Past Questions — 2022

In semiconductor physics, mobility (μ) is a measure of how quickly a charge carrier (electron or hole) can move through a metal or semiconductor when acted upon by an electric field. Mathematically, it is defined as the ratio of the drift velocity (vd) to the applied electric field (E). The formula is expressed as: μ = vd / E. - Option (a) is incorrect because diffusion velocity is driven by a concentration gradient, not by an external electric field. - Option (c) and (d) are incorrect because mobility describes the rate of flow (velocity), not the total distance or displacement traveled.

A Zener diode is specifically designed to operate in the reverse breakdown region. The 'Zener effect' occurs in heavily doped p-n junctions where the depletion region is very narrow. Due to the high doping concentration, a very strong electric field is created across this narrow junction when reverse-biased. This strong field allows electrons to 'tunnel' directly from the valence band of the p-side to the conduction band of the n-side (or vice versa), a quantum mechanical phenomenon known as Zener Tunneling. Why other options are incorrect: - b. Thermionic emission: This refers to the emission of electrons from a heated filament or cathode (used in vacuum tubes), not the breakdown mechanism of a diode. - c. Diffusion: Diffusion is the primary mechanism for current flow in a forward-biased diode, where carriers move from regions of high concentration to low concentration. - d. Hopping: Hopping conduction typically occurs in disordered materials or semiconductors with high impurity levels at very low temperatures, but it is not the operating principle of a Zener diode.

The given voltage expression is v(t) = 5 + 5sin(314t + π/6). This is a composite signal consisting of a DC component and an AC sinusoidal component. 1. Identify the components: - DC component (Vdc) = 5 V - AC peak voltage (Vm) = 5 V 2. Calculate the RMS value of the AC component (Vrms_ac): For a pure sine wave, Vrms_ac = Vm / √2 Vrms_ac = 5 / √2 ≈ 3.535 V 3. Calculate the total RMS value (Vrms_total): The total RMS value of a signal containing both DC and AC components is given by the square root of the sum of the squares of the DC value and the RMS value of the AC components: Vrms_total = √(Vdc² + Vrms_ac²) Vrms_total = √(5² + (5/√2)²) Vrms_total = √(25 + 25/2) Vrms_total = √(25 + 12.5) Vrms_total = √37.5 4. Final Calculation: √37.5 ≈ 6.1237 V Comparing this result to the options, 6.12 V is the correct value. Option (a) only considers the DC part, option (b) is incorrect, and option (d) is the peak-to-peak value of the AC part added to DC, which is not the RMS value.

Norton's Theorem is a fundamental network theorem used to simplify complex linear circuits. It states that any linear circuit containing several voltage sources and resistances can be replaced by an equivalent circuit consisting of a single current source (Norton current, In) in parallel with a single impedance (Norton impedance, Zn). Step-by-step breakdown: 1. Norton's Theorem focuses on the current delivered to a load. 2. The Norton equivalent circuit consists of a constant current source (In) connected in parallel with an equivalent resistance or impedance (Zn). 3. This is the dual of Thévenin's Theorem, which replaces a network with a voltage source in series with an impedance. Analysis of options: - Option (a) is incorrect because a current source in series with an impedance is not the standard Norton equivalent (and would be impractical as the impedance wouldn't affect the current flow from the source). - Option (b) is correct as it describes the standard Norton equivalent circuit. - Option (c) describes a Thévenin equivalent circuit, not a Norton equivalent. - Option (d) is identical to option (b) in the provided text, but the conceptual answer remains 'In parallel with a current source'.

A covalent bond is a type of chemical bond that occurs when two atoms share one or more pairs of valence electrons to achieve a stable electron configuration (usually an octet). In the context of semiconductor physics, elements like Silicon (Si) and Germanium (Ge) have four valence electrons and form covalent bonds with four neighboring atoms to create a stable crystal lattice structure. Analysis of options: - Option (a) describes an 'Ionic Bond', where electrons are transferred from a metal to a non-metal. - Option (b) is the correct definition of a 'Covalent Bond'. - Option (c) is incorrect because while the number of shared electrons can vary (single, double, or triple bonds), the fundamental mechanism is the sharing of electrons between specific atoms to reach stability, not a 'variable number of atoms' in a random sense. - Option (d) is physically impossible; nuclei are held together by strong nuclear forces in the nucleus, not by chemical bonding between atoms.

The principle behind heating instruments (like electric heaters, irons, and toasters) is Joule Heating. According to Joule's Law, the heat produced (H) in a conductor is given by the formula H = I²Rt, where I is the current, R is the resistance, and t is the time. To generate a significant amount of heat from an electrical current, a material with high electrical resistivity is required. Materials such as Nichrome (an alloy of nickel and chromium) are used because they have high resistivity and a high melting point, allowing them to convert electrical energy into thermal energy efficiently without melting. Analysis of other options: - a. LED light: These are semiconductor devices (p-n junctions) that rely on electroluminescence, not high resistivity for heat generation. - c. Tube light starter: A starter typically uses a bimetallic strip and a neon bulb to create a momentary short circuit; while it involves specific material properties, the primary function of the device is switching, not utilizing high resistivity for its core operation. - d. Cool down instruments: Cooling systems (like refrigerators or ACs) rely on thermodynamic cycles and refrigerants, not the electrical resistivity of materials to function.

A Zener diode is a semiconductor device. In electronics, a linear device is one where the output is directly proportional to the input (following Ohm's Law, V = IR, where resistance is constant). A non-linear device is one where the relationship between voltage and current is not a straight line. 1. **V-I Characteristics**: The current-voltage (V-I) characteristic curve of a Zener diode is highly non-linear. It remains almost zero until the threshold voltage is reached, and then it increases exponentially in forward bias. In reverse bias, it remains nearly zero until it reaches the Zener breakdown voltage (Vz), at which point the current increases sharply while the voltage remains almost constant. 2. **Why other options are incorrect**: - **Amplifier (b)**: A Zener diode is a passive device used for voltage regulation; it cannot amplify a signal (which requires an active device like a transistor or Op-Amp). - **Linear (c)**: As explained, its V-I curve is not a straight line passing through the origin. - **Transistor (d)**: A Zener diode is a two-terminal diode, whereas a transistor is a three-terminal device (e.g., BJT or MOSFET) used for switching or amplification.

KCL and Ohm’s law is the best answer because it directly satisfies the fact, principle, or condition asked in the question.

In metals, electrical resistivity is primarily caused by the scattering of conduction electrons as they move through the crystal lattice. According to the Drude model and quantum mechanical theories of conductivity, the resistivity (ρ) is proportional to the frequency of collisions between electrons and the lattice ions. As the temperature of a metal increases, the thermal energy causes the ions in the lattice to vibrate with greater amplitude around their equilibrium positions (these vibrations are quantized as phonons). These increased lattice vibrations increase the effective 'cross-section' for collisions, making it more likely that a conduction electron will be scattered. Therefore, the mean free path of the electrons decreases, leading to an increase in resistivity. Why other options are incorrect: - Options (a) and (b): In metals, the concentration of conduction electrons (electron density) is essentially constant and does not change significantly with temperature. This is a key difference between metals and semiconductors (where carrier density increases with temperature). - Option (c): While it is true that collisions increase, 'collision of electrons' usually refers to electron-electron scattering. While electron-electron scattering occurs, the dominant mechanism for the temperature dependence of resistivity in metals is electron-phonon scattering (electron-lattice interaction), which is directly caused by the lattice vibrations described in option (d). Option (d) provides the fundamental physical cause.

In a Bipolar Junction Transistor (BJT), the collector-base junction is reverse-biased during normal operation. For 'high-speed' transistors, the device is designed with very high doping concentrations in the base and collector regions to reduce transit time and decrease the width of the depletion region. When the doping concentration is sufficiently high, the depletion region becomes extremely narrow. In such cases, the electric field across the junction becomes intense enough to pull electrons directly from the valence band to the conduction band (field emission), which is the characteristic of Zener breakdown. This occurs at lower voltages than the impact ionization required for Avalanche breakdown. Since the collector-base junction is the one subjected to high reverse bias in these applications, the breakdown is attributed to the Zener mechanism at the base-collector junction.

A two-port network is said to be reciprocal if the ratio of the response at one port to the excitation at the other port remains the same when the positions of the excitation and response are interchanged. In terms of network parameters, the conditions for reciprocity are as follows: 1. Z-parameters: A network is reciprocal if Z12 = Z21. 2. Y-parameters: A network is reciprocal if Y12 = Y21. This is because the admittance matrix relates currents to voltages, and for a reciprocal network, the transfer admittance from port 1 to 2 must equal the transfer admittance from port 2 to 1. 3. ABCD-parameters: A network is reciprocal if AD - BC = 1 (or BC - AD = -1). 4. h-parameters: A network is reciprocal if h12 = -h21. Evaluating the options: - Option (a) Z11 = Z22: This is the condition for symmetry, not reciprocity. - Option (b) Y12 = Y21: This is the correct condition for reciprocity. - Option (c) BC - AD = -1: While this is mathematically equivalent to AD - BC = 1, the standard definition for reciprocity in ABCD parameters is AD - BC = 1. However, in many contexts, this is considered correct. But comparing it to the fundamental Y-parameter definition, Y12 = Y21 is the most direct and standard representation of reciprocity among the choices provided. - Option (d) h12 = h21: The condition for reciprocity in h-parameters is h12 = -h21, not h12 = h21.

Kirchhoff's laws are based on fundamental physical principles of conservation: Kirchhoff's Current Law (KCL) is based on the conservation of charge, and Kirchhoff's Voltage Law (KVL) is based on the conservation of energy. These laws describe the behavior of currents and voltages at nodes and around loops in a lumped-element circuit. Because they are derived from conservation laws, they do not depend on the nature of the components (whether they are linear resistors or non-linear diodes/transistors) or whether the components are time-invariant. As long as the circuit can be treated as a 'lumped' system (where the physical dimensions are much smaller than the wavelength of the signals), KCL and KVL remain valid for both linear and non-linear circuits. Therefore, option (d) is correct, while options (a), (b), and (c) are too restrictive.

In a Bipolar Junction Transistor (BJT), the leakage current (specifically the reverse saturation current) is highly sensitive to temperature changes. The leakage current $I_{CEO}$ (Collector-to-Emitter leakage current with the base open) is caused by the flow of minority carriers across the reverse-biased junction. As temperature increases, more electron-hole pairs are thermally generated in the semiconductor material, leading to a significant increase in the concentration of minority carriers. A general rule of thumb in semiconductor physics is that the reverse saturation current approximately doubles for every 10°C rise in temperature. Analyzing the other options: - (a) Output resistance typically decreases or remains relatively stable compared to the exponential rise of leakage current. - (c) Current gain ($\beta$) generally increases slightly with temperature because the current gain is proportional to the mobility of carriers and the base transport factor, though this effect is less 'noticeable' or dominant than the leakage current surge. - (d) While AC current gain may change, the most immediate and critical thermal instability in a CE configuration is the increase in $I_{CEO}$, which can lead to thermal runaway.

To solve this problem, we use the fundamental properties of planar graphs and their duals. 1. **Given Data**: - Number of nodes (V) = 4 - Number of elements/edges (E) = 5 2. **Euler's Formula for Planar Graphs**: For a connected planar graph, the relationship between vertices (V), edges (E), and faces (F) is given by: V - E + F = 2 Substituting the given values: 4 - 5 + F = 2 -1 + F = 2 F = 3 3. **Understanding Faces and Meshes**: The total number of faces (F) includes the 'outer' (infinite) face. The number of meshes (M) in the original network is the number of bounded faces: M = F - 1 = 3 - 1 = 2 4. **Properties of the Dual Network**: In graph theory, the dual of a planar graph is constructed such that: - The number of nodes in the dual network (V') equals the number of faces in the original network (F). - The number of edges in the dual network (E') equals the number of edges in the original network (E). - The number of faces in the dual network (F') equals the number of nodes in the original network (V). 5. **Calculating Meshes in the Dual Network**: The number of meshes in the dual network (M') is the number of bounded faces of the dual graph: M' = F' - 1 Since F' = V (from the dual property), we have: M' = V - 1 M' = 4 - 1 = 3 Wait, let's re-evaluate the definition of 'mesh' in the context of network topology for this specific exam. In many electrical network textbooks, the number of independent meshes is given by the formula: M = E - (V - 1). For the original network: M = 5 - (4 - 1) = 5 - 3 = 2. For the dual network: Nodes (V') = F = 3 Edges (E') = E = 5 Meshes (M') = E' - (V' - 1) = 5 - (3 - 1) = 5 - 2 = 3. Correction: Let's re-read the question. 'The number of meshes in its dual network'. Original: V=4, E=5. Dual: V'=3, E'=5. Meshes in Dual = E' - V' + 1 = 5 - 3 + 1 = 3. However, looking at the standard relationship: The number of meshes in the original network equals the number of nodes in the dual network minus one, and the number of nodes in the original network minus one equals the number of meshes in the dual network. Nodes of original (V) = 4. Meshes of dual (M') = V - 1 = 4 - 1 = 3. Wait, let me re-calculate. Original: V=4, E=5, F=3. Dual: V'=3, E'=5, F'=4. Meshes in Dual = F' - 1 = 4 - 1 = 3. Looking at the options provided (a. 4, b. 3, c. 2, d. 1), the calculated value is 3. Re-checking the logic: Original Graph G: V=4, E=5. Faces F = E - V + 2 = 5 - 4 + 2 = 3. Dual Graph G*: V* = F = 3, E* = E = 5. Faces F* = E* - V* + 2 = 5 - 3 + 2 = 4. Meshes in G* = F* - 1 = 4 - 1 = 3. Therefore, the correct option is b.

To find the final value of a signal given its Laplace transform Y(s), we typically use the Final Value Theorem (FVT). The Final Value Theorem states that: lim_{t→∞} y(t) = lim_{s→0} sY(s), provided that all poles of sY(s) lie in the left half of the s-plane (LHP). 1. Given Y(s) = 1 / (s(s - 1)). 2. Let's examine the poles of Y(s). The poles are the values of s for which the denominator is zero: s = 0 and s = 1. 3. For the Final Value Theorem to be applicable, the system must be stable, meaning all poles of sY(s) must have negative real parts. 4. Let's find sY(s): sY(s) = s * [1 / (s(s - 1))] = 1 / (s - 1). 5. The pole of sY(s) is at s = 1. Since this pole is in the Right Half Plane (RHP), the system is unstable. 6. When a system has a pole in the RHP, the time-domain signal y(t) grows exponentially. 7. To verify, we can find the inverse Laplace transform of Y(s) using partial fraction expansion: Y(s) = 1 / (s(s - 1)) = -1/s + 1/(s - 1). Taking the inverse Laplace transform: y(t) = -1 + e^{t} for t ≥ 0. 8. As t approaches infinity, y(t) = lim_{t→∞} (-1 + e^{t}) = infinity. Therefore, the final value is infinity. Options a, b, and c are incorrect because they assume the FVT can be blindly applied without checking stability, or they represent finite values which do not match the exponential growth of the signal.

In an n-channel JFET, pinch-off occurs when the depletion regions meet, effectively restricting the current flow. The pinch-off voltage (Vp) is defined as the value of VDS at which pinch-off occurs when VGS = 0. However, when a gate-to-source voltage (VGS) is applied, the voltage required at the drain to reach pinch-off changes. The condition for pinch-off is given by the relationship: VDS(pinch-off) = Vp + VGS. Given that the pinch-off voltage (Vp) is 4 V and the drain-to-source voltage at which pinch-off occurs (VDS) is 1 V, we substitute these values into the formula: 1 V = 4 V + VGS. Solving for VGS: VGS = 1 V - 4 V = -3 V. In the context of JFETs, VGS for an n-channel device is typically negative to narrow the channel. The magnitude of the required voltage is 3 V. Therefore, the pinch-off occurs for VGS = -3 V (represented as 3 V in the options).

The Free Electron Theory (specifically the Drude model and the Sommerfeld model) is based on the fundamental assumption that the valence electrons in a metal are 'free' to move throughout the volume of the metal. To simplify the physics, the theory assumes that the interior of the metal is a region of constant potential (often taken as zero), meaning the electrons do not experience any internal forces or periodic potentials from the ion cores while moving. The only potential barrier exists at the surface of the metal, which prevents the electrons from escaping (the work function). Why other options are incorrect: - b. Sinusoidal potential: This describes the periodic potential used in the 'Nearly Free Electron Model' or 'Bloch's Theorem' to account for the crystal lattice, but the basic Free Electron Theory ignores this. - c. Square-wave potential: This is not a standard assumption for the bulk behavior of electrons in metals. - d. A superconductor: This is a state of matter/property of certain materials at low temperatures, not a type of potential that electrons are subjected to within the general free electron theory.

In material science, vacant lattice sites are known as vacancies (a type of point defect). In halide crystals (such as NaCl or KCl), these vacancies play a critical role in the transport of matter and charge: 1. Diffusion: For an atom or ion to move through a crystal lattice, it typically requires an adjacent empty site to jump into. Therefore, the rate of diffusion is directly proportional to the concentration of vacancies. 2. Ionic Conductivity: In ionic crystals, electrical conduction occurs via the movement of ions. Ions can migrate through the lattice by hopping from one lattice site to a neighboring vacant site. Without these vacancies, the ions would be locked in place, and the crystal would be a near-perfect insulator. Plastic deformation is primarily caused by the movement of dislocations (line defects), and transparency is an optical property related to the electronic band gap, not point defects. Thus, vacancies specifically facilitate diffusion and ionic conductivity.