Current Electricity Set-3

Q1. Two wires X and Y of equal length $L$ and cross-sectional area $A$ have resistivities near temperature $T_0$ given by

with $\alpha,\beta>0$ and $\rho_X(T_0)=\rho_Y(T_0)=\rho_0$. At $T_0$ both wires carry the same current $I$. For a small increase $\Delta T>0$ (first‑order in $\Delta T$), which statement about the change in Joule heating $P=I^2R$ in the two wires is correct?

Q2. A battery has emf $E=12\ \text{V}$ and internal resistance $r=2\ \Omega$. When connected to an external resistor $R$ the steady current is $2.0\ \text{A}$. The value of $R$ is

Q3. A copper wire of radius $1.0\ \text{mm}$ carries a steady current $I=5.0\ \text{A}$. With electron density $n=8.5\times10^{28}\ \text{m}^{-3}$ and electron charge $e=1.6\times10^{-19}\ \text{C}$, the magnitude of the drift speed $v_d$ of electrons in the wire is closest to

Q4. Assertion: If two cells of emf $E_1$ and $E_2$ with internal resistances $r_1$ and $r_2$ are connected in series opposing, the equivalent emf is $|E_1-E_2|$ and the equivalent internal resistance is $|r_1-r_2|$.

Reason: When cells are connected in series their emfs add algebraically (opposing gives a difference), but the internal resistances always add ($r_{\rm eq}=r_1+r_2$) irrespective of polarity.

Q5. A cell with emf $E=12\ \text{V}$ and internal resistance $r=3\ \Omega$ supplies a variable external resistance $R$. The power delivered to $R$ is $P(R)=\dfrac{E^2R}{(R+r)^2}$ and the $P$ vs $R$ curve has a single maximum. The value of $R$ at the peak and the maximum power $P_{\rm max}$ are:

Q6. A uniform cylindrical conductor has length $L=1.0\ \text{m}$, cross-sectional area $A=1.0\ \text{mm}^2$, and resistivity $\rho=1.6\times10^{-8}\ \Omega\cdot\text{m}$. Its electrical resistance is

Q7. A battery of emf $E=12\ \text{V}$ and internal resistance $r=2\ \Omega$ is connected in series with an ammeter of internal resistance $r_A=1\ \Omega$ and an external resistor $R=6\ \Omega$. A voltmeter of internal resistance $r_V=4\ \Omega$ is connected across $R$ to measure the potential difference across $R$. The voltmeter will read (rounded to two decimal places)

Q8. Assertion: In a uniform steady conducting wire connected to a DC source the current density $\mathbf{J}$ is the same at all cross-sections.

Reason: Charge conservation (Kirchhoff's current law) in steady state implies net current through any cross-section is the same; for a uniform cross-section this means $\mathbf{J}$ is the same everywhere.

Q9. Two cells with emfs $E_1=1.5\ \text{V},\ r_1=1.0\ \Omega$ and $E_2=1.2\ \text{V},\ r_2=2.0\ \Omega$ are connected in parallel (positive to positive) and then connected to an external load $R=10\ \Omega$. The current through the load (approx.) is

Q10. A uniform metallic wire of resistance $R$ is stretched uniformly to twice its original length while its volume remains constant. Neglecting any change in resistivity, the new resistance becomes