Practice Questions

Propose a material mentioned in the chapter that would be ideal for constructing wire-bound standard resistors for laboratory use and justify your choice based on its resistivity-temperature characteristics.

Define electric current and name its SI unit.

Identify the primary charge carriers responsible for current in (a) metals and (b) electrolytic solutions.

Demonstrate that the unit of mobility ($\mu$) can be expressed as $\text{m}^2 \text{V}^{-1} \text{s}^{-1}$.

Define resistivity of a material and state its SI unit.

A battery with an emf of $6$ V and an internal resistance of $0.5 \, \Omega$ is connected to an external resistor of $5.5 \, \Omega$. Calculate the terminal voltage of the battery.

Propose a reason why connecting the terminals of a car battery (e.g., $12$ V emf, $0.05 \Omega$ internal resistance) directly with a thick copper wire is extremely dangerous, using the concept of maximum current.

Define the internal resistance of an electric cell.

Justify why electric current is considered a scalar quantity, even though it is often represented with an arrow in circuit diagrams.

Examine why alloys like manganin and constantan are used for making standard resistors.

A heating element is rated $2100$ W at $230$ V. Calculate the resistance of the element and the current it draws.

Recall the mathematical statement of Ohm's law.

A carbon resistor has a resistance of $500 \, \Omega$ at $20.0^{\circ}$C. Calculate its resistance at $80.0^{\circ}$C. The temperature coefficient of resistivity for carbon is $\alpha = -5.0 \times 10^{-4} \, (^{\circ}\text{C})^{-1}$.

In a Wheatstone bridge circuit with resistors $P=10 \, \Omega$, $Q=20 \, \Omega$, $R=15 \, \Omega$, and $S=30 \, \Omega$, a battery of $5$ V is connected across the bridge. Calculate the current drawn from the battery.

A wire of resistance $R$ is stretched to double its original length. Assuming the volume and resistivity of the material remain constant, calculate the new resistance of the wire in terms of $R$.

Two cells, one with emf $1.5$ V and internal resistance $0.2 \, \Omega$, and the other with emf $2.0$ V and internal resistance $0.3 \, \Omega$, are connected in parallel. Calculate the equivalent emf and equivalent internal resistance of the combination.

Critique the statement: "Ohm's law, $V=IR$, is a fundamental law of nature applicable to all materials."

List the factors on which the resistance of a conductor depends and recall the formula that relates them.

Examine why Ohm's law is not universally applicable to all conducting materials, providing an example of a non-ohmic device.

Analyze the effect on the balancing length in a meter bridge experiment if the positions of the galvanometer and the cell are interchanged.

Explain the term 'drift velocity' of free electrons in a metallic conductor.

Describe the vector relationship between current density (j), conductivity (σ), and electric field (E).

Explain Kirchhoff's two rules for analyzing electrical circuits. Name the fundamental conservation laws on which these rules are based.

A cell has an emf of $1.5$ V and an internal resistance of $0.1 \Omega$. Recall the formula to find the maximum current it can provide and calculate this value.

Recall the balance condition for a Wheatstone bridge and draw a labelled circuit diagram.

Define the temperature coefficient of resistivity, $\alpha$.

Recall the basic formula for electric power ($P$) dissipated in a conductor. List two other forms of this formula using Ohm's Law.

Two wires, one made of copper and the other of nichrome, have the same length and the same resistance. Analyze which wire is thicker and provide a justification based on their resistivities.

Compare the power loss in transmission lines when power is transmitted at $220$ V versus $22,000$ V. Assume the power to be delivered is $11$ kW and the resistance of the transmission lines is $2.0 \, \Omega$.

Evaluate the claim that the drift velocity of electrons is solely responsible for the speed at which a light bulb turns on when a switch is closed.

Formulate a mathematical model to justify why transmitting electrical power at high voltage is more efficient. Derive the expression for power loss and explain its dependence on transmission voltage.

Propose a modification to the standard series combination of two cells with emfs $\varepsilon_1$ and $\varepsilon_2$ (where $\varepsilon_1 > \varepsilon_2$) and internal resistances $r_1$ and $r_2$, such that the equivalent emf of the combination is $\varepsilon_1 - \varepsilon_2$. Justify your proposal.

For the circuit shown in Figure 3.17 of the source document, create a set of three independent equations using Kirchhoff's rules that would allow you to solve for the currents $I_1$, $I_2$, and $I_3$. Justify the choice of loops.

You are given two copper wires of the same length, but wire A has a diameter twice that of wire B. Design a simple experiment to prove which wire has lower resistance and justify which wire would be a better choice for a heating element.

Formulate a set of Kirchhoff's rule equations for the Wheatstone bridge circuit shown in Figure 3.18 of the source document when it is unbalanced ($I_g \neq 0$). Use these equations to derive the balance condition $\frac{R_2}{R_1} = \frac{R_4}{R_3}$ when $I_g$ is set to zero.

Design a simple platinum resistance thermometer. Given that the resistance of a platinum wire is $R_0$ at the ice point ($0^\circ\text{C}$) and $R_{100}$ at the steam point ($100^\circ\text{C}$), formulate a linear equation to determine an unknown temperature $t$ (in Celsius) from a measured resistance $R_t$. Justify the assumption of linearity and evaluate its limitations.

Explain the terms electromotive force (emf) and terminal voltage of a cell. Describe the condition under which the terminal voltage is equal to the emf.

A student proposes a circuit with three identical cells, each with emf $\varepsilon$ and internal resistance $r$. To get maximum current through an external resistor $R$, they suggest connecting all cells in parallel. Evaluate this proposal for two distinct cases: (a) $R \gg r$ and (b) $R \ll r$.

For the circuit shown, apply Kirchhoff's rules to solve for the currents $I_1$, $I_2$, and $I_3$ flowing through the resistors $R_1=10\, \Omega$, $R_2=5\, \Omega$, and $R_3=20\, \Omega$. The batteries have emfs $\varepsilon_1 = 6$ V and $\varepsilon_2 = 12$ V.

Calculate the drift velocity $v_d$ of electrons in an aluminum wire with a cross-sectional area of $2.5 \times 10^{-6} \text{ m}^2$ carrying a current of $5$ A. Assume one conduction electron per atom. The density of aluminum is $2.7 \times 10^3 \text{ kg/m}^3$ and its atomic mass is $27.0$ u.

Evaluate the effect of increasing temperature on the conductivity of a semiconductor versus a metal. Justify the difference in their behavior using the formula $\sigma = \frac{ne^2\tau}{m}$.

Critique the simple model for electron drift velocity, $v_d = \frac{eE\tau}{m}$, by identifying at least two major assumptions made in its derivation and explaining their implications on the model's accuracy.

Design an experiment using a Wheatstone bridge to determine the temperature coefficient of resistivity, $\alpha$, for an unknown metallic wire. Formulate the necessary equations and justify the procedure.

Summarize the effect of increasing temperature on the resistivity of (a) metals, and (b) semiconductors. Provide a brief reason for the behavior in metals.