Question

an ac current I= Io sin wt produces certain heat H in a resistor R over a time T=2π/w write the value of Dc current that would produce the same heat in the same resistor in the same time​

Answer 1

Given:

An ac current I= Io sin wt produces certain heat H in a resistor R over a time T=2π/w

To find:

Value of Dc current that would produce the same heat in the same resistor in the same time

Calculation:

Total heat by AC current:

$\rm{H} = \displaystyle \rm{\int {I}^{2} R \: dt}$

$\displaystyle = > \rm{H = \int { \{I_{0} \sin( \omega t) \}}^{2} R \: dt}$

$\displaystyle = > \rm{H = \int { I_{0}}^{2} R \: { \sin}^{2}( \omega t) dt}$

$\displaystyle = > \rm{H = ({ I_{0}}^{2} R) \int \: { \sin}^{2}( \omega t) dt}$

$\displaystyle = > \rm{H = \dfrac{({ I_{0}}^{2} R)}{2} \int \: \{1 - \cos( 2\omega t) \}dt}$

$\displaystyle = > \rm{H = \dfrac{({ I_{0}}^{2} R)}{2} \int_{0}^{ \frac{2\pi}{ \omega} } \: \{1 - \cos( 2\omega t) \}dt}$

$\displaystyle = > \rm{H = \dfrac{({ I_{0}}^{2} R)}{2} \times \{ \frac{2\pi}{ \omega} - \dfrac{\sin(4\pi)}{2 \omega} \} }$

$\displaystyle = > \rm{H = \dfrac{({ I_{0}}^{2} R)}{2} \times \{ \frac{2\pi}{ \omega} - 0 \} }$

$\displaystyle = > \rm{H = \dfrac{({ I_{0}}^{2} R)}{2} \times ( \frac{2\pi}{ \omega} ) }$

$\displaystyle = > \rm{H = \dfrac{({ I_{0}}^{2} R\pi)}{ \omega} }$

Let DC current be I ;

Now , heat by DC current:

$\displaystyle = > \rm{H = I^{2} R \: t}$

Since both currents are same:

$\therefore \: \rm{ \dfrac{({ I_{0}}^{2} R\pi)}{ \omega} = {I}^{2} R t }$

$= > \: \rm{ \dfrac{({ I_{0}}^{2} R\pi)}{ \omega} = {I}^{2} R \times ( \dfrac{2\pi}{ \omega} ) }$

$\rm{ = > {I}^{2} = \dfrac{ { I_{0} }^{2} }{2} }$

$\rm{ = > I = \dfrac{ I_{0} }{ \sqrt{2} } }$

So, final answer is:

$\boxed{ \bf{I = \dfrac{ I_{0} }{ \sqrt{2} } } }$