CUET UG Physics Booster Test – 3 Material Properties and Power

Ohm's law states that voltage is directly proportional to current, producing a straight-line V-I graph. Therefore, linear Ohm's law behavior corresponds to voltage remaining proportional to current. Non-linear deviation occurs when this proportionality breaks down. Sign-dependent behavior is observed in devices such as diodes where reversing the voltage polarity changes the current drastically. Non-unique V-I behavior occurs in materials such as GaAs where a single current value may correspond to multiple voltage values. Hence the correct matching is: 1-d, 2-c, 3-b, 4-a

In sign-dependent devices such as semiconductor diodes, reversing the polarity of the applied voltage changes the current dramatically. A forward bias permits substantial current flow, whereas a reverse bias allows only a very small current. This violates the simple linear relation between voltage and current assumed in Ohm's law. Statement 3 is incorrect because electrical power can still be dissipated as heat whenever current flows through the device.

For an ohmic conductor, reversing the applied voltage simply reverses the current direction while maintaining the same magnitude. Thus equal positive and negative voltages produce equal current magnitudes. A diode behaves differently. Under forward bias it conducts strongly, while under reverse bias the current is extremely small. Therefore the magnitudes of forward and reverse currents are unequal. Hence the condition ∣I_r∣≠∣I_0∣ indicates non-ohmic behavior.

Gallium Arsenide (GaAs) is a semiconductor material that exhibits electrical behavior significantly different from that of ordinary metallic conductors. In certain operating regions, particularly those associated with negative differential resistance, an increase in the applied voltage can lead to a decrease in current. This unusual behavior is observed in devices such as Gunn diodes fabricated using GaAs.

Because the V-I characteristic is non-linear, a single current value may sometimes correspond to more than one voltage value. This means that the voltage-current relationship is not unique throughout the entire operating range. Such behavior differs fundamentally from the straight-line V-I graph observed for ohmic conductors.

Furthermore, GaAs does not obey Ohm's law globally because its resistance is not constant over all applied voltages. Instead, its V-I curve contains non-linear regions where the ratio V/I changes continuously. Therefore, all three statements correctly describe the electrical characteristics of Gallium Arsenide, making Option D the correct answer.

Using the microscopic current relation, I=neAv_d Therefore, v_d=I/neA Substituting the values, v_d=1.5/(8.5×10^(28))(1.6×10^(-19))(1.0×10^(-7))v_d≈1.1×10^(-3) m/s This extremely small drift speed demonstrates that large electric currents result from the enormous number of free electrons rather than from rapid electron motion.

Insulators such as ceramic, rubber, glass, and plastics possess extremely high resistivities compared to metallic conductors. Their resistivities are often 10^(18)times greater than those of metals. This occurs because electrons remain tightly bound to their parent atoms and cannot move freely under ordinary electric fields. Since free charge carriers are practically absent, electrical conduction is extremely difficult. As a result, insulators are widely used for electrical isolation and protection. Therefore, option A correctly describes the fundamental difference between conductors and insulators.

The temperature coefficient of resistivity αis used in the approximate relation ρ_T=ρ_0[1+α(T-T_0)] It represents the fractional change in resistivity per unit change in temperature. This approximation works only over a limited range of temperatures where the resistivity-temperature graph is approximately linear. For most metallic conductors, αis positive, meaning resistivity increases as temperature rises.

The relation ρ_T=ρ_0[1+α(T-T_0)] is a first-order linear approximation describing how resistivity varies with temperature. It assumes approximately linear behavior around the reference temperature T_0. At T=T_0, ρ_T=ρ_0 which confirms the meaning of the reference resistivity. The equation is not intended to describe resistivity accurately over all temperatures and certainly does not represent a higher-order non-linear polynomial relation.

In metals, the number density of free electrons remains nearly constant with temperature. However, increasing temperature increases lattice vibrations and the random thermal motion of electrons. As the lattice vibrates more intensely, electrons experience more frequent collisions. This reduces the average relaxation time τ. Since conductivity depends on τ, σ=ne^2τ/m a decrease in τreduces conductivity and increases resistivity. Thus all three statements are correct.

The relation ρ_T=ρ_0[1+α(T-T_0)] is only an approximate linear description valid over a moderate temperature range. At very low temperatures, experimental observations show significant deviations from this linear behavior. The resistivity-temperature curve bends away from a straight line, indicating that a constant value of αcan no longer accurately describe the material. Statement 3 is incorrect because resistivity does not suddenly become completely independent of temperature.

The temperature dependence of resistivity for metallic conductors is expressed by the relation: ρ = ρ₀ [1 + α(T − T₀)] where ρ is the resistivity at temperature T, ρ₀ is the resistivity at reference temperature T₀, and α is the temperature coefficient of resistivity. The coefficient α indicates how strongly the resistivity changes with temperature. For ordinary metals such as copper, α has a significant positive value, causing resistivity to increase noticeably with temperature. Nichrome is a special alloy used in heating elements because its resistivity remains nearly constant even when subjected to large temperature variations. If the measured fractional change in resistivity remains extremely small despite a large increase in temperature, the multiplying factor α must itself be very small. Mathematically, only a value of α close to zero can produce negligible changes in resistivity over a substantial temperature range. This property makes nichrome suitable for heating coils, electric irons, and resistance wires because the resistance remains relatively stable during operation.

Precision resistors are expected to maintain a constant resistance value under varying environmental conditions. According to NCERT, alloys such as manganin and constantan are preferred for constructing standard resistors because their resistivity changes very little with temperature. This property is represented by a very small temperature coefficient of resistivity. When current passes through a resistor, heating occurs due to the Joule heating effect. If the material had a large temperature coefficient, the resistance would change significantly as temperature rises, causing measurement errors. Manganin and constantan overcome this limitation because their resistance remains almost unchanged over a wide temperature range. As a result, they are extensively used in resistance boxes, bridge circuits, laboratory standards, and precision electrical instruments. The stability of resistance is far more important for standard resistors than simply having low resistance. Therefore, these alloys are selected because they provide accurate and reproducible resistance values independent of ordinary temperature fluctuations.

In intrinsic semiconductors, conductivity depends strongly on the concentration of charge carriers. As temperature increases, more electrons gain sufficient energy to cross the forbidden energy gap from the valence band to the conduction band. This process creates additional electron-hole pairs and substantially increases the carrier density n. Although the relaxation time τ may decrease slightly because of increased lattice vibrations, the increase in carrier concentration is much greater. Consequently, the overall conductivity increases with temperature. This behavior is opposite to that observed in metallic conductors, where conductivity generally decreases as temperature rises. The third statement is incorrect because conduction in semiconductors occurs through electrons and holes moving within a crystal lattice. Electrolytic conduction involves ions moving through a solution. Therefore, semiconductor conduction and electrolytic conduction are fundamentally different mechanisms.

For a semiconductor, σ ∝ nτ Let initial conductivity be σ₁ and final conductivity be σ₂. Given: σ₂ = 2σ₁ Also, relaxation time decreases by 20%: τ₂ = 0.8τ₁ Using the conductivity relation: σ₂/σ₁ = (n₂τ₂)/(n₁τ₁) Substituting values: 2 = (n₂ × 0.8τ₁)/(n₁τ₁) 2 = 0.8(n₂/n₁) n₂/n₁ = 2/0.8 n₂/n₁ = 2.5 Therefore, the carrier density must increase by a factor of 2.5. This result illustrates a key NCERT concept: in semiconductors, the rise in carrier concentration with temperature dominates over the reduction in relaxation time, producing an overall increase in conductivity. Unit Verification n₂/n₁ is a ratio of carrier densities and is dimensionless.

According to NCERT, a steady current in a conductor requires the presence of an electric field inside the conductor. The electric field is related to the potential gradient through the relation E = −dV/dx. When the electric potential decreases continuously along a conductor, a non-zero electric field exists within the material. This electric field exerts force on free electrons, causing them to acquire a drift velocity opposite to the field direction. The collective drift of electrons constitutes electric current. The existence of a potential drop also indicates that electrical energy is continuously being converted into thermal energy because of collisions between electrons and lattice ions. Thus, a conductor carrying current is not an equipotential region. Instead, a finite potential gradient must exist to maintain the electric field required for current flow. This principle forms the basis of Ohm's law and explains why resistance causes a voltage drop in practical circuits.

The electric potential energy associated with a charge depends on the electric potential at its location. The change in potential energy is defined as: ΔUₚₒₜ = Ufinal − Uinitial For a charge ΔQ moving from point A to point B, ΔUₚₒₜ = ΔQ[V(B) − V(A)] When the charge moves from a higher potential to a lower potential, V(B) < V(A). Therefore, the quantity [V(B) − V(A)] becomes negative. As a result, the potential energy decreases and ΔUₚₒₜ becomes negative. This decrease in potential energy supplies the energy ultimately converted into heat within the conductor. For a steady current I flowing for time Δt, the charge transferred is ΔQ = IΔt. Using the voltage drop V = V(A) − V(B), ΔUₚₒₜ = −IVΔt Thus, options A, B and D are consistent with NCERT concepts of electric potential energy and energy dissipation. The incorrect statement is the assertion that the change in potential energy is always positive.

In a conductor, electrons are continuously accelerated by the internal electric field. If collisions were absent, the decrease in potential energy would be completely converted into kinetic energy, giving: ΔK = −ΔUₚₒₜ However, real conductors contain lattice ions that constantly scatter the drifting electrons. These frequent collisions prevent electrons from continuously accelerating and gaining unlimited kinetic energy. Instead, electrons attain a steady average drift velocity. During each collision, part of the energy acquired from the electric field is transferred to the lattice ions. This transferred energy increases the vibrational motion of the atoms in the conductor. The enhanced lattice vibration manifests as heating of the conductor, which is the microscopic origin of Joule heating. Thus, all three statements correctly describe the NCERT explanation of energy transfer and resistance in conducting materials.

The heat produced in a resistor is given by: ΔW = IVΔt This expression assumes steady current flow, which means electrons possess a constant average drift velocity maintained by repeated collisions with lattice ions. Without these collisions, current would not remain steady. The energy supplied by the electric field is transferred through electron-ion interactions to the atomic lattice. Consequently, ΔW represents the energy converted into thermal energy within the conductor. Using Ohm's law: V = IR For constant resistance, P = IV = V²/R Therefore, ΔW = (V²/R)Δt This shows that heat generated varies as the square of the applied voltage when resistance remains constant. Hence all three statements accurately describe the physical interpretation of Joule heating.

Electrical power in a resistor is expressed as: P = IV At the microscopic level, the electric field performs work on free electrons. These electrons gain energy while drifting through the conductor. However, because of frequent collisions with lattice ions, the gained energy is repeatedly transferred to the lattice. This transferred energy increases the vibrational motion of the atoms within the conductor. The enhanced vibrations manifest as thermal energy, causing the conductor to heat up. This process is known as Ohmic loss or Joule heating. Therefore, the macroscopic power consumed by a resistor corresponds directly to the rate at which energy is transferred from drifting electrons to lattice vibrations. This microscopic interpretation provides the physical basis of power dissipation in resistive electrical devices.

For transmission of electrical power, P = VI Therefore, I = P/V Power lost in transmission cables is P₍c₎ = I²R₍c₎ Substituting I = P/V, P₍c₎ = (P/V)²R₍c₎ P₍c₎ = P²R₍c₎/V² This relation shows that transmission losses are inversely proportional to the square of the transmission voltage. If the voltage is doubled, the loss becomes one-fourth. If the voltage is increased ten times, the loss becomes one-hundredth. Therefore, electric power is transmitted at very high voltages using step-up transformers. The reduced current minimizes I²R losses, improves efficiency and decreases unnecessary heating of transmission lines.