Current Electricity Set-5

Q1. A uniform metallic wire of length $L$ and resistance $R$ is stretched uniformly to length $2L$ without changing its volume. The new resistance of the wire is:

Q2. Two identical cells each of emf $E$ and internal resistance $r$ are connected to the same external resistor $R$ in two ways: (i) in series and (ii) in parallel (polarities aligned). For which condition on $R$ does the parallel connection deliver more power to $R$ than the series connection?

Q3. In a potentiometer the potential gradient along the wire is $10\,\text{mV/cm}$. An unknown cell gives a balancing length $120\,\text{cm}$ when open-circuited. When a $10\,\Omega$ resistor is connected across the cell, the balancing length reduces to $96\,\text{cm}$. The internal resistance of the cell is:

Q4. A straight wire of length $L$ and constant cross-sectional area has resistivity varying linearly along its length from $\rho_0$ at $x=0$ to $5\rho_0$ at $x=L$. A potential difference $V$ is applied across the two ends. The electric potential at the midpoint $x=L/2$ measured from the end at $x=0$ equals:

Q5. A battery of emf $E$ and internal resistance $r$ supplies a resistor $R_1=5\,\Omega$ and the measured current is $I_1=1.5\,\text{A}$. When the resistor is replaced by $R_2=10\,\Omega$, the measured current becomes $I_2=1.0\,\text{A}$. The values of $E$ and $r$ are:

Q6. A copper wire of cross-sectional area $1\ \text{mm}^2$ carries a steady current of $2\ \text{A}$. If the number density of free electrons in copper is $n=8.5\times10^{28}\ \text{m}^{-3}$ and the electron charge is $e=1.6\times10^{-19}\ \text{C}$, the magnitude of the drift velocity $v_d$ of electrons in the wire is (use $I=nAe v_d$):

Q7. A battery of emf $12\ \text{V}$ and internal resistance $1\ \Omega$ is connected to a resistor $R_1=2\ \Omega$ in series, and then to a parallel combination of $R_2=4\ \Omega$ and $R_3=12\ \Omega$ (i.e. $R_2\parallel R_3$). The power dissipated in $R_3$ is:

Q8. A $10\,\mu\text{F}$ capacitor is charged from a $12\ \text{V}$ battery through $R_1=100\ \text{k}\Omega$. The switch is closed at $t=0$; after $t=1.0\ \text{s}$ the switch is opened and immediately the capacitor is connected across $R_2=10\ \text{k}\Omega$ (negligible switching time). The voltage across the capacitor immediately after the switch is moved (i.e. at $t=1.0^+$ s) is:

Q9. Two identical cells each of emf $E$ and internal resistance $r$ are used to supply power to an external load. When the two cells are connected in series the equivalent emf and internal resistance are $2E$ and $2r$; when connected in parallel (identical ideal cells) the equivalent emf and internal resistance are $E$ and $r/2$. Let $P_{\text{series}}^{\max}$ and $P_{\text{parallel}}^{\max}$ be the maximum power deliverable to an external load in the series and parallel cases respectively. The ratio $\dfrac{P_{\text{series}}^{\max}}{P_{\text{parallel}}^{\max}}$ is:

Q10. Two resistors $R_1$ and $R_2$ are in series with a battery of emf $E$ and internal resistance $r$. Initially (switch open) both resistors are in series and the power dissipated in $R_1$ is $P$. When the switch is closed $R_2$ is short-circuited and the power dissipated in $R_1$ becomes $4P$. If $R_1=5\ \Omega$ and $R_2=11\ \Omega$, the internal resistance $r$ of the battery is:

Q11. A uniform metallic wire of initial length $L$ and resistance $R$ is stretched uniformly to a new length $2L$ without any loss of material. What is the new resistance of the wire?

Q12. A cell of emf $\mathcal{E}$ and internal resistance $r$ is connected to an external resistor $R$. When $R=10\ \Omega$ the terminal voltage is $6\ \text{V}$; when $R=5\ \Omega$ the terminal voltage is $5\ \text{V}$. The values of $\mathcal{E}$ and $r$ are:

Q13. A cylindrical wire of length $L$ has radius varying linearly from $r$ at end $A$ to $2r$ at end $B$ (so $r(x)=r(1+x/L)$). The resistivity is uniform ($\rho$). A potential difference $V$ is applied across $A$ and $B$. The potential difference between $A$ and the midpoint of the wire is:

Q14. Assertion (A): Two cells with unequal emfs $\mathcal{E}_1>\mathcal{E}_2$ and finite internal resistances $r_1,r_2$ are connected in parallel (positive to positive, negative to negative). If no external load is connected, a current will flow between the cells.
Reason (R): The difference in their open-circuit terminal potentials drives a circulating current; finite internal resistances allow this current to flow from the higher-emf cell into the lower-emf cell.

Q15. In a potentiometer experiment the same driver cell (unknown internal resistance $r_D$) and the same potentiometer wire are used to find null lengths $l_1$ and $l_2$ for two test cells of emfs $e_1$ and $e_2$ (each test cell may have its own internal resistance). Which relation between the balancing lengths is correct?