Class 12 Physics Chapter 3 Current Electricity

Series Limitations, Complex Circuit Analysis

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To solve this problem, we need to analyze the impact of a faulty cell and compare it with the effect of having cells in reverse polarity:

– In the assertion, when one cell has zero emf in a series connection, it acts as a simple resistor rather than contributing to the overall emf. However, without considering additional factors such as internal resistance, the presence of other cells can still maintain some potential difference across the terminals, albeit reduced from the expected sum.

– The reason states that total emf diminishment occurs solely due to reverse polarity. While reverse polarity does reduce effective emf by subtracting cell values, having a cell with zero emf does not eliminate the contribution of others unless the internal resistances completely offset available voltage.

Thus, while both statements are true independently, the reason doesn’t specifically address the scenario described in the assertion where a non-contributing cell leads to operational failure; it focuses on a different cause for emf reduction.

Therefore, option b) “Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.” is correct.

Key Concept: Real-world Application, Internal Resistance

c) 2 A
[Solution Description] To achieve maximum power transfer across the load, the load resistance R must be equal to the internal resistance r of the battery – known as the condition for maximum power transfer.

Calculating the total resistance $R_t = R + r = 11 + 1 = 12 \, \Omega$.

The total current I can be calculated using Ohm’s Law:

$I = \frac{V}{R_t} = \frac{24}{12} = 2 \, \text{A}.$

Thus, the current flowing through the circuit should be 2 A.

Key Concept: Current Calculation

c) 5 A

[Solution Description]

The formula to calculate current $I$ is given by $I = \frac{q}{t}$, where $q$ is the charge and $t$ is the time. Here, $q = 15 \, \text{C}$ and $t = 3 \, \text{s}$.

$I = \frac{15 C}{3 s} = 5 \, \text{A}$

Therefore, the current is 5 Amperes.

Key Concept: Simple Application

b) 3 A
[Solution Description] According to KCL, the sum of currents entering a junction equals the sum of currents leaving the junction. Here, 4 A enters and 1 A leaves, so the remaining current leaving should be $4 – 1 = 3 A.$

Key Concept: Temperature Effect

b) Decreases

[Solution Description] In semiconductors, with an increase in temperature, the number of free electrons increases due to thermal excitation. This increase results in decreased resistivity because resistivity depends inversely on the number of free electrons.

Key Concept: Simple Calculation

d) 30 $\Omega$

[Solution Description] To balance the Wheatstone bridge, we use the equation $\frac{R_1}{R_2} = \frac{R_3}{R_4}$. Substituting the given values:

$\frac{5}{10} = \frac{15}{R_4}$

Solving for $R_4$:

$R_4 = \frac{15 \times 10}{5} = 30 \, \Omega$

Therefore, $R_4$ should be 30 $\Omega$ for the bridge to be balanced.

Key Concept: Real-world Application, Internal Resistance

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that internal resistance is significant in devices powered by batteries for long durations. This is true because high internal resistance can lead to greater energy losses as heat, reducing the overall efficiency and operational time of the device.

The reason explains that increased current leads to a higher voltage drop across the internal resistance of the battery. This is also true because according to Ohm’s Law for the internal resistance ($V_{\text{drop}} = Ir_{\text{internal}}$), where $I$ is the current and $r_{\text{internal}}$ is the internal resistance, an increase in current results in a proportionate increase in the voltage lost within the battery itself.

Therefore, both the Assertion and the Reason are true, and the Reason correctly explains why internal resistance affects device efficiency and lifespan.

Key Concept: Basic Formula

c) $P = I^2 R$
[Solution Description] The power loss in AC circuits can be calculated using the formula derived from Ohm’s Law, which accounts for resistive losses. It is represented by $P = I^2 R$, where $P$ is the power loss, $I$  is the current, and R  is the resistance.

Key Concept: Basic Definition

c) Total opposition to current flow
[Solution Description] Impedance in an AC circuit represents the total opposition to current flow, which is a combination of resistance and reactance.

Key Concept: Resistivity Formula

d) $rT = r0 [1 + a (T-T0)]$

[Solution Description] The resistivity of a metallic conductor changes with temperature and is given by $rT = r0 [1 + a (T-T0)]$, where $r0$ is the resistivity at some reference temperature $T0$, $a$ is the temperature coefficient, and $T$ is the current temperature.

Key Concept: Basic Concept

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states the formula for the equivalent emf when two cells are connected in parallel. This formula is correct and derived from the principle that in parallel, potentials across all components remain the same. The reason provides the formula for the equivalent internal resistance, which is also correct according to the formulas provided for cells in parallel. However, the reason does not explain why the assertion about the equivalent emf is true. Therefore, both the assertion and reason are true, but the reason is not the correct explanation of the assertion.

Key Concept: Dimensional Influence

c) Assertion is true, but Reason is false.
[Solution Description]
The resistance $R$ of a conductor is given by the formula

$R = \rho \frac{l}{A}$

where $\rho$ is the resistivity, $l$ is the length, and $A$ is the cross-sectional area. According to this equation, resistance is directly proportional to the length ($l$), so if the length is doubled, the resistance also doubles, making the Assertion true.

However, the Reason states that resistance is proportional to the cross-sectional area, which is incorrect because resistance is inversely proportional to the area ($A$). Thus, the Reason is false.

Therefore, the correct option is (c): Assertion is true, but Reason is false.

Key Concept: Basic Concept

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Energy conservation in electrical circuits implies that energy provided must be equal to energy used, hence the assertion that KVL is based on this concept is true. Also, according to KVL, the sum of the electromotive forces (EMF) or voltage sources in any closed circuit is equal to the sum of potential drops (voltage drops). Hence, both statements are true and Reason correctly explains Assertion.

Key Concept: Basic Series Property

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]

The assertion states that in a series circuit, when one resistor fails, the whole circuit stops functioning. This is true because there is no alternative pathway for the current to pass through if one component is broken or missing. The reason provided is also true as it explains why this phenomenon occurs: series circuits consist of a single path for current flow. Thus, both the assertion and the reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Series Combination

b) $e_1 + e_2$

[Solution Description] When two cells are connected in series, their equivalent EMF is given by the sum of their individual EMFs, i.e., $e_{\text{eq}} = e_1 + e_2$.

Key Concept: Field Dependency

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion claims that Ohm’s Law applies universally, which is incorrect as it is known to be valid only for ohmic materials where resistance remains constant. Certain materials exhibit non-ohmic behavior, where the resistance may vary with changes in temperature, electric field, or other factors, leading to a non-linear V-I relationship. The reason is true and highlights how materials like semiconductors can have their V-I characteristics affected by varying electric fields.

Key Concept: Charge Flow

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion correctly states that in a solid conductor, electric current is primarily carried by negatively charged electrons. This aligns with the concept that electrons are free to move under the influence of an electric field in solids.

The reason, however, is incorrect because it suggests that both positive and negative charges move in solid conductors. In reality, within solid conductors, only negatively charged electrons are mobile; the lattice ions or positive charges remain mostly stationary.

Thus, the Assertion is true, but the Reason is false.

Key Concept: Simple Calculation

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion statement “The equivalent emf of a series combination of cells is the sum of their individual emfs” is true because in a series connection, the total voltage is simply the addition of all the voltages since they are connected end to end.

However, the reason “The potential difference across each cell in series is the same” is false. In a series circuit, the potential difference across each component depends on its resistance or internal characteristics and is not necessarily the same for each element.

Thus, the correct answer is that the assertion is true but the reason provided is false.

Key Concept: Basic Formula

c) $V = IR$

[Solution Description]

Ohm’s Law states the relationship between voltage ($V$), current ($I$), and resistance ($R$) in an electrical circuit. The formula is given by:

$V = IR$

This equation expresses that voltage is equal to the product of current and resistance.

Key Concept: Mobility Definition

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] Both the assertion and reason correctly define and represent the concept of mobility in conductors. Mobility, denoted by $\mu$, is indeed the drift velocity ($v_d$) per unit electric field ($E$), which is clearly shown through the equation $\mu = \frac{v_d}{E}$. Thus, both statements are true and the reason provides the correct explanation of the assertion.

Key Concept: Power Efficiency, Voltage Transformation

c) 95.87%
[Solution Description] To find the efficiency, we need to calculate the power loss in the transmission lines and subtract it from the generated power. The current in the transmission line can be found using:
$I = \frac{P_{\text{generated}}}{V_{\text{transmission}}} = \frac{500 \times 10^6}{220 \times 10^3} \approx 2272.73 \, \text{A}$

Power loss due to ohmic heating in the transmission line is given by $P_c = I^2 R$,
$P_c = (2272.73)^2 \times 4 = 20.65 \times 10^6 \, \text{W}$

Efficiency can thus be calculated as:
$\text{Efficiency} = \left(1 – \frac{P_c}{P_{\text{generated}}}\right) \times 100 = \left(1 – \frac{20.65 \times 10^6}{500 \times 10^6}\right) \times 100 \approx 95.87%$

Therefore, the efficiency of this transmission system is approximately 95.87%.

Key Concept: Resistance Impact

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] The formula for power dissipation across a resistor is given by $P = \frac{V^2}{R}$. Here, $P$ denotes the power dissipated, $V$ represents voltage, and $R$ indicates resistance. According to this formula, as resistance ($R$) increases while keeping voltage ($V$) constant, the power dissipated ($P$) decreases because they are inversely related. Thus, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Basic Formula Application

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
Both the assertion and reason are correct expressions for power loss in a conductor. The formula $P = I^2 R$ is derived from the basic relationship between current ($I$), resistance ($R$), and power ($P$). Similarly, the formula $P = V^2R$ comes from substituting Ohm’s law ($V = IR$) into the power formula. Therefore, while both statements are true, the reason provided does not directly explain why the assertion is true.

Key Concept: Ohmic Loss

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description] The power loss in a resistor can be expressed as $P = I^2R$. However, this formula is not a direct derivation of Ohm’s Law, but rather an expression obtained when considering both voltage and current relationships in resistive circuits. Ohm’s Law, given by $V = IR$, helps relate voltage, current, and resistance, but does not directly derive the power loss equation. Therefore, although both statements are true, the reason is not the correct explanation of the assertion.

Key Concept: Basic Relation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] The assertion states that resistance is inversely proportional to cross-sectional area, which is true according to the formula $R = \rho \frac{l}{A}$. This means if the cross-sectional area ($A$) decreases, the resistance ($R$) will increase, provided other factors remain constant. The reason correctly explains this relationship using the formula. Therefore, both the assertion and reason are true and the reason is the correct explanation of the assertion.

Key Concept: Formula Application

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] To find the equivalent resistance $R_{eq}$ of two resistors in parallel, we use the formula:

$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$

Substituting the given resistor values:

$\frac{1}{R_{eq}} = \frac{1}{6} + \frac{1}{12}$

Calculating each term separately:

$\frac{1}{6} = 0.1667, \quad \frac{1}{12} = 0.0833$

Adding these terms gives:

$\frac{1}{R_{eq}} = 0.1667 + 0.0833 = 0.25$

Taking the reciprocal to find $R_{eq}$:

$R_{eq} = \frac{1}{0.25} = 4 \, \Omega$

Therefore, the assertion that the equivalent resistance is 4 $\Omega$ is true, and the reason correctly explains how it is calculated.

Key Concept: Resistivity Equation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve this, we need to understand the resistivity equation provided. The formula $r_T = r_0 [1 + a (T–T_0)]$ describes how the resistivity ($r_T$) of a metallic conductor changes with temperature. Here, $r_0$ is the resistivity at a reference temperature $T_0$, and $a$ is the temperature coefficient of resistivity.

The assertion states that this is the resistivity equation for materials, which is true as it captures the linear relationship between temperature and resistivity in metals.

The reason points out that the equation’s applicability is limited to a specific temperature range. This is also true because most materials exhibit non-linear behavior outside the specified range; thus, the linear approximation ceases to be accurate. Since both statements are correct and the explanation aligns correctly with the assertion, the correct response is option (a).

Key Concept: Material Property

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion states that resistivity is a property of the material, which is true because resistivity is inherent to the material and does not depend on its shape or size.

The reason claims that resistivity depends on the dimensions of the conductor, which is false. While resistance depends on the length and cross-sectional area of the conductor, resistivity is independent of these geometric factors and only varies with material type and conditions like temperature.

Thus, while the assertion is correct, the reason provided is incorrect.

Key Concept: Basic Definition

d) m$^2$/Vs

[Solution Description]

The definition states that mobility $\mu$ is measured in terms of $m^2/Vs$. This corresponds to its formula where drift velocity ($v_d$) with units of m/s is divided by electric field ($E$) with units of V/m, resulting in units of $m^2/Vs$.

Key Concept: Ohm’s Law Basics

b) $V = RI$

[Solution Description] Ohm’s Law states that the voltage $V$ across a conductor is equal to the product of the current $I$ flowing through it and its resistance $R$. The mathematical representation is $V = IR$.