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The time constant τ L also tells us how quickly the induced voltage decays. At t = τ L , the magnitude of the induced voltage is

| V L ( τ L ) | = ε e −1 = 0.37 ε = 0.37 V ( 0 ) .

The voltage across the inductor therefore drops to about 37 % of its initial value after one time constant. The shorter the time constant τ L , the more rapidly the voltage decreases.

After enough time has elapsed so that the current has essentially reached its final value, the positions of the switches in [link] (a) are reversed, giving us the circuit in part (c). At t = 0 , the current in the circuit is I ( 0 ) = ε / R . With Kirchhoff’s loop rule, we obtain

I R + L d I d t = 0 .

The solution to this equation is similar to the solution of the equation for a discharging capacitor, with similar substitutions. The current at time t is then

I ( t ) = ε R e t / τ L .

The current starts at I ( 0 ) = ε / R and decreases with time as the energy stored in the inductor is depleted ( [link] ).

The time dependence of the voltage across the inductor can be determined from V L = L ( d I / d t ) :

V L ( t ) = ε e t / τ L .

This voltage is initially V L ( 0 ) = ε , and it decays to zero like the current. The energy stored in the magnetic field of the inductor, L I 2 / 2 , also decreases exponentially with time, as it is dissipated by Joule heating in the resistance of the circuit.

The graph of I versus t. The value of I at t equal to 0 is epsilon I R. I decreases with time till the curve reaches 0. At t equal to tau subscript L, the value of I is 0.37 epsilon I R.
Time variation of electric current in the RL circuit of [link] (c). The induced voltage across the coil also decays exponentially.

An RL Circuit with a source of emf

In the circuit of [link] (a), let ε = 2.0 V , R = 4.0 Ω , and L = 4.0 H . With S 1 closed and S 2 open ( [link] (b)), (a) what is the time constant of the circuit? (b) What are the current in the circuit and the magnitude of the induced emf across the inductor at t = 0 , at t = 2.0 τ L , and as t ?

Strategy

The time constant for an inductor and resistor in a series circuit is calculated using [link] . The current through and voltage across the inductor are calculated by the scenarios detailed from [link] and [link] .

Solution

  1. The inductive time constant is
    τ L = L R = 4.0 H 4.0 Ω = 1.0 s .
  2. The current in the circuit of [link] (b) increases according to [link] :
    I ( t ) = ε R ( 1 e t / τ L ) .

    At t = 0 ,
    ( 1 e t / τ L ) = ( 1 1 ) = 0 ; so I ( 0 ) = 0 .

    At t = 2.0 τ L and t , we have, respectively,
    I ( 2.0 τ L ) = ε R ( 1 e −2.0 ) = ( 0.50 A ) ( 0.86 ) = 0.43 A ,

    and
    I ( ) = ε R = 0.50 A .

    From [link] , the magnitude of the induced emf decays as
    | V L ( t ) | = ε e t / τ L .

    At t = 0 , t = 2.0 τ L , and as t , we obtain
    | V L ( 0 ) | = ε = 2.0 V , | V L ( 2.0 τ L ) | = ( 2.0 V ) e −2.0 = 0.27 V and | V L ( ) | = 0 .

Significance

If the time of the measurement were much larger than the time constant, we would not see the decay or growth of the voltage across the inductor or resistor. The circuit would quickly reach the asymptotic values for both of these. See [link] .

Figures a, b and c show the oscilloscope traces of voltage versus time of the voltage across source, the voltage across the inductor and the voltage across the resistor respectively. Figure a is a square wave varying from minus 12 volts to plus 12 volts, with a period from minus 10 ms to minus 0.001 ms. Figure b shows a square wave varying from minus 6 volts to plus 6 volts with a spike of 16 volts at the beginning of every crest and a spike of minus 16 volts at the beginning of every trough. The period is the same as that in figure a. Figure c shows a square wave varying from minus 0.3 to plus 0.3 volts, with spikes going out of the trace area in the positive direction at the beginnings of every crest and trough. The period of the wave is from minus 9.985 to plus 0.015 ms.
A generator in an RL circuit produces a square-pulse output in which the voltage oscillates between zero and some set value. These oscilloscope traces show (a) the voltage across the source; (b) the voltage across the inductor; (c) the voltage across the resistor.
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An RL Circuit without a source of emf

After the current in the RL circuit of [link] has reached its final value, the positions of the switches are reversed so that the circuit becomes the one shown in [link] (c). (a) How long does it take the current to drop to half its initial value? (b) How long does it take before the energy stored in the inductor is reduced to 1.0 % of its maximum value?

Strategy

The current in the inductor will now decrease as the resistor dissipates this energy. Therefore, the current falls as an exponential decay. We can also use that same relationship as a substitution for the energy in an inductor formula to find how the energy decreases at different time intervals.

Solution

  1. With the switches reversed, the current decreases according to
    I ( t ) = ε R e t / τ L = I ( 0 ) e t / τ L .

    At a time t when the current is one-half its initial value, we have
    I ( t ) = 0.50 I ( 0 ) so e t / τ L = 0.50 ,

    and
    t = [ ln ( 0.50 ) ] τ L = 0.69 ( 1.0 s ) = 0.69 s ,

    where we have used the inductive time constant found in [link] .
  2. The energy stored in the inductor is given by
    U L ( t ) = 1 2 L [ I ( t ) ] 2 = 1 2 L ( ε R e t / τ L ) 2 = L ε 2 2 R 2 e −2 t / τ L .

    If the energy drops to 1.0 % of its initial value at a time t , we have
    U L ( t ) = ( 0.010 ) U L ( 0 ) or L ε 2 2 R 2 e −2 t / τ L = ( 0.010 ) L ε 2 2 R 2 .

    Upon canceling terms and taking the natural logarithm of both sides, we obtain
    2 t τ L = ln ( 0.010 ) ,

    so
    t = 1 2 τ L ln ( 0.010 ) .

    Since τ L = 1.0 s , the time it takes for the energy stored in the inductor to decrease to 1.0 % of its initial value is
    t = 1 2 ( 1.0 s ) ln ( 0.010 ) = 2.3 s .

Significance

This calculation only works if the circuit is at maximum current in situation (b) prior to this new situation. Otherwise, we start with a lower initial current, which will decay by the same relationship.

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Questions & Answers

Using Kirchhoff's rules, when choosing your loops, can you choose a loop that doesn't have a voltage?
Michael Reply
how was the check your understand 12.7 solved?
Bysteria Reply
Who is ISSAAC NEWTON
LOAK Reply
he's the father of 3 newton law
Hawi
he is Chris Issaac's father :)
Ethem
how to name covalent bond
Bryan Reply
Who is ALEXANDER BELL
LOAK
what do you understand by the drift voltage
Brunelle Reply
what do you understand by drift velocity
Brunelle
nothing
Gamal
well when you apply a small electric field to a conductor that causes to add a little velocity to charged particle than usual, which become their average speed, that is what we call a drift.
graviton
drift velocity
graviton
what is an electromotive force?
Danilo Reply
It is the amount of other forms of energy converted into electrical energy per unit charge that flow through it.
Brunelle
How electromotive force is differentiated from the terminal voltage?
Danilo
in the emf power is generated while in the terminal pd power is lost.
Brunelle
what is then chemical name of NaCl
Sagar Reply
sodium chloride
Azam
sodium chloride
Brunelle
How can we differentiate between static point and test charge?
Comfort Reply
Wat is coplanar in physics
Humble Reply
two point charges +30c and +10c are separated by a distance of 80cm,compute the electric intensity and force on a +5×10^-6c charge place midway between the charges
Tijani Reply
0.0844kg
Humble
what is the difference between temperature and heat
Ishom Reply
Heat is the condition or quality of being hot While Temperature is ameasure of cold or heat, often measurable with a thermometer
Abdul
Temperature is the one of heat indicators of materials that can be measured with thermometers, and Heat is the quantity of calor content in material that can be measured with calorimetry.
Gamma
the average kinetic energy of molecules is called temperature. heat is the method or mode to transfer energy to molecules of an object but randomly, while work is the method to transfer energy to molecules in such manner that every molecules get moved in one direction.
Adnan
2. A brass rod of length 50cm and diameter 3mm is joined to a steel rod of the same length and diameter. What is the change in length of the combined rod at 250°c( degree Celsius) if the original length are 40°c(degree Celsius) is there at thermal stress developed at the junction? The end of the rod are free to expand (coefficient of linear expansion of brass = 2.0×10^-5, steel=1.2×10^-5k^1)
TONEJIT Reply
A charge insulator can be discharged by passing it just above a flame. Explain.
Mudassar Reply
of the three vectors in the equation F=qv×b which pairs are always at right angles?
DRASHTI Reply
what is an ideal gas?
Justine Reply
What is meant by zero Kelvin ?
Justine
Why does water cool when put in the pot ?
Justine
when we pour the water in a vessel(pot) the hot body(water) loses its heat to the surrounding in order to maintain thermal equilibrium.Thus,water cools.
rupendra
when we drop water in the pot, the pot body loses heat to surrounded in order to maintain thermal equilibrium thus,water cool.
Srabon
my personal opinion ideal gas means doesn't exist any gas that obey all rules that is made for gases, like when get the temp of a gas lower, it's volume decreases.since the gas will convert to liquid when the temp get lowest.. so you can imagine it, but you can't get a gas at the lowest T
Adnan
Edit An ideal gas is a theoretically gascomposed of many randomly moving point particles whose only interactions are perfectly elastic collisions.
Gamma
ideal gases are real gases at low temperature
Brunelle
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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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