Current Electricity

A steady current $I$ flows through a metallic conductor of non-uniform cross-section. Which of the following quantities remains constant along the length of the conductor?

The relationship between current $I$ and drift velocity $v_d$ is given by $I = nAev_d$. What does 'n' represent?

Ohm's law is valid only when:

The resistance of a wire of length $L$ and area of cross-section $A$ is $R$. If the wire is stretched to double its length keeping the volume constant, the new resistance will be:

In the given circuit, the potential difference across the resistor $R$ is measured. If the drift velocity of electrons in the resistor is $v_d$, how does $v_d$ change if the potential difference across it is doubled, assuming the temperature remains constant?

The current density $J$ in a conductor is related to the electric field $E$ and conductivity $\sigma$ by the relation:

A potential difference of $V$ is applied across a conductor at temperature $T$. If the temperature is increased to $T'$, and $V$ is kept constant, what happens to the product of conductivity ($\sigma$) and resistivity ($\rho$)?

A composite cylindrical wire is formed by a core of metal A (resistivity $\rho_A$) and an outer shell of metal B (resistivity $\rho_B$). Both have the same length $L$. If the core radius is $R$ and the outer radius of the shell is $2R$, and the drift velocity of electrons in the core is $v_A$ and in the shell is $v_B$ when a battery is connected across the ends, what is the ratio $v_A / v_B$ assuming both metals have the same number density of free electrons $n$?

In a semiconductor research lab, a thin-film resistor is subjected to an increasing electric field. Due to high-energy collisions, the relaxation time $\tau$ is found to be inversely proportional to the square root of the drift velocity ($v_d$), such that $\tau = k/\sqrt{v_d}$. If the current density is $J = n e v_d$, what is the relationship between the applied Electric Field ($E$) and the current density ($J$) for this material?

A specialized copper conductor in a particle accelerator is designed with a non-uniform cross-section. The radius of the wire increases linearly from $r_1$ at one end to $r_2$ at the other over a length $L$. If a steady potential difference $V$ is maintained across the ends, and the relaxation time of electrons is $\tau$, how does the drift velocity $v_d$ of the electrons vary along the length $x$ (where $0 \le x \le L$ and $x=0$ is at the end with radius $r_1$)?

A conductor of length $l$ has a resistance $R$. If its area of cross-section is $A$, then its conductivity $\sigma$ is:

The conductivity of a semiconductor at absolute zero temperature is:

Which of the following factors does NOT affect the resistivity of a wire?

What is the resistivity of a substance that allows current to flow with zero opposition?

In a metallic conductor, the relationship between the current density ($J$) and the drift velocity ($v_d$) of free electrons is defined by the equation $J = n e v_d$. If a graph is plotted with $J$ on the vertical axis and $v_d$ on the horizontal axis, which of the following statements correctly describes the behavior of the slope of this graph as the temperature of the conductor is increased?

Two wires of the same material have lengths in the ratio $1:3$ and diameters in the ratio $2:3$. The ratio of their conductivities is:

If $R$ is resistance, $\rho$ is resistivity, $L$ is length and $A$ is cross-sectional area, then conductivity $\sigma$ is equal to:

For a material with $n$ charge carriers per unit volume, each of charge $q$ and mass $m$, the conductivity $\sigma$ is related to the mean free path $\lambda$ and average speed $v$ as:

The resistance of a conductor is $5 \Omega$ at $50^{\circ}C$ and $6 \Omega$ at $100^{\circ}C$. Its resistance at $0^{\circ}C$ is:

In a laboratory experiment, the resistance of a copper wire is measured as $R_1$ at $T_1$ and $R_2$ at $T_2$. If the linear thermal expansion coefficient of copper is $\gamma$ and the temperature coefficient of resistivity is $\alpha$, the ratio $R_2/R_1$ is approximately:

Which of the following is the correct mathematical form of Kirchhoff's Junction Rule?

When traversing a battery from its negative terminal to its positive terminal, the emf is taken as:

A junction is a point in a circuit where at least how many conductors meet?

While applying Kirchhoff's loop rule, if we travel in the direction of the current through a resistor $R$, the potential change is taken as:

Using Kirchhoff's rules, find the potential of point X if the bottom wire is grounded (0V).

In a closed loop, if the sum of emf is 12V and there are two resistors of $2 \Omega$ and $4 \Omega$ in series, the current is:

Two batteries of 6V and 4V with internal resistances $1 \Omega$ each are connected in parallel. The effective emf is:

A bridge rectifier circuit is simulated using a DC bridge for a 'worst-case' leakage test. The bridge has resistors $R_1=2\,\Omega$, $R_2=4\,\Omega$, $R_3=3\,\Omega$, and $R_4=6\,\Omega$ in a diamond shape. A $20\,V$ source is applied across one pair of opposite vertices. If a $10\,\Omega$ resistor is placed across the other pair of vertices, calculate the current through this $10\,\Omega$ resistor.

A telecommunications backup system involves a charging circuit where a $110\,V$ DC supply charges a lead-acid battery ($E=12\,V$, $r=0.5\,\Omega$) through a current-limiting resistor of $10\,\Omega$. Parallel to the battery, a cooling fan with a resistance of $20\,\Omega$ is connected. Find the charging current entering the battery branch.

In an aircraft's electrical system, three wires meet at a common junction point J. Wire 1 carries a current of $I_1$ into the junction from a $28\,V$ source through a $7\,\Omega$ resistor. Wire 2 carries $I_2$ away from the junction to a $14\,V$ source through a $2\,\Omega$ resistor. Wire 3 is grounded through a $4\,\Omega$ resistor. Calculate the potential at junction J.

Which principle is the Wheatstone bridge based on?

In a Meter Bridge, the error due to 'end resistance' can be minimized by:

The Wheatstone bridge is named after:

When the Wheatstone bridge is not balanced, the current in the galvanometer flows from:

Calculate the equivalent resistance between points A and C in the given bridge circuit if $P=10\Omega, Q=10\Omega, R=10\Omega, S=10\Omega$ and $G=50\Omega$.

What happens to the balance point of a Wheatstone bridge if the source voltage is doubled?

In the circuit shown, if no current flows through the $10\Omega$ resistor, the value of $X$ is:

A laboratory technician is calibrating a specialized sensor designed to measure temperature fluctuations in a chemical reactor. The sensor is integrated into a Wheatstone bridge circuit. At a reference temperature, the bridge is balanced. However, during an experiment, the resistance of the sensor $R_s$ changes due to heat. The bridge consists of three known resistors: $P = 100\, \Omega$, $Q = 100\, \Omega$, and $R = 100\, \Omega$, along with the sensor $S$. A sensitive galvanometer with resistance $G = 15\, \Omega$ is connected across the bridge. If the sensor resistance $S$ shifts to $101\, \Omega$ and a $2\, V$ battery with negligible internal resistance is used, what is the approximate current flowing through the galvanometer?

A student constructs a bridge where the arms $P$, $Q$, $R$, and $S$ have resistances of $2\ \Omega$, $2\ \Omega$, $2\ \Omega$, and $3\ \Omega$ respectively. A galvanometer of resistance $10\ \Omega$ is connected across the bridge. If a $6V$ battery with negligible internal resistance is connected across the bridge, calculate the current flowing through the galvanometer using Kirchhoff's laws or bridge unbalanced equations.

An engineer is checking a bridge balance for a strain gauge application. Initially, $P=1000\,\Omega$, $Q=1000\,\Omega$, $R=1000\,\Omega$, and the strain gauge $S=1000\,\Omega$. Under mechanical stress, the resistance $S$ changes by $\Delta S$. If the output voltage across the bridge (originally balanced) is $25\,mV$ when the input voltage is $10\,V$, find the change in resistance $\Delta S$. (Assume high-impedance measurement).

The sensitivity of a potentiometer can be increased by which of the following methods?

A potentiometer wire has resistance $40 \Omega$ and length $10 m$. It is connected to a $2 V$ battery. The potential gradient is:

What is the potential drop across a $10 m$ long potentiometer wire if the potential gradient is $0.1 V/m$?

The terminal voltage of a cell is greater than its emf when:

A potentiometer wire of length $10\text{ m}$ has a resistance of $20\text{ \Omega}$. It is connected in series with a battery of $5\text{ V}$ and a resistance of $30\text{ \Omega}$. What is the potential gradient along the wire?

Two cells of emfs $E_1$ and $E_2$ ($E_1 > E_2$) are connected in series as shown. They are balanced against $400\text{ cm}$ of a potentiometer wire. When the polarity of $E_2$ is reversed, the balance length becomes $100\text{ cm}$. Find the ratio $E_1/E_2$.

For a cell, the graph of terminal voltage $V$ vs current $I$ is a straight line with:

A battery of 6 cells, each of e.m.f. $2\,V$ and internal resistance $0.2\,\Omega$, are connected in series but one cell is wrongly connected (reversed). This battery is used to charge a $4\,V$ accumulator using an external resistance of $10\,\Omega$. What is the potential difference across the terminals of the wrongly connected cell during charging?

A potentiometer circuit is set up with a $2.0\,V$ driver cell. A long wire $AB$ of $400\,cm$ has a resistance of $8\,\Omega$. A battery of $1.5\,V$ with internal resistance $0.5\,\Omega$ is being tested. If a $0.5\,V$ standard source is balanced at $100\,cm$, but the driver cell starts to deplete and its voltage drops by $10\%$, by how much will the balance point for the $0.5\,V$ source shift?

Two identical cells, each of e.m.f. $1.5\,V$ and internal resistance $r$, are connected in parallel. This combination is then used to send current through an external resistor of $2.5\,\Omega$. A potentiometer measures the terminal voltage of this parallel combination as $1.25\,V$. What is the internal resistance $r$ of each individual cell?

In the $V-I$ characteristic of a GaAs (Gallium Arsenide) semiconductor, there is a specific region where the current decreases as the voltage increases. This region is known as:

In the context of Ohm's law limitations, what happens to the resistance of a metallic conductor when the current passing through it becomes very high, leading to significant heating?

Which material is used to demonstrate the negative resistance region in the $V-I$ curve?

The $V-I$ characteristic of an Ohmic conductor is a:

Consider a conductor whose resistance increases significantly due to the heating effect of current. If the voltage $V$ is doubled, the current $I$ becomes less than double. This limitation of Ohm's law is primarily due to:

Consider a water voltameter. The V-I characteristic shows that current starts flowing only after the back e.m.f. is overcome. This represents a limitation of Ohm's law because:

According to the limitations of Ohm's law, for some materials, the relation between $V$ and $I$ is not unique. This means that for the same value of current $I$, there may be more than one value of voltage $V$. Which material typically demonstrates this?

An engineer is designing a high-power incandescent lighting system. They notice that at the moment of switching on, the 'inrush current' is much higher than the steady-state operating current. If they plot the $V$-$I$ graph for the filament, it curves towards the voltage axis at higher values. What physical process causes this specific limitation of Ohm's Law?

A laboratory technician is testing a thyristor, a complex semiconductor device. During the test, the technician finds that for a single value of current $I$, there are multiple possible operating voltages $V$, depending on the history of the device's bias. This behavior is a primary limitation of Ohm's Law. Which characteristic of the device is being described?

An experimentalist is studying a Gallium Arsenide ($GaAs$) semiconductor device used in a high-speed communication circuit. While increasing the applied voltage, they observe a peculiar region where the current actually decreases as the electric field increases. Based on the provided I-V characteristic curve of $GaAs$, what is the physical implication of the region marked 'AB'?

The coefficient $\alpha$ for most metals is of the order of:

Resistance of a wire at $20 ^\circ C$ is $20 \Omega$ and at $500 ^\circ C$ is $60 \Omega$. At what temperature will its resistance be $25 \Omega$?

If temperature of a conductor increases, the frequency of collision of electrons:

Which of the following is an alloy used for making standard resistances?

A Copper rod and a Silicon rod both have a resistance of $50 \, \Omega$ at room temperature ($300 \, K$). If both rods are cooled down to the temperature of liquid nitrogen ($77 \, K$), what will be the observation regarding their final resistances $R_{Cu}$ and $R_{Si}$?

In some precision applications, the resistance $R_t$ of a conductor at temperature $t$ is modeled by the quadratic equation $R_t = R_0(1 + \alpha t + \beta t^2)$, where $\alpha$ and $\beta$ are constants. Given that the temperature coefficient of resistance at a specific temperature $t$ is defined as $\frac{1}{R_t} \frac{dR_t}{dt}$, which of the following represents its expression?

A wire of resistance $5 \, \Omega$ at $50^\circ C$ and $6 \, \Omega$ at $100^\circ C$ has a resistance at $0^\circ C$ of:

Superconductivity is a phenomenon where the resistivity of a material becomes exactly zero below a certain critical temperature. Above this temperature, the material usually behaves as:

If $\rho_1$ and $\rho_2$ are the resistivities of a material at temperatures $T_1$ and $T_2$ respectively, then the temperature coefficient $\alpha$ (at $T_1$) is:

For a conductor, if $\alpha = 0.005 \, ^\circ C^{-1}$, the percentage increase in resistance for a $20 \, ^\circ C$ rise in temperature is:

For a resistor $R$, if the current $I$ is tripled, the power $P$ becomes:

Which of the following is equivalent to 1 Watt?

What is the power of a circuit if $10 \text{ J}$ of work is done in $2 \text{ s}$?

The efficiency of an electric motor is given by the ratio of:

In the circuit shown below, a battery of emf $E$ and internal resistance $r$ is connected to an external resistor $R$. A switch $S$ allows another identical resistor $R$ to be connected in parallel with the first resistor. If the resistance $R$ is exactly twice the internal resistance ($R = 2r$), find the ratio of the total power dissipated in the external circuit when the switch $S$ is closed ($P_c$) to when it is open ($P_o$).

A DC source of $6\text{ V}$ is connected to a series combination of two resistors $2\text{ }\Omega$ and $4\text{ }\Omega$. The power dissipated in the $4\text{ }\Omega$ resistor is:

Electrical energy is converted into heat in a resistor. This phenomenon is known as:

A source of EMF $E$ and internal resistance $r$ is connected to an external resistance $R$. The graph of power $P$ vs current $I$ is a:

The heat produced in a resistor $R$ when a sinusoidal current $I = I_0 \sin(\omega t)$ flows through it for one full cycle ($T = 2\pi/\omega$) is:

If power $P$ is delivered to a load via a line of resistance $R_l$, the efficiency of transmission is:

In a closed circuit, the current flows from:

When two cells of EMF $2 V$ and $4 V$ are connected in series correctly (supporting each other), the net EMF is:

Which of the following is correct for a cell in series combination?

A cell of EMF $2 V$ and internal resistance $1 \Omega$ is connected to a $1 \Omega$ resistor. The terminal voltage is:

Two cells of EMF $E_1$ and $E_2$ are connected in parallel. If $E_1 > E_2$, then the equivalent EMF $E_{eq}$ is:

In a potentiometer, the balance point is found at $l$. If the current through the potentiometer wire is increased, the balance point $l$ will:

For a given cell, the internal resistance $r$ is:

Two batteries, one of EMF $18$ V and internal resistance $2$ $\Omega$ and the other of EMF $12$ V and internal resistance $1$ $\Omega$, are connected in parallel with their like poles together. The combination is connected to an external resistance of $10$ $\Omega$. The current through the $10$ $\Omega$ resistor is:

A cell of EMF $E$ and internal resistance $r$ is used to send current through a resistance $R$. The plot of terminal voltage $V$ vs $R$ is:

A set of $n$ identical cells, each of EMF $E$ and internal resistance $r$, are connected in series. When the set is short-circuited, the current is $I$. If the cells are connected in parallel and short-circuited, the current will be:

Cells are connected in series to increase the:

A mixed grouping of cells has $n$ cells in series in each row and $m$ such rows in parallel. The total number of cells is:

Which of the following remains constant in a parallel combination of identical cells?

If $n$ identical cells are connected in series, the total internal resistance is:

Two cells of emfs $E_1 = 6\text{ V}$ and $E_2 = 4\text{ V}$ with internal resistances $r_1 = 1\, \Omega$ and $r_2 = 2\, \Omega$ respectively are connected in series but with opposing polarities to an external resistance $R = 7\, \Omega$. Calculate the potential difference across the terminals of the second cell ($E_2$).

In the circuit shown, two identical cells of emf $E = 2\text{ V}$ and internal resistance $r = 1\, \Omega$ each are connected in parallel to an external resistor $R = 1.5\, \Omega$. What is the current $I$ flowing through the resistor $R$?

Three identical cells of emf $E$ and internal resistance $r$ are connected in series. If the polarity of one cell is reversed, the equivalent internal resistance is:

A solar array simulator consists of 12 identical cells, each of EMF $E$ and internal resistance $r$. When all cells are connected in series, the current through a fixed load $R$ is $I$. When all cells are connected in parallel, the current through the same load is also $I$. Determine the relationship between the load resistance $R$ and the internal resistance $r$ of a single cell.

In a laboratory experiment, two non-identical cells with EMFs $E_1 = 6\text{ V}, r_1 = 1\text{ }\Omega$ and $E_2 = 4\text{ V}, r_2 = 2\text{ }\Omega$ are connected in parallel such that their like terminals are together. This combination is then connected across an external resistance $R = 5\text{ }\Omega$. Calculate the potential difference across the terminals of the second cell ($E_2$).

A high-precision sensor requires exactly $4\,V$ to operate. You have a collection of cells with $E=1.5\,V$ and $r=0.2\,\Omega$ each. You connect 4 such cells in series. To ensure the voltage stays at exactly $4\,V$ when a current of $2\,A$ is drawn by the sensor, you must add an additional group of $n$ cells in parallel to the existing series string. How many cells $n$ should be in the parallel bank if the parallel bank itself is then connected in series with the original 4 cells?