# 23.10 Rl circuits

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• 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 $L$ 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.

When the switch is first moved to position 1 (at $t=0$ ), the current is zero and it eventually rises to ${I}_{0}=\text{V/R}$ , where $R$ is the total resistance of the circuit. The opposition of the inductor $L$ 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

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 ${I}_{0}=\text{V/R}$ with a characteristic time constant     $\tau$ for an RL circuit, given by

$\tau =\frac{L}{R}\text{,}$

where $\tau$ has units of seconds, since $\text{1}\phantom{\rule{0.25em}{0ex}}\text{H}\text{=}\text{1}\phantom{\rule{0.25em}{0ex}}\text{Ω}\text{·}\text{s}$ . In the first period of time $\tau$ , the current rises from zero to $0\text{.}\text{632}{I}_{0}$ , since $I={I}_{0}\left(1-{e}^{-1}\right)={I}_{0}\left(1-0\text{.}\text{368}\right)=0\text{.}\text{632}{I}_{0}$ . The current will go 0.632 of the remainder in the next time $\tau$ . 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 $\tau$ . In just a few multiples of the time $\tau$ , the final value is very nearly achieved, as the graph in [link] (b) illustrates.

The characteristic time $\tau$ depends on only two factors, the inductance $L$ and the resistance $R$ . The greater the inductance $L$ , the greater $\tau$ is, which makes sense since a large inductance is very effective in opposing change. The smaller the resistance $R$ , the greater $\tau$ 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 $L$ and small $R$ —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, $\left(\text{1/2}\right){\text{LI}}_{0}^{2}$ , 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 ${I}^{2}R$ . Once again the behavior is exponential, and $I$ is found to be

#### Questions & Answers

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effiom
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then what
Amera
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aliyu
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Richard
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Jason
is a one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimensions.
Praise
what is a classical electrodynamics?
Marga
what is dynamics
Marga
dynamic is the force that stimulates change or progress within the system or process
Oze
what is the formula to calculate wavelength of the incident light
David Reply
if a spring is is stiffness of 950nm-1 what work will be done in extending the spring by 60mmp
Hassan Reply
State the forms of energy
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machanical
Ridwan
Word : Mechanical wave Definition : The waves, which need a material medium for their propagation, e.g., Sound waves. \n\nOther Definition: The waves, which need a material medium for their propagation, are called mechanical waves. Mechanical waves are also called elastic waves. Sound waves, water waves are examples of mechanical waves.t Definition: wave consisting of periodic motion of matter; e.g. sound wave or water wave as opposed to electromagnetic wave.h
Clement Reply
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Akinpelu
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Akinpelu Reply
a wave which require material medium for its propagation
syed
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Samuel Reply
watt
Okoli
Am I correct
Okoli
it can be in kilowatt, megawatt and so
Femi
yes
Femi
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Jaheim
kW
Akinpelu
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Samuel
SI.unit of power is.watt=j/c.but kw.and Mw are bigger.umots
syed
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aish Reply
study of matter and its nature
Akinpelu
The word physics comes from a Greek word Physicos which means Nature.The Knowledge of Nature. It is branch of science which deals with the matter and energy and interaction between them.
Uniform
why in circular motion, a tangential acceleration can change the magnitude of the velocity but not its direction
Syafiqah Reply
reasonable
Femi
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MUSTAPHA
What is a wave
Mutuma Reply
Tramsmission of energy through a media
Mateo
is the disturbance that carry materials as propagation from one medium to another
Akinpelu
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Akinpelu
find the triple product of (A*B).C given that A =i + 4j, B=2i - 3j and C = i + k
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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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