Current Electricity
NCERT Class 12 Physics - Chapter 3 | High-Yield Concepts & Advanced Problems
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Question 1
A steady direct current flows through a metallic wire whose cross-sectional area increases monotonically from the left end to the right end. Which of the following macroscopic quantities remains strictly constant along the length of this non-uniform wire?
Concept Involved: Conservation of charge in steady-state current systems (NCERT Section 3.4).
Explanation: Under steady-state conditions, charge cannot accumulate at any segment of an isolated conductor. Therefore, the total number of charges entering any cross-section per unit time must equal the number of charges leaving it. This makes the total electric current (I) completely constant along the entire length of the wire. However, current density is defined as J = I / A. Since the area (A) increases from left to right, J must decrease. Furthermore, because J = σE, the internal electric field (E) must also decrease. Finally, since drift velocity is given by v_d = I / (nAe), it varies inversely with the area and will decrease along the wire. Thus, only the total electric current remains constant.
Common Misconception: Students often assume that because Ohm's law applies locally, the electric field or current density remains uniform throughout a continuous homogeneous material, ignoring the geometric constraints.
Question 2
A long cylindrical metallic wire of uniform resistance R is mechanically stretched such that its total length is increased by exactly 10% while maintaining a constant temperature. Assuming the material density and total volume remain perfectly invariant, what is the ratio of its new electrical resistivity to its initial electrical resistivity?
Concept Involved: Intrinsic vs. Extrinsic material properties (NCERT Section 3.2).
Explanation: Electrical resistivity (ρ) is an intrinsic property of a material that depends solely on its microscopic composition, chemical nature, and temperature. It is defined by the relation ρ = m / (n · e^2 · τ), where m is electron mass, n is charge carrier concentration, e is elementary charge, and τ is relaxation time. Mechanical stretching alters only the extrinsic geometric parameters of the conductor—such as its individual length (L) and cross-sectional area (A)—which will modify its total macroscopic resistance (R = ρL/A). Because the chemical composition and temperature remain completely fixed during the stretching process, the resistivity itself remains absolutely unchanged. Therefore, the ratio of new resistivity to initial resistivity is exactly 1:1.
Common Misconception: Students often confuse total structural resistance R (which increases to 1.21R due to the length increase and simultaneous area decrease) with electrical resistivity ρ, leading them to incorrectly select option A.
Question 3
A pure metallic conductor and an intrinsic semiconductor are both heated simultaneously from an initial temperature of 300 K to a final temperature of 400 K. What are the resulting changes in their respective electrical conductivities?
Concept Involved: Microscopic temperature dependence of conductivity (NCERT Section 3.9).
Explanation: Electrical conductivity is given by σ = (n · e^2 · τ) / m. In a metallic conductor, the number density of free electrons (n) is exceptionally high and stays practically constant with temperature. As a metal is heated, its lattice ions vibrate with greater amplitude, increasing the frequency of collisions with moving electrons. This shortens the average relaxation time (τ), causing the metal's conductivity to decrease. Conversely, in an intrinsic semiconductor, n is small at room temperature but increases exponentially with temperature as thermal energy breaks covalent bonds and releases free carriers. This exponential surge in n easily dominates the slight decrease in τ. Consequently, heating causes the conductivity of the metal to decrease and that of the semiconductor to increase.
Common Misconception: Forgetting that conductivity is the mathematical reciprocal of resistivity, causing students to invert the behavior of these two distinct properties.
Question 4
A specific non-ohmic electronic component displays a current-voltage characteristic curve where, over a certain operating range, the electric current drops as the applied voltage is increased. What can be concluded about the dynamic resistance of the device in this specific region?
Concept Involved: Limitations and extensions of Ohm's Law (NCERT Section 3.5).
Explanation: For non-ohmic materials and modern semiconductor devices (such as a tunnel diode or a Gallium Arsenide crystal structure), the relationship between voltage V and current I is non-linear. The localized resistance at any point on this characteristic curve is defined as the dynamic resistance, given by R_dynamic = ΔV / ΔI. In regions where an increase in the applied potential difference (ΔV > 0) results in a corresponding reduction in the net current flow (ΔI < 0), the ratio yields a negative value. This behavior is called negative dynamic resistance and is useful in amplification systems and high-frequency oscillators.
Common Misconception: Students often believe that electrical resistance must always be a static, positive value, failing to realize that dynamic resistance represents a localized rate of change rather than a simple V/I ratio.
Question 5
At a constant temperature, an external voltage source is connected across a uniform copper wire, creating an internal electric field E. If the magnitude of this electric field is doubled, how does it affect the average relaxation time τ of the conduction electrons?
Concept Involved: Microscopic mechanics of electron transport and drift velocity (NCERT Section 3.4).
Explanation: The relaxation time τ is defined as the average time interval separating two consecutive collisions of a conduction electron with the localized ionic lattice of the metal. It depends on the mean free path λ and the root-mean-square thermal velocity of the electrons, given by τ ≈ λ / v_thermal. At a constant temperature, the thermal velocity is around 10^6 m/s. The drift velocity added by an external electric field is very small, typically only 10^-4 m/s. Because this drift velocity is negligible compared to the thermal velocity, doubling the electric field has virtually no effect on the total speed of the electrons or their collision frequency. Therefore, the average relaxation time remains practically unchanged.
Common Misconception: It is common to incorrectly apply the formula v_dist = (e · E · τ) / m by treating τ as a dependent variable that changes with the field, rather than recognizing it as an intrinsic property of the material's thermal state.
Question 6
A current density vector inside a structured conductor is defined as J = (3i + 4j) A/m². What is the net electric current passing through an internal flat reference plane whose area vector is given by A = 5i m²?
Concept Involved: Vector properties of current density and scalar current definition (NCERT Section 3.2).
Explanation: Electric current is a scalar quantity defined as the flux of the current density vector J through a given surface area. Mathematically, this relation is expressed as the vector dot product: I = J · A. Given the vectors J = 3i + 4j and A = 5i, we compute the dot product by multiplying their corresponding components: I = (3 · 5) + (4 · 0) = 15 A. The component of the current density vector perpendicular to the surface area (along the i direction) contributes to the current passing through it, while the parallel component (along the j direction) does not cross the surface.
Common Misconception: Treating electric current as a vector or adding the components of the current density directly without evaluating the geometric orientation via the dot product.
Question 7
Two discrete resistors have resistances R&sub1; and R&sub2; at 0°C, with temperature coefficients of resistance α&sub1; (α&sub1; > 0) and α&sub2; (α&sub2; < 0) respectively. If these two components are connected together in a simple series arrangement, what mathematical relation must hold to ensure the total equivalent resistance remains entirely independent of small temperature variations?
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