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== 14.1#h(0.6em)Mutual Inductance


  #strong[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 #emph[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 #emph[mutually induced emf], and the phenomenon that occurs is known as #strong[mutual inductance (#emph[M])]. As an example, let’s consider two tightly wound coils . Coils 1 and 2 have #math.equation(block: false, alt: "N sub 1")[$N_(1)$] and #math.equation(block: false, alt: "N sub 2")[$N_(2)$] turns and carry currents #math.equation(block: false, alt: "I sub 1")[$I_(1)$] and #math.equation(block: false, alt: "I sub 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 #math.equation(block: false, alt: "Φ sub 21 ,")[$"Φ"_(21) ,$] whereas the flux through a single turn of coil 1 due to the magnetic field of #math.equation(block: false, alt: "I sub 2")[$I_(2)$] is #math.equation(block: false, alt: "Φ sub 12 .")[$"Φ"_(12) .$]


  #figure(figph[Figure shows the cross sections of two coils. In each one, the cross sections of the wire of the coil are shown as two circles, one at the top and the other at the bottom. Dots in the upper circles and crosses in the lower ones indicate the direction of flow of current. Coil 1 has field lines labeled B1 passing from between the two circles, going right. Some of these pass through coil 2, which is smaller than coil 1.], alt: "Figure shows the cross sections of two coils. In each one, the cross sections of the wire of the coil are shown as two circles, one at the top and the other at the bottom. Dots in the upper circles and crosses in the lower ones indicate the direction of flow of current. Coil 1 has field lines labeled B1 passing from between the two circles, going right. Some of these pass through coil 2, which is smaller than coil 1.", caption: [Some of the magnetic field lines produced by the current in coil 1 pass through coil 2.])


  The mutual inductance #math.equation(block: false, alt: "M sub 21")[$M_(21)$] of coil 2 with respect to coil 1 is the ratio of the flux through the #math.equation(block: false, alt: "N sub 2")[$N_(2)$] turns of coil 2 produced by the magnetic field of the current in coil 1, divided by that current, that is,


  #math.equation(block: true, alt: "M sub 21 equals the fraction N sub 2 Φ sub 21 over I sub 1 .")[$M_(21) = frac(N_(2) "Φ"_(21), I_(1)) .$]
  Similarly, the mutual inductance of coil 1 with respect to coil 2 is


  #math.equation(block: true, alt: "M sub 12 equals the fraction N sub 1 Φ sub 12 over I sub 2 .")[$M_(12) = frac(N_(1) "Φ"_(12), I_(2)) .$]
  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 #emph[M] is called the #strong[henry (H)] in honor of Joseph #strong[Henry] (1799–1878), an American scientist who discovered induced emf independently of Faraday. Thus, we have #math.equation(block: false, alt: "1 H equals 1 V times s/A")[$1 #h(0.2em) "H" = 1 #h(0.2em) "V" · "s/A"$]. From  and , we can show that #math.equation(block: false, alt: "M sub 21 equals M sub 12 ,")[$M_(21) = M_(12) ,$] so we usually drop the subscripts associated with mutual inductance and write


  #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: true, alt: "M equals the fraction N sub 2 Φ sub 21 over I sub 1 equals the fraction N sub 1 Φ sub 12 over I sub 2 .")[$M = frac(N_(2) "Φ"_(21), I_(1)) = frac(N_(1) "Φ"_(12), I_(2)) .$]
]


  The emf developed in either coil is found by combining #strong[Faraday’s law] and the definition of mutual inductance. Since #math.equation(block: false, alt: "N sub 2 Φ sub 21")[$N_(2) "Φ"_(21)$] is the total flux through coil 2 due to #math.equation(block: false, alt: "I sub 1")[$I_(1)$], we obtain


  #math.equation(block: true, alt: "ε sub 2 equals minus the fraction d over d t open parenthesis N sub 2 Φ sub 21 close parenthesis equals minus the fraction d over d t open parenthesis M I sub 1 close parenthesis equals − M the fraction d I sub 1 over d t")[$ε_(2) = − frac(d, d t) ( N_(2) "Φ"_(21) ) = − frac(d, d t) ( M I_(1) ) = "−" M frac(d I_(1), d t)$]
  where we have used the fact that #emph[M] is a time-independent constant because the geometry is time-independent. Similarly, we have


  #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: true, alt: "ε sub 1 equals − M the fraction d I sub 2 over d t .")[$ε_(1) = "−" M frac(d I_(2), d t) .$]
]


  In , we can see the significance of the earlier description of mutual inductance (#emph[M]) as a geometric quantity. The value of #emph[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.  defines the mutual inductance in terms of properties in the circuit, whereas the previous definition of mutual inductance in  is defined in terms of the magnetic flux experienced, regardless of circuit elements. You should be careful when using  and  because #math.equation(block: false, alt: "ε sub 1 and ε sub 2")[$ε_(1) #h(0.2em) "and" #h(0.2em) ε_(2)$] do not necessarily represent the total emfs in the respective coils. Each coil can also have an emf induced in it because of its #emph[self-inductance] (self-inductance will be discussed in more detail in a later section).


  A large mutual inductance #emph[M] may or may not be desirable. We want a transformer to have a large mutual inductance. But an appliance, such as an electric clothes dryer, can induce a dangerous emf on its metal case if the mutual inductance between its coils and the case is large. One way to reduce mutual inductance is to counter-wind coils to cancel the magnetic field produced .


  #figure(figph[Figure a shows a heating coil within a metal case of a clothes dryer. Figure b shows the same coil, enlarged. The coil is wound on a cylinder in such a way that one wire is wound all the way to the other side, twisted around and wound all the way back. Thus, two adjacent windings have current flowing in opposite directions.], alt: "Figure a shows a heating coil within a metal case of a clothes dryer. Figure b shows the same coil, enlarged. The coil is wound on a cylinder in such a way that one wire is wound all the way to the other side, twisted around and wound all the way back. Thus, two adjacent windings have current flowing in opposite directions.", caption: [The heating coils of an electric clothes dryer can be counter-wound so that their magnetic fields cancel one another, greatly reducing the mutual inductance with the case of the dryer.])


  #strong[Digital signal processing] is another example in which mutual inductance is reduced by counter-winding coils. The rapid on/off emf representing 1s and 0s in a digital circuit creates a complex time-dependent magnetic field. An emf can be generated in neighboring conductors. If that conductor is also carrying a digital signal, the induced emf may be large enough to switch 1s and 0s, with consequences ranging from inconvenient to disastrous.


  #examplebox("Example 1")[Mutual Inductance][
shows a coil of #math.equation(block: false, alt: "N sub 2")[$N_(2)$] turns and radius #math.equation(block: false, alt: "R sub 2")[$R_(2)$] surrounding a long solenoid of length #math.equation(block: false, alt: "l sub 1 ,")[$l_(1) ,$] radius #math.equation(block: false, alt: "R sub 1 ,")[$R_(1) ,$] and #math.equation(block: false, alt: "N sub 1")[$N_(1)$] turns. (a) What is the mutual inductance of the two coils? (b) If #math.equation(block: false, alt: "N sub 1 equals 500 turns")[$N_(1) = 500 #h(0.2em) "turns"$], #math.equation(block: false, alt: "N sub 2 equals 10 turns")[$N_(2) = 10 #h(0.2em) "turns"$], #math.equation(block: false, alt: "R sub 1 equals 3.10 cm")[$R_(1) = 3.10 #h(0.2em) "cm"$], #math.equation(block: false, alt: "l sub 1 equals 75.0 cm")[$l_(1) = 75.0 #h(0.2em) "cm"$], and the current in the solenoid is changing at a rate of 200 A/s, what is the emf induced in the surrounding coil?


    #figure(figph[Figure shows a solenoid, in the form of a long coil with a small diameter, that is concentrically arranged with another, bigger coil. The radius of the solenoid is R1 and that of the coil is R2. The length of the solenoid is l1.], alt: "Figure shows a solenoid, in the form of a long coil with a small diameter, that is concentrically arranged with another, bigger coil. The radius of the solenoid is R1 and that of the coil is R2. The length of the solenoid is l1.", caption: [A solenoid surrounded by a coil.])


    Strategy
There is no magnetic field outside the solenoid, and the field inside has magnitude #math.equation(block: false, alt: "B sub 1 equals μ sub 0 open parenthesis N sub 1 / l sub 1 close parenthesis I sub 1")[$B_(1) = μ_(0) ( N_(1) "/" l_(1) ) I_(1)$] and is directed parallel to the solenoid’s axis. We can use this magnetic field to find the magnetic flux through the surrounding coil and then use this flux to calculate the mutual inductance for part (a), using . We solve part (b) by calculating the mutual inductance from the given quantities and using  to calculate the induced emf.


    Solution


    + The magnetic flux #math.equation(block: false, alt: "Φ sub 21")[$"Φ"_(21)$] through the surrounding coil is
#math.equation(block: true, alt: "Φ sub 21 equals B sub 1 π R 1 2 equals the fraction μ sub 0 N sub 1 I sub 1 over l sub 1 π R 1 2 .")[$"Φ"_(21) = B_(1) π R_(1)^(2) = frac(μ_(0) N_(1) I_(1), l_(1)) π R_(1)^(2) .$]
Now from , the mutual inductance is
#math.equation(block: true, alt: "M equals the fraction N sub 2 Φ sub 21 over I sub 1 equals open parenthesis the fraction N sub 2 over I sub 1 close parenthesis open parenthesis the fraction μ sub 0 N sub 1 I sub 1 over l sub 1 close parenthesis π R 1 2 equals the fraction μ sub 0 N sub 1 N sub 2 π R 1 2 over l sub 1 .")[$M = frac(N_(2) "Φ"_(21), I_(1)) = ( frac(N_(2), I_(1)) ) ( frac(μ_(0) N_(1) I_(1), l_(1)) ) π R_(1)^(2) = frac(μ_(0) N_(1) N_(2) π R_(1)^(2), l_(1)) .$]
+ Using the previous expression and the given values, the mutual inductance is
#math.equation(block: true, alt: "M, equals the fraction open parenthesis 4 π times 10 to the power −7 T times m/A close parenthesis open parenthesis 500 close parenthesis open parenthesis 10 close parenthesis π open parenthesis 0.0310 m close parenthesis squared over 0.750 m; equals 2.53 times 10 to the power −5 H .")[$M & = frac(( 4 π #h(0.2em) × #h(0.2em) 10^(−7) #h(0.2em) "T" · "m/A" ) ( 500 ) ( 10 ) π attach(( 0.0310 #h(0.2em) "m" ), t: 2), 0.750 #h(0.2em) "m") \  & = 2.53 #h(0.2em) × #h(0.2em) 10^(−5) #h(0.2em) "H" .$]
Thus, from , the emf induced in the surrounding coil is
#math.equation(block: true, alt: "ε sub 2, equals − M the fraction d I sub 1 over d t equals − open parenthesis 2.53 times 10 to the power −5 H close parenthesis open parenthesis 200 A/s close parenthesis; equals −5.06 times 10 to the power −3 V .")[$ε_(2) & = "−" M frac(d I_(1), d t) = "−" ( 2.53 #h(0.2em) × #h(0.2em) 10^(−5) #h(0.2em) "H" ) ( 200 #h(0.2em) "A/s" ) \  & = −5.06 #h(0.2em) × #h(0.2em) 10^(−3) #h(0.2em) "V" .$]


    Significance
Notice that #emph[M] in part (a) is independent of the radius #math.equation(block: false, alt: "R sub 2")[$R_(2)$] of the surrounding coil because the solenoid’s magnetic field is confined to its interior. In principle, we can also calculate #emph[M] by finding the magnetic flux through the solenoid produced by the current in the surrounding coil. This approach is much more difficult because #math.equation(block: false, alt: "Φ sub 12")[$"Φ"_(12)$] is so complicated. However, since #math.equation(block: false, alt: "M sub 12 equals M sub 21 ,")[$M_(12) = M_(21) ,$] we do know the result of this calculation.
]


  #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
A current #math.equation(block: false, alt: "I open parenthesis t close parenthesis equals open parenthesis 5.0 A close parenthesis sin open parenthesis open parenthesis 120 π rad/s close parenthesis t close parenthesis")[$I ( t ) = ( 5.0 #h(0.2em) "A" ) #h(0.2em) "sin" #h(0.2em) ( ( 120 π #h(0.2em) "rad/s" ) t )$] flows through the solenoid of part (b) of . What is the maximum emf induced in the surrounding coil?


    
    #solutionbox[
#math.equation(block: true, alt: "4.77 times 10 to the power −2 V")[$4.77 #h(0.2em) × #h(0.2em) 10^(−2) #h(0.2em) "V"$]
]
]


  
    
=== Summary


    - Inductance is the property of a device that expresses how effectively it induces an emf in another device.
- Mutual inductance is the effect of two devices inducing emfs in each other.
- A change in current #math.equation(block: false, alt: "d I sub 1 / d t")[$attach(d I, b: 1) "/" d t$] in one circuit induces an emf #math.equation(block: false, alt: "open parenthesis ε sub 2 close parenthesis")[$( ε_(2) )$] in the second:
#math.equation(block: true, alt: "ε sub 2 equals minus M the fraction d I 1 over d t ,")[$ε_(2) = − M frac(d I 1, d t) ,$]
where #emph[M] is defined to be the mutual inductance between the two circuits and the minus sign is due to Lenz’s law.
- Symmetrically, a change in current #math.equation(block: false, alt: "d I sub 2 / d t")[$attach(d I, b: 2) "/" d t$] through the second circuit induces an emf #math.equation(block: false, alt: "open parenthesis ε sub 1 close parenthesis")[$( ε_(1) )$] in the first: 
#math.equation(block: true, alt: "ε sub 1 equals minus M the fraction d I sub 2 over d t ,")[$ε_(1) = − M frac(d I_(2), d t) ,$]
where #emph[M] is the same mutual inductance as in the reverse process.


  
  
    
=== Conceptual Questions


    
      Show that #math.equation(block: false, alt: "N Φ sub m / I")[$N "Φ"_("m") "/" I$] and #math.equation(block: false, alt: "ε / open parenthesis d I / d t close parenthesis ,")[$ε "/" ( d I "/" d t ) ,$] which are both expressions for self-inductance, have the same units.


    
    #solutionbox[
#math.equation(block: true, alt: "the fraction Wb over A equals the fraction T times m squared over A equals the fraction V times s over A equals the fraction V over A/s")[$frac("Wb", "A") = frac("T" · "m"^(2), "A") = frac("V" · "s", "A") = frac("V", "A/s")$]
]


    
      A 10-H inductor carries a current of 20 A. Describe how a 50-V emf can be induced across it.


    
    
      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?


    
    #solutionbox[
The induced current from the 12-V battery goes through an inductor, generating a large voltage.
]


    
      When the current through a large inductor is interrupted with a switch, an arc appears across the open terminals of the switch. Explain.


    
  
  
    
=== Problems


    
      When the current in one coil changes at a rate of 5.6 A/s, an emf of #math.equation(block: false, alt: "6.3 times 10 to the power −3 V")[$6.3 #h(0.2em) × #h(0.2em) 10^(−3) #h(0.2em) "V"$] is induced in a second, nearby coil. What is the mutual inductance of the two coils?


    
    
      An emf of #math.equation(block: false, alt: "9.7 times 10 to the power −3 V")[$9.7 #h(0.2em) × #h(0.2em) 10^(−3) #h(0.2em) "V"$] is induced in a coil while the current in a nearby coil is decreasing at a rate of 2.7 A/s. What is the mutual inductance of the two coils?


    
    #solutionbox[
#math.equation(block: true, alt: "M equals 3.6 times 10 to the power −3 H")[$M = 3.6 #h(0.2em) × #h(0.2em) 10^(−3) #h(0.2em) "H"$]
]


    
      Two coils close to each other have a mutual inductance of 32 mH. If the current in one coil decays according to #math.equation(block: false, alt: "I equals I sub 0 e to the power − α t")[$I = I_(0) e^("−" α t)$], where #math.equation(block: false, alt: "I sub 0 equals 5.0 A")[$I_(0) = 5.0 #h(0.2em) "A"$] and #math.equation(block: false, alt: "α equals 2.0 times 10 cubed s to the power −1 ,")[$α = 2.0 #h(0.2em) × #h(0.2em) 10^(3) #h(0.2em) "s"^(−1) ,$] what is the emf induced in the second coil immediately after the current starts to decay? At #math.equation(block: false, alt: "t equals 1.0 times 10 to the power −3 s ?")[$t = 1.0 #h(0.2em) × #h(0.2em) 10^(−3) #h(0.2em) "s" ?$]


    
    
      A coil of 40 turns is wrapped around a long solenoid of cross-sectional area #math.equation(block: false, alt: "7.5 times 10 to the power −3 m squared .")[$7.5 #h(0.2em) × #h(0.2em) 10^(−3) #h(0.2em) "m"^(2) .$] The solenoid is 0.50 m long and has 500 turns. (a) What is the mutual inductance of this system? (b) The outer coil is replaced by a coil of 40 turns whose radius is three times that of the solenoid. What is the mutual inductance of this configuration?


    
    #solutionbox[
a. #math.equation(block: false, alt: "3.8 times 10 to the power −4 H")[$3.8 #h(0.2em) × #h(0.2em) 10^(−4) #h(0.2em) "H"$]; b. #math.equation(block: false, alt: "3.8 times 10 to the power −4 H")[$3.8 #h(0.2em) × #h(0.2em) 10^(−4) #h(0.2em) "H"$]
]


    
      A 600-turn solenoid is 0.55 m long and 4.2 cm in diameter. Inside the solenoid, a small #math.equation(block: false, alt: "open parenthesis 1.1 cm times 1.4 cm close parenthesis ,")[$( 1.1 #h(0.2em) "cm" #h(0.2em) × #h(0.2em) 1.4 #h(0.2em) "cm" ) ,$] single-turn rectangular coil is fixed in place with its face perpendicular to the long axis of the solenoid. What is the mutual inductance of this system?


    
    
      A toroidal coil has a mean radius of 16 cm and a cross-sectional area of #math.equation(block: false, alt: "0.25 cm squared")[$attach(0.25 #h(0.2em) "cm", t: 2)$]; it is wound uniformly with 1000 turns. A second toroidal coil of 750 turns is wound uniformly over the first coil. Ignoring the variation of the magnetic field within a toroid, determine the mutual inductance of the two coils.


    
    #solutionbox[
#math.equation(block: true, alt: "M sub 21 equals 2.3 times 10 to the power −5 H")[$M_(21) = 2.3 #h(0.2em) × #h(0.2em) 10^(−5) #h(0.2em) "H"$]
]


    
      A solenoid of #math.equation(block: false, alt: "N sub 1")[$N_(1)$] turns has length #math.equation(block: false, alt: "l sub 1")[$l_(1)$] and radius #math.equation(block: false, alt: "R sub 1 ,")[$R_(1) ,$] and a second smaller solenoid of #math.equation(block: false, alt: "N sub 2")[$N_(2)$] turns has length #math.equation(block: false, alt: "l sub 2")[$l_(2)$] and radius #math.equation(block: false, alt: "R sub 2")[$R_(2)$]. The smaller solenoid is placed completely inside the larger solenoid so that their long axes coincide. What is the mutual inductance of the two solenoids?