Chapter 30: Q78P (page 899)

At time t = 0, a 12.0 V potential difference is suddenly applied to the leads of a coil of inductance 23.0 mHand a certain resistance R. At time t = 0.150 ms, the current through the inductor is changing at the rate of 280 A/s. Evaluate R.

Short Answer

Expert verified

The value for R will be $95 Ω$ .

Step by step solution

Given

$L = 0.023 H$ .
$ε = 12 V$ .
$t = 0.00015$ .
$\frac{d i}{d t} = 280 A/s$ .

Understanding the concept

We have the equation for the rise in the current. We can take the derivative for the current to evaluate the value for R.

Formula:

$i = \frac{ε}{R} (1 - e^{- t / τ_{L}})$

$τ_{L} = \frac{L}{R}$

Step 3: Evaluate the value for R

We have,

$i = \frac{ε}{R} (1 - e^{- \frac{t}{τ_{L}}})$

We take derivative for both sides with respect to t, we get,

$\frac{d i}{d t} = \frac{d (\frac{ε}{R} (1 - e^{- \frac{t}{τ_{L}}}))}{d t}$

We have,

$\begin{array}{rcl} τ_{L} & = & \frac{L}{R} \\ \frac{d i}{d t} & = & \frac{d (\frac{ε}{R} (1 - e^{- \frac{t}{(\frac{L}{R})}}))}{d t} \\ \frac{d i}{d t} & = & \frac{ε R}{R L} \times e^{- \frac{t}{(\frac{L}{R})}} \\ \frac{d i}{d t} & = & \frac{ε}{L} \times e^{- \frac{t}{(\frac{L}{R})}} \\ \frac{L}{ε} \times \frac{d i}{d t} & = & e^{- \frac{t}{(\frac{L}{R})}} \\ e^{\frac{- R t}{L}} & = & \frac{0.023 \times 280}{12} \\ e^{\frac{- R t}{L}} & = & 0.537 \end{array}$

Taking natural log at both sides, we get,

$\begin{array}{rcl} - \frac{R t}{L} & = & \log 0.536 \\ \frac{R t}{L} & = & 0.62 \\ R & = & \frac{0.62 \times L}{t} \\ = & \frac{0.62 \times 0.023}{0.00015} \\ = & 95.0 Ω \end{array}$

Therefore, the value of resistance is $95.0 Ω$

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!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

At time t = 0, a 12.0 V potential difference is suddenly applied to the leads of a coil of inductance 23.0 mHand a certain resistance R. At time t = 0.150 ms, the current through the inductor is changing at the rate of 280 A/s. Evaluate R.

Short Answer

Step by step solution

Given

Understanding the concept

Step 3: Evaluate the value for R

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Wave Optics

Electrostatics

Energy Physics

Space Physics

Famous Physicists

Electricity and Magnetism

Study anywhere. Anytime. Across all devices.