Subiectul III

În figura alăturată este reprezentată schema unui circuit electric în care se cunosc: $E=14\ V$ , $r =3\ \Omega$ , $R=1,5\ \Omega$ , $R=2,5\ \Omega$ . Rezistenţa electrică a conductoarelor de legătură se neglijează. Determinaţi:

puterea totală dezvoltată de baterie când întrerupătorul este deschis;
energia consumată de circuitul exterior în timpul $t = 10\ min$ când întrerupătorul este deschis;
randamentul transferului de energie de la baterie către circuitul exterior când întrerupătorul este deschis;
valoarea rezistenței electrice a rezistorului ştiind că valoarea puterii disipate în circuitul exterior nu se modifică prin închiderea sau deschiderea întrerupătorului .

Rezolvare:

Calculăm puterea totală dezvoltată de baterie atunci când întrerupătorul este deschis.

$P_{tot}=E\cdot I$

$I=\frac{E}{R_{ext}+r}$

$\Leftrightarrow I=\frac{E}{R+R_1+r}$

$\begin{align*} \Rightarrow P_{tot}&=E\cdot\frac{E}{R+R_1+r}\\\\ &=\frac{E^2}{R+R_1+r}\\\\ &=\frac{14^2}{1,5+2,5+3}\\\\ &=\frac{196}{7}\\\\ &=28\ W \end{align*}$

$\begin{align*} \Leftrightarrow P_{tot}=28\ W \end{align*}$ .

Avem:

$\begin{align*} W&=U\cdot I\cdot t\\ &=I\cdot R_{ext}\cdot I \cdot t\\ &=I^2\cdot R_{ext}\cdot t \end{align*}$

$\begin{align*} R_{ext}&=R+R_1\\ &=1,5+2,5\\ &=4\ \Omega \end{align*}$

Scriem legea lui Ohm pentru întregul circuit:

$\begin{align*} I&=\frac{E}{R+R_1+r}\\\\ &=\frac{14}{1,5+2,5+3}\\\\ &=\frac{14}{7}\\\\ & =2\ A \end{align*}$

$\begin{align*} t&=10\ min\\ &= 10\cdot 60 \ s\\ &=600\ s \end{align*}$

$\begin{align*} \Rightarrow W&=2^2\cdot 4\cdot 600\\ &=4\cdot 2400\\ & =9600\ J\\ &=9,6\ kJ \end{align*}$

$\begin{align*} \Leftrightarrow W=9,6\ kJ \end{align*}$ .

Determinăm randamentul transferului de energie de la baterie către circuitul exterior când întrerupătorul este deschis.

$\begin{align*} \eta&=\frac{P_{ext}}{P_{totala}}\\\\ &=\frac{U\cdot I}{E\cdot I}\\\\ &=\frac{I\cdot R_{ext}\cdot I}{I\cdot (R_{ext}+r)\cdot I}\\\\ & =\frac{R_{ext}}{R_{ext}+r}\\\\ & =\frac{4}{4+3}\\\\ &=\frac{4}{7}\\\\ & \simeq 0,57\\\\ & =57\% \end{align*}$

$\begin{align*} \Leftrightarrow \eta =57\% \end{align*}$ .

Avem:

$\begin{align*} P&=U\cdot I\\ & =I\cdot R_{ext}\cdot I\\ & =I^2\cdot R_{ext} \end{align*}$

$\begin{align*} P'&=U'\cdot I'\\ & =I'\cdot R'_{ext}\cdot I'\\ &=I'^2\cdot R'_{ext} \end{align*}$

$\begin{align*} P=P' \end{align*}$

$\begin{align*} \Rightarrow I^2\cdot R_{ext}=I'^2\cdot R'_{ext} \end{align*}$

$\begin{align*} &I=\frac{E}{R_{ext}+r}\\\\ &I'=\frac{E}{R'_{ext}+r} \end{align*}$

$\begin{align*} &\Rightarrow \left (\frac{E}{R_{ext}+r} \right )^2\cdot R_{ext}=\left (\frac{E}{R'_{ext}+r} \right )^2\cdot R'_{ext} \\\\&\Leftrightarrow \frac{E^2}{(R_{ext}+r)^2}\cdot R_{ext}=\frac{E^2}{(R'_{ext}+r)^2}\cdot R'_{ext} \\\\&\Rightarrow \frac{R_{ext}}{(R_{ext}+r)^2}=\frac{R'_{ext}}{(R'_{ext}+r)^2} \\\\&\Rightarrow R_{ext}\cdot (R'_{ext}+r)^2=R'_{ext}\cdot (R_{ext}+r)^2 \\\\&\Leftrightarrow R_{ext}\cdot R'^2_{ext} + 2\cdot R_{ext}\cdot R'_{ext}\cdot r +R_{ext}\cdot r^2=R'_{ext}\cdot R^2_{ext}+2\cdot R'_{ext}\cdot R_{ext}\cdot r +R'_{ext}\cdot r^2 \\\\&\Rightarrow R_{ext}\cdot R'^2_{ext}+R_{ext}\cdot r^2=R'_{ext}\cdot R^2_{ext}+R'_{ext}\cdot r^2\ \ \ (1) \end{align*}$

Împărţim relaţia $\begin{align*} (1) \end{align*}$ la $\begin{align*} (R_{ext}\cdot R'_{ext}) \end{align*}$ :

$\begin{align*} &\Rightarrow \frac{R_{ext}\cdot R'^2_{ext}}{R_{ext}\cdot R'_{ext}}+\frac{R_{ext}\cdot r^2}{R_{ext}\cdot R'_{ext}}=\frac{R'_{ext}\cdot R^2_{ext}}{R_{ext}\cdot R'_{ext}}+\frac{R'_{ext}\cdot r^2}{R_{ext}\cdot R'_{ext}} \\\\&\Rightarrow R'_{ext}+\frac{r^2}{R'_{ext}}=R_{ext}+\frac{r^2}{R_{ext}} \\\\&\Rightarrow R'_{ext}-R_{ext}=r^2\cdot \left ( \frac{1}{R_{ext}}-\frac{1}{R'_{ext}} \right ) \\\\&\Rightarrow R'_{ext}-R_{ext}=r^2\cdot \frac{R'_{ext}-R_{ext}}{R'_{ext}\cdot R_{ext}}\ \Big|\cdot \frac{R'_{ext}\cdot R_{ext}}{R'_{ext}- R_{ext}} \\\\&\Rightarrow \left (R'_{ext}-R_{ext} \right )\cdot \frac{R'_{ext}\cdot R_{ext}}{R'_{ext}-R_{ext}}=r^2 \\\\&\Rightarrow R'_{ext}\cdot R_{ext}=r^2 \end{align*}$

$\begin{align*}\Rightarrow R'_{ext}&=\frac{r^2}{R_{ext}}\\\\ &=\frac{r^2}{R+R_1}\\\\ &=\frac{9}{1,5+2,5}\\\\ &=\frac{9}{4}\\\\ &=2,25\ \Omega \end{align*}$

$\begin{align*} R'_{ext}=R+R'_{ep} \end{align*}$

$\begin{align*} \Rightarrow R'_{ep}&=R'_{ext}-R\\ &=2,25-1,5\\ &=0,75\ \Omega \end{align*}$

$\begin{align*} &\frac{1}{R'_{ep}}=\frac{1}{R_1}+\frac{1}{R_2} \\\\&\Rightarrow \frac{1}{R_2}=\frac{1}{R'_{ep}}-\frac{1}{R_1} \\\\&\Rightarrow \frac{1}{R_2}=\frac{R_1-R'_{ep}}{R_1\cdot R'_{ep}} \end{align*}$

$\begin{align*} \Rightarrow R_2&=\frac{R_1\cdot R'_{ep}}{R_1-R'_{ep}}\\\\ &=\frac{2,5\cdot 0,75}{2,5-0,75}\\\\ &=\frac{1,875}{1,75}\\\\ &\simeq 1,07\ \Omega \end{align*}$

$\begin{align*} \Leftrightarrow R_2\simeq 1,07\ \Omega \end{align*}$ .