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Learning objectives

By the end of this section, you will be able to:

  • Calculate the current in an RL circuit after a specified number of characteristic time steps.
  • Calculate the characteristic time of an RL circuit.
  • Sketch the current in an RL circuit over time.

We know that the current through an inductor L size 12{L} {} cannot be turned on or off instantaneously. The change in current changes flux, inducing an emf opposing the change (Lenz’s law). How long does the opposition last? Current will flow and can be turned off, but how long does it take? [link] shows a switching circuit that can be used to examine current through an inductor as a function of time.

Part a of the figure shows an inductor connected in series with a resistor. The arrangement is connected across a cell by an on and off switch with two positions. When in position one, the battery, resistor, and inductor are in series and a current is established. In position two, the battery is removed and the current stops eventually because of energy loss in the resistor. Part b of the diagram shows the graph when the switch is in position one. It shows a graph for current growth verses time. The current is along the Y axis and the time is along the X axis. The graph shows a smooth rise from origin to a maximum value I zero corresponding to Y axis and value four tau on X axis. Part c of the diagram shows the graph when the switch is in position two. It shows a graph for current decay verses time is shown. The current is along the Y axis and the time is along the X axis. The graph is decreasing curve from a value I zero on Y axis, touching the X axis at a point where value of time equals four tau.
(a) An RL circuit with a switch to turn current on and off. When in position 1, the battery, resistor, and inductor are in series and a current is established. In position 2, the battery is removed and the current eventually stops because of energy loss in the resistor. (b) A graph of current growth versus time when the switch is moved to position 1. (c) A graph of current decay when the switch is moved to position 2.

When the switch is first moved to position 1 (at t = 0 size 12{t=0} {} ), the current is zero and it eventually rises to I 0 = V/R size 12{I rSub { size 8{0} } = ital "V/R"} {} , where R is the total resistance of the circuit. The opposition of the inductor L size 12{L} {} is greatest at the beginning, because the amount of change is greatest. The opposition it poses is in the form of an induced emf, which decreases to zero as the current approaches its final value. The opposing emf is proportional to the amount of change left. This is the hallmark of an exponential behavior, and it can be shown with calculus that

I = I 0 ( 1 e t / τ )     (turning on), size 12{I=I rSub { size 8{0} } \( 1 - e rSup { size 8{ - t/τ} } \) } {}

is the current in an RL circuit when switched on (Note the similarity to the exponential behavior of the voltage on a charging capacitor). The initial current is zero and approaches I 0 = V/R size 12{I rSub { size 8{0} } = ital "V/R"} {} with a characteristic time constant     τ for an RL circuit, given by

τ = L R , size 12{τ= { {L} over {R} } } {}

where τ size 12{τ} {} has units of seconds, since 1 H = 1 Ω · s . In the first period of time τ size 12{τ} {} , the current rises from zero to 0 . 632 I 0 size 12{0 "." "632"I rSub { size 8{0} } } {} , since I = I 0 ( 1 e 1 ) = I 0 ( 1 0 . 368 ) = 0 . 632 I 0 size 12{I=I rSub { size 8{0} } \( 1 - e rSup { size 8{ - 1} } \) =I rSub { size 8{0} } \( 1 - 0 "." "368" \) =0 "." "632"I rSub { size 8{0} } } {} . The current will go 0.632 of the remainder in the next time τ size 12{τ} {} . A well-known property of the exponential is that the final value is never exactly reached, but 0.632 of the remainder to that value is achieved in every characteristic time τ size 12{τ} {} . In just a few multiples of the time τ size 12{τ} {} , the final value is very nearly achieved, as the graph in [link] (b) illustrates.

The characteristic time τ size 12{τ} {} depends on only two factors, the inductance L size 12{L} {} and the resistance R size 12{R} {} . The greater the inductance L size 12{L} {} , the greater τ size 12{τ} {} is, which makes sense since a large inductance is very effective in opposing change. The smaller the resistance R size 12{R} {} , the greater τ size 12{τ} {} is. Again this makes sense, since a small resistance means a large final current and a greater change to get there. In both cases—large L size 12{L} {} and small R size 12{R} {} —more energy is stored in the inductor and more time is required to get it in and out.

When the switch in [link] (a) is moved to position 2 and cuts the battery out of the circuit, the current drops because of energy dissipation by the resistor. But this is also not instantaneous, since the inductor opposes the decrease in current by inducing an emf in the same direction as the battery that drove the current. Furthermore, there is a certain amount of energy, ( 1/2 ) LI 0 2 size 12{ \( "1/2" \) ital "LI" rSub { size 8{0} } rSup { size 8{2} } } {} , stored in the inductor, and it is dissipated at a finite rate. As the current approaches zero, the rate of decrease slows, since the energy dissipation rate is I 2 R size 12{ I rSup { size 8{2} } R} {} . Once again the behavior is exponential, and I is found to be

Questions & Answers

Determine the total force and the absolute pressure on the bottom of a swimming pool 28.0m by 8.5m whose uniform depth is 1 .8m.
Henny Reply
for the answer to complete, the units need specified why
muqaddas Reply
That's just how the AP grades. Otherwise, you could be talking about m/s when the answer requires m/s^2. They need to know what you are referring to.
Kyle
Suppose a speck of dust in an electrostatic precipitator has 1.0000×1012 protons in it and has a net charge of –5.00 nC (a very large charge for a small speck). How many electrons does it have?
Alexia Reply
how would I work this problem
Alexia
how can you have not an integer number of protons? If, on the other hand it supposed to be 1e12, then 1.6e-19C/proton • 1e12 protons=1.6e-7 C is the charge of the protons in the speck, so the difference between this and 5e-9C is made up by electrons
Igor
what is angular velocity
Obaapa Reply
angular velocity can be defined as the rate of change in radian over seconds.
Fidelis
Why does earth exert only a tiny downward pull?
Mya Reply
hello
Islam
Why is light bright?
Abraham Reply
what is radioactive element
Attah Reply
an 8.0 capacitor is connected by to the terminals of 60Hz whoes rms voltage is 150v. a.find the capacity reactance and rms to the circuit
Aisha Reply
thanks so much. i undersooth well
Valdes Reply
what is physics
Nwafor Reply
is the study of matter in relation to energy
Kintu
physics can be defined as the natural science that deals with the study of motion through space,time along with its related concepts which are energy and force
Fidelis
a submersible pump is dropped a borehole and hits the level of water at the bottom of the borehole 5 seconds later.determine the level of water in the borehole
Obrian Reply
what is power?
aron Reply
power P = Work done per second W/ t. It means the more power, the stronger machine
Sphere
e.g. heart Uses 2 W per beat.
Rohit
A spherica, concave shaving mirror has a radius of curvature of 32 cm .what is the magnification of a persons face. when it is 12cm to the left of the vertex of the mirror
Alona Reply
did you solve?
Shii
1.75cm
Ridwan
my name is Abu m.konnek I am a student of a electrical engineer and I want you to help me
Abu
the magnification k = f/(f-d) with focus f = R/2 =16 cm; d =12 cm k = 16/4 =4
Sphere
what do we call velocity
Kings
A weather vane is some sort of directional arrow parallel to the ground that may rotate freely in a horizontal plane. A typical weather vane has a large cross-sectional area perpendicular to the direction the arrow is pointing, like a “One Way” street sign. The purpose of the weather vane is to indicate the direction of the wind. As wind blows pa
Kavita Reply
hi
Godfred
what about the wind vane
Godfred
If a prism is fully imersed in water then the ray of light will normally dispersed or their is any difference?
Anurag Reply
the same behavior thru the prism out or in water bud abbot
Ju
If this will experimented with a hollow(vaccum) prism in water then what will be result ?
Anurag
Practice Key Terms 1

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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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