Question

The voltage $v$ across a device and the current $i$ through it are \[v(t)=5 \cos 2 t \mathrm{V}, \quad i(t)=10\left(1-e^{-0.5 t}\right) \mathrm{A}\] Calculate: (a) the total charge in the device at $t=1$ s (b) the power consumed by the device at $t=1$ s.

Name: Problem 11, Chapter 1: Fundamentals of Electric Circuits
Uploaded: 2022-01-16T07:08:38-08:00
Duration: 1 min 22 s
Channel: Varsha Aggarwal
Description: The voltage v across a device and the current i through it are v(t)=5 cos 2 t V, i(t)=10(1-e^-0.5 t) A Calculate: (a) the total charge in the device at t=1 s (b) the power consumed by the device at t=1 s.

   The voltage $v$ across a device and the current $i$ through it are
\[v(t)=5 \cos 2 t \mathrm{V}, \quad i(t)=10\left(1-e^{-0.5 t}\right) \mathrm{A}\]
Calculate:
(a) the total charge in the device at $t=1$ s
(b) the power consumed by the device at $t=1$ s.

Fundamentals of Electric Circuits

Charles K.… 3rd Edition

Chapter 1, Problem 11

Instant Answer

Solved by Expert Varsha Aggarwal

01/16/2022

Step 1

So, we need to calculate the integral of $i(t)$ from $0$ to $1$ s. The integral of $i(t)$ is given by: \[\int i(t) dt = \int_0^1 10(1-e^{-0.5t}) dt\] Show more…

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The voltage $v$ across a device and the current $i$ through it are \[v(t)=5 \cos 2 t \mathrm{V}, \quad i(t)=10\left(1-e^{-0.5 t}\right) \mathrm{A}\] Calculate: (a) the total charge in the device at $t=1$ s (b) the power consumed by the device at $t=1$ s.

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Key Concepts

Instantaneous Power in Electrical Circuits

The instantaneous power in an electrical circuit is given by the product of the instantaneous voltage and the instantaneous current. This concept is essential for determining how much energy per unit time is being consumed or delivered by a device at any given moment. Calculating power at a specific time requires evaluating both the voltage and the current at that particular moment, reflecting the dynamic energy interactions within the circuit.

Integration of Current to Determine Charge

Charge, defined as the amount of electrical energy transferred over time, can be calculated by integrating the current with respect to time. This concept is fundamental in circuit analysis where current, representing how fast charge flows, is used to determine the net charge accumulated in a device by summing up infinitesimal contributions of current over a time interval.

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