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23.9 Inductance  (Page 4/10)

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Section summary

  • Inductance is the property of a device that tells how effectively it induces an emf in another device.
  • Mutual inductance is the effect of two devices in inducing emfs in each other.
  • A change in current Δ I 1 / Δ t size 12{ΔI rSub { size 8{1} } /Δt} {} in one induces an emf emf 2 size 12{"emf" rSub { size 8{2} } } {} in the second:
    emf 2 = − M Δ I 1 Δ t , size 12{"emf" rSub { size 8{2} } = - M { {ΔI rSub { size 8{1} } } over {Δt} } } {}
    where M is defined to be the mutual inductance between the two devices, and the minus sign is due to Lenz’s law.
  • Symmetrically, a change in current Δ I 2 / Δ t size 12{ΔI rSub { size 8{2} } /Δt} {} through the second device induces an emf emf 1 size 12{"emf" rSub { size 8{1} } } {} in the first:
    emf 1 = − M Δ I 2 Δ t , size 12{"emf" rSub { size 8{1} } = - M { {ΔI rSub { size 8{2} } } over {Δt} } } {}
    where M is the same mutual inductance as in the reverse process.
  • Current changes in a device induce an emf in the device itself.
  • Self-inductance is the effect of the device inducing emf in itself.
  • The device is called an inductor, and the emf induced in it by a change in current through it is
    emf = − L Δ I Δ t , size 12{"emf"= - L { {ΔI} over {Δt} } } {}
    where L size 12{L} {} is the self-inductance of the inductor, and Δ I / Δ t size 12{ΔI/Δt} {} is the rate of change of current through it. The minus sign indicates that emf opposes the change in current, as required by Lenz’s law.
  • The unit of self- and mutual inductance is the henry (H), where 1 H = 1 Ω ⋅ s size 12{1`H=1` %OMEGA cdot s} {} .
  • The self-inductance L size 12{L} {} of an inductor is proportional to how much flux changes with current. For an N size 12{N} {} -turn inductor,
    L = N Δ Φ Δ I . size 12{L=N { {ΔΦ} over {ΔI} } } {}
  • The self-inductance of a solenoid is
    L = μ 0 N 2 A ℓ (solenoid), size 12{L= { {μ rSub { size 8{0} } N rSup { size 8{2} } A} over {ℓ} } } {}
    where N size 12{N} {} is its number of turns in the solenoid, A size 12{A} {} is its cross-sectional area, ℓ size 12{ℓ} {} is its length, and μ 0 = 4π × 10 −7 T ⋅ m/A size 12{μ rSub { size 8{0} } =4π times "10" rSup { size 8{"-7"} } `T cdot "m/A"} {} is the permeability of free space.
  • The energy stored in an inductor E ind size 12{E rSub { size 8{"ind"} } } {} is
    E ind = 1 2 LI 2 . size 12{E rSub { size 8{"ind"} } = { {1} over {2} } ital "LI" rSup { size 8{2} } } {}

Conceptual questions

How would you place two identical flat coils in contact so that they had the greatest mutual inductance? The least?

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How would you shape a given length of wire to give it the greatest self-inductance? The least?

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Verify, as was concluded without proof in [link] , that units of T ⋅ m 2 / A = Ω ⋅ s = H size 12{T cdot m rSup { size 8{2} } /A= %OMEGA cdot s=H} {} .

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Problems&Exercises

Two coils are placed close together in a physics lab to demonstrate Faraday’s law of induction. A current of 5.00 A in one is switched off in 1.00 ms, inducing a 9.00 V emf in the other. What is their mutual inductance?

1.80 mH

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If two coils placed next to one another have a mutual inductance of 5.00 mH, what voltage is induced in one when the 2.00 A current in the other is switched off in 30.0 ms?

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The 4.00 A current through a 7.50 mH inductor is switched off in 8.33 ms. What is the emf induced opposing this?

3.60 V

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A device is turned on and 3.00 A flows through it 0.100 ms later. What is the self-inductance of the device if an induced 150 V emf opposes this?

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Starting with emf 2 = − M Δ I 1 Δ t size 12{"emf" rSub { size 8{2} } = - M { {ΔI rSub { size 8{1} } } over {Δt} } } {} , show that the units of inductance are ( V ⋅ s ) /A = Ω ⋅ s size 12{ \( V cdot s \) "/A"= %OMEGA cdot s} {} .

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Camera flashes charge a capacitor to high voltage by switching the current through an inductor on and off rapidly. In what time must the 0.100 A current through a 2.00 mH inductor be switched on or off to induce a 500 V emf?

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A large research solenoid has a self-inductance of 25.0 H. (a) What induced emf opposes shutting it off when 100 A of current through it is switched off in 80.0 ms? (b) How much energy is stored in the inductor at full current? (c) At what rate in watts must energy be dissipated to switch the current off in 80.0 ms? (d) In view of the answer to the last part, is it surprising that shutting it down this quickly is difficult?

(a) 31.3 kV

(b) 125 kJ

(c) 1.56 MW

(d) No, it is not surprising since this power is very high.

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(a) Calculate the self-inductance of a 50.0 cm long, 10.0 cm diameter solenoid having 1000 loops. (b) How much energy is stored in this inductor when 20.0 A of current flows through it? (c) How fast can it be turned off if the induced emf cannot exceed 3.00 V?

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A precision laboratory resistor is made of a coil of wire 1.50 cm in diameter and 4.00 cm long, and it has 500 turns. (a) What is its self-inductance? (b) What average emf is induced if the 12.0 A current through it is turned on in 5.00 ms (one-fourth of a cycle for 50 Hz AC)? (c) What is its inductance if it is shortened to half its length and counter-wound (two layers of 250 turns in opposite directions)?

(a) 1.39 mH

(b) 3.33 V

(c) Zero

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The heating coils in a hair dryer are 0.800 cm in diameter, have a combined length of 1.00 m, and a total of 400 turns. (a) What is their total self-inductance assuming they act like a single solenoid? (b) How much energy is stored in them when 6.00 A flows? (c) What average emf opposes shutting them off if this is done in 5.00 ms (one-fourth of a cycle for 50 Hz AC)?

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When the 20.0 A current through an inductor is turned off in 1.50 ms, an 800 V emf is induced, opposing the change. What is the value of the self-inductance?

60.0 mH

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How fast can the 150 A current through a 0.250 H inductor be shut off if the induced emf cannot exceed 75.0 V?

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Integrated Concepts

A very large, superconducting solenoid such as one used in MRI scans, stores 1.00 MJ of energy in its magnetic field when 100 A flows. (a) Find its self-inductance. (b) If the coils “go normal,” they gain resistance and start to dissipate thermal energy. What temperature increase is produced if all the stored energy goes into heating the 1000 kg magnet, given its average specific heat is 200 J/kg·ºC ?

(a) 200 H

(b) 5.00ºC

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Unreasonable Results

A 25.0 H inductor has 100 A of current turned off in 1.00 ms. (a) What voltage is induced to oppose this? (b) What is unreasonable about this result? (c) Which assumption or premise is responsible?

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Questions & Answers

how did you get 1640
Noor Reply
If auger is pair are the roots of equation x2+5x-3=0
Peter Reply
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
MATTHEW Reply
420
Sharon
from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28
Salma
Devon is 32 32​​ years older than his son, Milan. The sum of both their ages is 54 54​. Using the variables d d​ and m m​ to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
Aaron Reply
find product (-6m+6) ( 3m²+4m-3)
SIMRAN Reply
-42m²+60m-18
Salma
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bill
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bill
-24m+3+3mÁ^2
Susan
i really want to learn
Amira
I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please
Amira
Hw did u arrive to this answer.
Aphelele
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Bajemah
-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18
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Jovelyn Reply
x=3-2y
Salma
y=x+3/2
Salma
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Enock
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Nandala
3x-12y=18
Kelvin
please why isn't that the 0is in ten thousand place
Grace Reply
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
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cameron Reply
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
mahnoor Reply
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00
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d=r×t the equation would be 8/r+24/r+4=3 worked out
Sheirtina
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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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