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Found in: Page 938

### Fundamentals Of Physics

Book edition 10th Edition
Author(s) David Halliday
Pages 1328 pages
ISBN 9781118230718

# What direct current will produce the same amount of thermal energy, in a particular resistor, as an alternating current that has a maximum value of 2.60 A?

The value of the direct current is 1.84 A.

See the step by step solution

## Step 1: Listing the given quantities:

The current, $\mathrm{i}=2.60\mathrm{A}$

## Step 2: Understanding the concepts of power dissipation:

Average power dissipated in resistance R when the current is AC and DC should be equated to find the direct current which will produce the same amount of thermal energy as an alternating current of 2.60 A

Formula:

Power Dissipation, ${\mathrm{P}}_{\mathrm{d}}={\mathrm{i}}_{\mathrm{d}}^{2}\mathrm{R}$

## Step 3: Calculation of the value of the direct current:

Average power dissipated in resistance R when current is AC.

${\mathrm{P}}_{\mathrm{avg}}={\mathrm{i}}_{\mathrm{rms}}^{2}\mathrm{R}$

The power is dissipated in resistor R when current is DC.

${\mathrm{P}}_{\mathrm{d}}={\mathrm{i}}_{\mathrm{d}}^{2}\mathrm{R}$

Both these powers produce the same thermal energy. Hence,

$\begin{array}{rcl}{\mathrm{P}}_{\mathrm{avg}}& =& {\mathrm{P}}_{\mathrm{d}}\\ {\mathrm{i}}_{\mathrm{rms}}^{2}\mathrm{R}& =& {\mathrm{i}}_{\mathrm{d}}^{2}\mathrm{R}\\ {\mathrm{i}}_{\mathrm{d}}& =& {\mathrm{i}}_{\mathrm{rms}}\\ & =& \frac{\mathrm{i}}{\sqrt{2}}\end{array}$

Substitute 2.60A for i in the above equation.

$\begin{array}{rcl}{\mathrm{i}}_{\mathrm{d}}& =& \frac{\mathrm{i}}{\sqrt{2}}\\ & =& \frac{2.6}{\sqrt{2}}\\ & =& 1.84\mathrm{A}\end{array}$

Hence, the value of the direct current is 1.84A.