Question

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?

Short answer

The value of the direct current is 1.84 A.

Step-by-step solution

Listing the given quantities:

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

Understanding the concepts of power dissipation:

Average power dissipated in resistance Rwhen the current isACand 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.