Chapter 3: Q 3.5-5E (page 121)

The power generated or dissipated by a circuit element equals the voltage across the element times the current through the element. Show that the power dissipated by a resistor equall²R, the power associated with an inductorequals the derivative of $\frac{1}{2} {LI}^{2}$ and the power associated with a capacitor equals the derivative of $\frac{1}{2} C {E_{c}}^{2}$ .

Short Answer

Expert verified

The power dissipated by a circuit is $P_{R} = I^{2} (t) R$ .

The power dissipated by an inductor is $P_{L} = \frac{1}{2} \frac{d}{dt} [{LI}^{2} (t)]$

The power associated by a capacitor is $P_{C} = \frac{1}{2} \frac{d}{dt} [C {E_{C}}^{2} (t)]$

Step by step solution

Evaluate the power dissipated by a circuit

The power is given by the equation P = I(t)V(t).

Since the voltage across resister is

$V (t) = E_{R} (t) = RI (t) P_{R} = I (t) RI (t) P_{R} = I^{2} (t) R$

Hence, the power dissipated by a circuit is role="math" localid="1664225724130" $P_{R} = I^{2} (t) R$ .

Find the power dissipated by an inductor.

Since the voltage across the inductor is

$V (t) = E_{L} (t) = L \frac{dI (t)}{dt} P_{L} = I (t) L \frac{dI (t)}{dt} = \frac{L}{2} 2 I (t) \frac{dI (t)}{dt} P_{L} = \frac{d}{dt} [\frac{{LI}^{2} (t)}{2}] P_{L} = \frac{1}{2} \frac{d}{dt} [{LI}^{2} (t)]$

Hence, the power dissipated by an inductor is $P_{L} = \frac{1}{2} \frac{d}{dt} [{LI}^{2} (t)]$

Determine the power associated by a capacitor

Since the voltage across the capacitor is

$V (t) = E_{C} (t) = \frac{1}{C} q (t) q (t) = C E_{c} (t) I (t) = \frac{dq (t)}{dt} = \frac{d [{CE}_{c} (t)]}{dt} P_{C} = \frac{d [{CE}_{C} (t)]}{dt} \frac{q (t)}{C} = \frac{C}{2} 2 E_{C} (t) \frac{{dE}_{C} (t)}{dt} P_{C} = \frac{d}{dt} [\frac{C {E_{C}}^{2} (t)}{2}] P_{C} = \frac{1}{2} \frac{d}{dt} [C {E_{C}}^{2} (t)]$

Hence, the power associated by a capacitor is $P_{C} = \frac{1}{2} \frac{d}{dt} [C {E_{C}}^{2} (t)]$

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 Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Short Answer

Step by step solution

Evaluate the power dissipated by a circuit

Find the power dissipated by an inductor.

Determine the power associated by a capacitor

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Pure Maths

Geometry

Decision Maths

Theoretical and Mathematical Physics

Applied Mathematics

Study anywhere. Anytime. Across all devices.