Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 27: Problem 52

The current in a series \(R L\) circuit rises to half its final value in \(7.6 \mathrm{s}\) What's the time constant?

Short Answer

Expert verified
The time constant for the circuit is approximately \( 10.97s \)

Step by step solution

01

Identify the given variables

The given variables from the exercise are: Time \( t = 7.6s \), Initial value \( V_0 = 1 \) (since it's relative to the final value), and Final value \( V_f = 2 \) (as the current doubles to reach half its final value).
02

Apply the formula for RL circuits

The formula is \( t = τ \ln \left(\frac{V_f}{V_0}\right) \). Substituting in the known variables gives us \( 7.6 = τ \ln \left(\frac{2}{1}\right) \).
03

Simplify the expression

Simplify the equation to find the time constant. The natural logarithm of 2, \( \ln(2) \), is approximately \( 0.693 \). So our equation becomes \( 7.6 = τ * 0.693 \).
04

Solve for the time constant

The last step is to solve for the time constant τ. To do this, both sides of the equation need to be divided by \(0.693\), to isolate τ on its own. The calculation is \(τ = 7.6 / 0.693\).
05

Calculate the time constant

Doing the calculation gives us \(τ ≈ 10.97s\). So the time constant for the circuit is approximately \(10.97s\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A 2.0 -A current is flowing in a \(20-\mathrm{H}\) inductor. A switch opens, interrupting the current in 1.0 ms. Find the induced emf in the inductor.

A magnetic field is given by \(\vec{B}=B_{0}\left(x / x_{0}\right)^{2} \hat{k},\) where \(B_{0}\) and \(x_{0}\) are constants. Find an expression for the magnetic flux through a square of side \(2 x_{0}\) that lies in the \(x\) -y plane with one corner at the origin and sides coinciding with the positive \(x\) - and \(y\) -axes.

A stent is a cylindrical tube, often made of metal mesh, that's inserted into a blood vessel to overcome a constriction. It's sometimes necessary to heat the stent after insertion to prevent cell growth that could cause the constriction to recur. One method is to place the patient in a changing magnetic field, so that induced currents heat the stent. Consider a stainless- steel stent 12 mm long by 4.5 mm diameter, with total resistance \(41 \mathrm{m} \Omega\). Treating the stent as a wire loop in the optimum orientation, find the rate of change of magnetic field needed for a heating power of \(250 \mathrm{mW}.\)

What's the current in a 10 -mH inductor storing \(50 \mu\) J of energy?

A square wire loop of side \(l\) and resistance \(R\) is pulled with constant speed \(v\) from a region of no magnetic field until it's fully inside a region of constant, uniform magnetic field \(\vec{B}\) perpendicular to the loop plane. The boundary of the ficld region is parallel to one side of the loop. Find an expression for the total work done by whatever is pulling the loop.
See all solutions

Recommended explanations on Physics Textbooks

Measurements

Read Explanation

Wave Optics

Read Explanation

Electricity

Read Explanation

Electric Charge Field and Potential

Read Explanation

Space Physics

Read Explanation

Fundamentals of Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free