Current Electricity Set-1

Q1. Two straight wires of the same material and length $L=1.0\ \mathrm{m}$ have circular cross-sectional areas $A_1=1.0\times10^{-6}\ \mathrm{m^2}$ and $A_2=2.0\times10^{-6}\ \mathrm{m^2}$. Their resistivity is $\rho=1.6\times10^{-8}\ \Omega\cdot\mathrm{m}$. The two wires are connected in parallel between common ends. The equivalent resistance between the ends is:

Q2. Two conductors X and Y were tested and the measured $(V,I)$ data (in SI units) are:
For X: $(1.0,\,0.50),\ (2.0,\,1.00),\ (3.0,\,1.50)$.
For Y: $(1.0,\,0.40),\ (2.0,\,0.75),\ (3.0,\,1.05)$.
Which one of the following statements is correct at $V=2.0\ \mathrm{V}$?

Q3. Assertion: When two identical resistive bulbs each of resistance $R$ are connected in series across the same ideal battery, the total rate of heat production in the two bulbs is half the rate when a single bulb (of resistance $R$) alone is connected across the same battery.
Reason: In series the equivalent resistance doubles so the circuit current halves and since total power in the bulbs is $P=I^2R_\text{total}$, the total power becomes half.

Q4. An ammeter with internal resistance $r_A=0.20\ \Omega$ and a voltmeter with internal resistance $R_V=10.0\ \mathrm{k}\Omega$ are used to measure an unknown resistance $R_x$. The voltmeter is connected across $R_x$ while the ammeter is in series with the combination. The instrument readings are $V=5.00\ \mathrm{V}$ and $I=0.520\ \mathrm{A}$. Assuming the supply is ideal, the value of $R_x$ is approximately:

Q5. At temperature increase $\Delta T=10\ \mathrm{K}$ a metal has resistivity $\rho_m(\Delta T)=\rho_0(1+\alpha\Delta T)$ with $\rho_0=1.00\times10^{-3}\ \Omega\cdot\mathrm{m}$ and $\alpha=4.0\times10^{-3}\ \mathrm{K^{-1}}$. A semiconductor has $\rho_s(\Delta T)=\rho_0\exp(-\beta\Delta T)$ with $\beta=0.10\ \mathrm{K^{-1}}$ and the same $\rho_0$ at $\Delta T=0$. At $\Delta T=10\ \mathrm{K}$ the ratio of their conductivities $\sigma_s/\sigma_m$ is closest to:

Q6. Assertion: For a cell of emf $E$ and internal resistance $r$, the power delivered to an external load $R$ is maximum when $R=r$.
Reason: At $R=r$ the current is $I=\dfrac{E}{2r}$ and the power delivered to the load equals the power dissipated inside the internal resistance.

Q7. A potentiometer wire has uniform potential gradient $k=10.0\ \mathrm{mV/cm}$. A cell gives balance length $l_1=120\ \mathrm{cm}$ when open-circuited. When a resistor $R=10.0\ \Omega$ is connected across the cell, the balance length reduces to $l_2=100\ \mathrm{cm}$. The internal resistance $r$ of the cell is:

Q8. Assertion: In a circuit with source internal resistance $r>0$ and load resistor $R$, connecting a voltmeter of finite resistance $R_V$ across the load decreases the current through $R$ compared to the case when the voltmeter is not connected.
Reason: The voltmeter draws an extra current in parallel which increases the voltage drop across $r$, lowering the terminal voltage across $R$ and therefore reducing the current through $R$.

Q9. Two identical cells, each of emf $E=1.50\ \mathrm{V}$ and internal resistance $r=1.00\ \Omega$, are connected in parallel and this combination is connected to an external resistor $R=2.00\ \Omega$. The current drawn from the parallel combination is:

Q10. Two identical lamps each of resistance $R=5.0\ \Omega$ are connected to a battery of emf $E=12.0\ \mathrm{V}$ with internal resistance $r=5.0\ \Omega$.
Case (I): lamps in series across the battery.
Case (II): lamps in parallel across the same battery.
The ratio (power in each lamp in Case II) / (power in each lamp in Case I) is: