An inductor is connected to the terminals of a battery that...

An inductor is connected to the terminals of a battery that has an emf of 12.0 $\mathrm{V}$ and negligible internal resistance. The current is 4.86 $\mathrm{mA}$ at 0.725 $\mathrm{ms}$ after the connection is completed. After a long time the current is 6.45 $\mathrm{mA}$ . What are (a) the resistance $R$ of the inductor and (b) the inductance $L$ of the inductor?

Key Concepts

RL Circuit Analysis

An RL circuit is a basic electrical circuit that includes a resistor (R) and an inductor (L) connected in series with a power source. It is used to study how current changes over time when the circuit is switched on, highlighting the interplay between resistive and inductive effects.

Time Constant

The time constant, represented as ? = L/R, is a measure of the response speed of an RL circuit. It determines how quickly the current in the circuit reaches its steady-state value after a change, such as closing a switch.

Exponential Transient Response

Upon energizing an RL circuit, the current does not instantly reach its steady-state value but follows an exponential growth described by I(t) = I_final * (1 - exp(-t/?)). This expression characterizes the transient behavior from the initial condition to the final steady state.

Steady-State Behavior

In the long-term after the transient effects have decayed, an RL circuit reaches steady state. At this point, the inductor acts like a short circuit, and the circuit current is determined solely by the resistor according to Ohm's law (I = emf/R), simplifying the analysis.

Transcript

00:01 For this problem we have an rl circuit.

00:05 We're told that at 0 .725 milliseconds there's a current of 4 .86 milliamps and the long time current as t goes to infinity is 6 .45mm.

00:18 For the first part we want to find what this resistance is.

00:21 So we'll want to recall that the current is a function of time for an rl circuit is given by our voltage, whatever that ems is here we're told that's 12 volts and then 1 minus e to the minus r over l our time constant multiplied by t so our long time current is just going to be given by v over r we're told what that is that's 6 .45 mll amps so that means that our resistance is going to be our voltage i'm going to call that i max because that's our maximum current so that's going to be 12 volts divided by 6 .45 millie amps.

01:15 And this gets us to three significant figures, 1 ,160 oms.

01:23 For the second part, we're going to find what our inductance is.

01:27 And for this, we need to utilize two things.

01:30 We need to utilize that information about the current being 4 .86 milliamps at 0 .725 milliseconds, and we need to utilize what our resistance is.

01:41 With both of those things, we can use this time equation for the current to solve for our inductees.

01:48 So let's see what this gets us.

01:53 First, i'm going to do some shorthand notation, and i'm not going to plug in anything to the end, just because it would be a little bit messy if i did.

02:03 So i'm going to call this t equals 25 milliseconds, 4 .86.

02:18 Millie amps i'm going to call this i tilda and i'm also going to call t equals 7 .25 milliseconds i'm going to call this t tilde and we'll see why this is useful doing this once once things start getting a little bit messy so we're told that i tilde v over r 1 minus e to the minus r over l t t tilda.

02:56 And we want to solve this equation for our l.

02:59 So let's start rearranging this.

03:03 This gets us 1 minus e to the r over l t t tilda.

03:08 We see that that is equal to i r or i tilda r over v.

03:18 I'm going to swoop this around here...

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