Question

A cell of emf $E$ and internal resistance $r$ drives a current $i$ through an external resistance $R$

• A. the cell supplies $Ei$ power
• B. heat is produced in R at the rate $Ei$
• C. heat is produced in $R$ at the rate $Ei\left(\frac{R}{R+r}\right)$
• D. heat is produced in the cell at the rate $Ei\left(\frac{r}{R+r}\right)$

Answer 1

Answer: (A), (C), (D)

The correct options are
A the cell supplies $Ei$ power
C heat is produced in $R$ at the rate $Ei\left(\frac{R}{R+r}\right)$
D heat is produced in the cell at the rate $Ei\left(\frac{r}{R+r}\right)$

Initially, the total current $i$, supplied by the cell is $($consider external resistance $R$ is not connected in the circuit.$)$
$i=\frac{E}{r}$ or $E=ir$.................................................................$\left(1\right)$
The power supplied by the cell is
$P=\frac{E^2}{r}$
$P=E\frac{E}{r}$
$P=Ei$ ..................................................................from$\left(1\right)$
When the resistance $R$ is connected in circuit the total current $i$, will now
$i=\frac{E}{R+r}$ or $E=i(R+r)$ .................................$\left(2\right)$
The heat produced in $R$ in time $t$ is
$Q_R=i^{2}Rt$
$Q_R=(\frac{E}{R+r}{)}^{2}Rt$
$Q_R=E^2\left[\frac{R}{(R+r{)}^2}\right]t$..........................................from $\left(2\right)$
$Q_R=E.\left(i\right(R+r\left)\right)\left[\frac{R}{(R+r{)}^2}\right]t$.......................................................................from $\left(2\right)$
$Q_R=Ei\left[\frac{R}{(R+r)}\right]t$
$\frac{Q_R}{t}=Ei\left[\frac{R}{(R+r)}\right]$
Hence, the heat is produced in R at the rate $Ei\left[\frac{R}{(R+r)}\right]$
Now, The heat produced in cell in time t is
$Q_c=i^{2}rt$
$Q_c={\left(\frac{E}{R+r}\right)}^{2}rt$
$Q_c=E^2\left[\frac{r}{(R+r{)}^2}\right]t$.........................................from $\left(2\right)$
$Q_c=E.\left(i\right(R+r\left)\right)\left[\frac{r}{(R+r{)}^2}\right]t$..............,,,,,....from $\left(2\right)$
$Q_c=Ei\left[\frac{r}{(R+r)}\right]t$
$\frac{Q_c}{t}=Ei\left[\frac{r}{(R+r)}\right]$
Hence, the heat is produced in cell at the rate $Ei\left[\frac{r}{(R+r)}\right]$