In Fig. 32-36, a 12.0 V ideal battery, a 20 Ωresistor, and an ideal inductor are connected by a

In Fig. 32-36, a 12.0 V ideal battery, a 20 Ωresistor, and...

In Fig. 32-36, a $12.0 \mathrm{~V}$ ideal battery, a $20 \Omega$ resistor, and an ideal inductor are connected by a switch at time $t=0 \mathrm{~s}$. At what rate is the battery transferring energy to the inductor's field at $t=1.61 \tau_{L}$ ?

Key Concepts

RL Circuit

An RL circuit contains a resistor and an inductor connected, often with a battery or other voltage source. This circuit is fundamental in analyzing how current builds up and decays over time due to the inductor’s opposition to changes in current, making it a classic example in transient analysis and basic circuit theory.

Transient Response

When a circuit is switched on, it does not reach its steady-state condition instantaneously. The transient response describes the time-dependent behavior of current and voltage as the inductor and resistor interact, typically resulting in an exponential increase or decrease of current, which is critical for understanding energy build-up in inductors.

Time Constant (?L)

The time constant for an RL circuit, defined as ?L = L/R, quantifies the rate at which the current approaches its steady-state value. It represents the characteristic time over which significant changes occur in the transient response, and evaluating circuit behavior at multiples of the time constant provides insight into how rapidly the system evolves.

Energy Stored in an Inductor

The energy stored in the magnetic field of an inductor is given by the expression (1/2)LI², where L is the inductance and I is the current. This concept is important to understand how energy is temporarily held within the magnetic field during transient events, and how that energy changes over time as the current builds or decays.

Rate of Energy Transfer (Power)

The rate at which energy is transferred to or from a circuit component is defined as power. In the context of an RL circuit, determining the instantaneous power delivered to the inductor's magnetic field involves examining the time derivative of the stored energy or calculating the product of the instantaneous voltage across the inductor and the current. This is crucial for analyzing how quickly energy is being stored during transient conditions.

Transcript

00:01 Okay, so for this problem, we have a resistor and inductor connected, and we want to know at what rate is the battery transferring energy to the inductor.

00:09 At this particular time, t equals 1 .61, tau, l.

00:18 Oh, is that tau l? yeah, tau inductor, tau l.

00:22 Okay, so here we go.

00:25 Let's do some derivatives.

00:26 So we were trying to evaluate d -u -d -t at t equals 1 .61 tau -l.

00:37 And then so that means we need to evaluate the derivative of the energy of an inductor.

00:46 So it's 1 half l, i squared.

00:51 And then we can plug in our formula for the current in a charging inductor, which is this v over r times 1 minus e to the minus t over tau.

01:06 So i'm basically i'm just going to plug this in here.

01:11 And then, oh yeah, we're giving some other given to all.

01:14 I'll write those down.

01:14 But anyway, i'm going to plug in this eye here and then evaluate do udt at t equals 1 .6 tau -l.

01:21 And then before i do that, let me just go ahead and write down the other givens.

01:24 So the emf, oh, i'll use voltage here.

01:38 So it's 12 .0 volts and r is 20 oms.

01:51 Do they actually give you the inductance? oh no, but luckily you don't need it.

01:59 Okay, because it's not really luck.

02:03 Okay, so anyway, so let's go ahead and evaluate u as a function of i.

02:09 So i can say that u is equal to one half l times this whole thing squared, so then that's b squared over r squared.

02:24 And then i'll just go ahead and expand it.

02:26 So 1 minus 2e to the minus t over tau, l, plus e to the minus 2 tau over, t or tau.

02:42 Okay, and let me get that other parenthesis.

02:45 Okay, wonderful...

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