{I}_{2}}.$

The emf developed in either coil is found by combining Faraday’s law    and the definition of mutual inductance. Since ${N}_{2}{\text{Φ}}_{21}$ is the total flux through coil 2 due to ${I}_{1}$ , we obtain

${\epsilon }_{2}=-\frac{d}{dt}\left({N}_{2}{\text{Φ}}_{21}\right)=-\frac{d}{dt}\left(M{I}_{1}\right)=\text{−}M\frac{d{I}_{1}}{dt}$

where we have used the fact that M is a time-independent constant because the geometry is time-independent. Similarly, we have

${\epsilon }_{1}=\text{−}M\frac{d{I}_{2}}{dt}.$

In [link] , we can see the significance of the earlier description of mutual inductance ( M ) as a geometric quantity. The value of M neatly encapsulates the physical properties of circuit elements and allows us to separate the physical layout of the circuit from the dynamic quantities, such as the emf and the current. [link] defines the mutual inductance in terms of properties in the circuit, whereas the previous definition of mutual inductance in [link] is defined in terms of the magnetic flux experienced, regardless of circuit elements. You should be careful when using [link] and [link] because ${\epsilon }_{1}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{\epsilon }_{2}$ do not necessarily represent the total emfs in the respective coils. Each coil can also have an emf induced in it because of its self-inductance (self-inductance will be discussed in more detail in a later section).

#### Questions & Answers

The properties of a system during a reversible constant pressure non-flow process at P= 1.6bar, changes from constant volume of 0.3m³/kg at 20°C to a volume of 0.55m³/kg at 260°C. its constant pressure process is 3.205KJ/kg°C Determine: 1. Heat added, Work done, Change in Internal Energy and Change in Enthalpy
Opeyemi Reply
please how do dey get 5/9 in the conversion of Celsius and Fahrenheit
Gwam Reply
what is copper loss
timileyin Reply
this is the energy dissipated(usually in the form of heat energy) in conductors such as wires and coils due to the flow of current against the resistance of the material used in winding the coil.
Henry
it is the work done in moving a charge to a point from infinity against electric field
Ashok Reply
what is the weight of the earth in space
peterpaul Reply
As w=mg where m is mass and g is gravitational force... Now if we consider the earth is in gravitational pull of sun we have to use the value of "g" of sun, so we can find the weight of eaeth in sun with reference to sun...
Prince
g is not gravitacional forcé, is acceleration of gravity of earth and is assumed constante. the "sun g" can not be constant and you should use Newton gravity forcé. by the way its not the "weight" the physical quantity that matters, is the mass
Jorge
Yeah got it... Earth and moon have specific value of g... But in case of sun ☀ it is just a huge sphere of gas...
Prince
Thats why it can't have a constant value of g ....
Prince
not true. you must know Newton gravity Law . even a cloud of gas it has mass thats al matters. and the distsnce from the center of mass of the cloud and the center of the mass of the earth
Jorge
please why is the first law of thermodynamics greater than the second
Ifeoma Reply
define electric image.obtain expression for electric intensity at any point on earthed conducting infinite plane due to a point charge Q placed at a distance D from it.
Mateshwar Reply
explain the lack of symmetry in the field of the parallel capacitor
Phoebe Reply
pls. explain the lack of symmetry in the field of the parallel capacitor
Phoebe
does your app come with video lessons?
Ahmed Reply
What is vector
Ajibola Reply
Vector is a quantity having a direction as well as magnitude
Damilare
tell me about charging and discharging of capacitors
Ahemen Reply
a big and a small metal spheres are connected by a wire, which of this has the maximum electric potential on the surface.
Bundi Reply
3 capacitors 2nf,3nf,4nf are connected in parallel... what is the equivalent capacitance...and what is the potential difference across each capacitor if the EMF is 500v
Prince Reply
equivalent capacitance is 9nf nd pd across each capacitor is 500v
santanu
four effect of heat on substances
Prince Reply
why we can find a electric mirror image only in a infinite conducting....why not in finite conducting plate..?
Rima Reply
because you can't fit the boundary conditions.
Jorge
what is the dimensions for VISCOUNSITY (U)
Branda

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Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
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