Class 12 Physics Chapter 7 Alternating Current

Key Concept: Advanced Circuit Analysis, Impedance and Reactance

b) 159 Hz
[Solution Description] The resonant frequency $$(f_0)$$ of a parallel RLC circuit is given by the formula:

$$
f_0 = \frac{1}{2\pi\sqrt{LC}}
$$

Substituting the given values:

$$
L = 100 \times 10^{-3} \, \text{H}, \quad C = 10 \times 10^{-6} \, \text{F}
$$

$$
f_0 = \frac{1}{2\pi\sqrt{(100 \times 10^{-3})(10 \times 10^{-6})}}
$$

Calculating further:

$$
f_0 = \frac{1}{2\pi\sqrt{1 \times 10^{-3}}}
$$

$$
f_0 = \frac{1}{2\pi \times 0.03162}
$$

$$
f_0 \approx 159 \, \text{Hz}
$$

Therefore, the resonant frequency is approximately 159 Hz.

Key Concept: Basic Concept

a) A rotating vector
[Solution Description] A phasor is a complex number representing the magnitude and phase of sinusoidal functions, usually in alternating current circuits. It can be seen as a rotating vector that provides information on the amplitude and phase angle.

Key Concept: Power Factor Concept

a) Maximum power dissipation
[Solution Description]
A power factor of 1 indicates that the voltage and current are in phase, leading to maximum power dissipation with no reactive power.

Key Concept: Construction and Working

c) Step-up transformer
[Solution Description] When the number of turns in the secondary coil $$(N_s)$$ is greater than the number of turns in the primary coil $$(N_p),$$ the transformer is called a step-up transformer. This increases the voltage from primary to secondary.

Key Concept: Advanced Phase Analysis, Power Calculation

c) $$0$$

[Solution Description] For a purely inductive circuit, the current lags the voltage by $$\frac{\pi}{2}$$. The expression for current is $$i(t) = i_m \sin(\omega t – \frac{\pi}{2})$$, where $$i_m = \frac{v_m}{X_L}$$. Given that $$v(t) = v_m \sin(\omega t)$$, we have $$v_m = 220 \sqrt{2} \, \text{V}$$. Next, calculate $$X_L = \omega L$$. Since L is not given, assume any value that cancels out later as it doesn’t affect calculation due to zero average power. Thus,

$$p(t) = v(t) \cdot i(t) = 220\sqrt{2} \sin(\omega t) \cdot i_m \sin(\omega t – \frac{\pi}{2})$$

At $$t = \frac{\pi}{4\omega}$$, we find:

$$p\left(\frac{\pi}{4\omega}\right) = 220\sqrt{2} \sin\left(\frac{\pi}{4}\right) \cdot i_m \sin\left(0\right) = 0$$

Hence the instantaneous power is zero.

Key Concept: Formula Application

b) $$12.57 \Omega$$
[Solution Description]
The formula for inductive reactance $$X_L$$ is given by:
$$
X_L = \omega L
$$
where $$\omega = 2\pi f$$ and $$L = 40$$ $$mH = 40 \times 10^{-3} H.$$

First, calculate $$\omega$$:
$$
\omega = 2\pi \times 50
$$
$$
\omega = 314.16 \, \text{rad/s}
$$

Now, calculate $$X_L$$:
$$
X_L = 314.16 \times 40 \times 10^{-3}
$$
$$
X_L = 12.57 \, \Omega
$$

Therefore, the inductive reactance is $$12.57 \Omega.$$

Key Concept: Complex Circuit Analysis, Advanced Reasoning

a) 0.54

[Solution Description] To find the power factor, evaluate the total impedance of the circuit and determine the phase angle.

Calculate the inductive reactance $$(X_L)$$ and capacitive reactance $$(X_C)$$:

$$X_L = 2 \pi f L = 2 \pi \times 50 \times 0.5 = 157.08 \, \Omega$$

$$X_C = \frac{1}{2 \pi f C} = \frac{1}{2 \pi \times 50 \times 100 \times 10^{-6}} \approx 31.83 \, \Omega$$

Net reactance X is given by $$X = X_L – X_C$$:

$$X = 157.08 – 31.83 = 125.25 \, \Omega$$

Calculate the net impedance Z using $$Z = \sqrt{R^2 + X^2}$$:

$$Z = \sqrt{80^2 + 125.25^2} \approx 149.49 \, \Omega$$

Phase angle $$\phi$$ is given by $$\tan \phi = \frac{X}{R}$$, hence:

$$\tan \phi = \frac{125.25}{80} \Rightarrow \phi = \tan^{-1}\left(\frac{125.25}{80}\right) \approx 57.47^\circ$$

Finally, power factor $$\cos \phi$$ is:

$$\cos \phi = \cos(57.47^\circ) \approx 0.54$$

Key Concept: Frequency Variation Impact, Resonance in Security Systems

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] At resonance, the impedance of an LCR circuit is minimized, hence the current is maximized according to $$i_m = \frac{v_m}{Z}$$. When the frequency deviates from its resonant value, the impedance increases due to the mismatch between capacitive and inductive reactances $$X_C \ne X_L$$ leading to a decrease in current amplitude since $$Z > R$$. This concept is utilized in metal detectors where metals alter the inductive properties, effectively shifting the resonance frequency and reducing the current at the original resonance frequency. Thus, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: RMS Values

b) 7.07 A

[Solution Description] To find the RMS current, we use the formula $$I_{rms} = \frac{i_m}{\sqrt{2}}$$. Here, $$i_m = 10\,A.$$

$$\begin{align*}
I_{\text{rms}} & = \frac{10}{\sqrt{2}} \\
& = \frac{10}{1.414} \\
& \approx 7.07\,\text{A}
\end{align*}$$

Key Concept: Phasor Diagram Analysis

c) They are 180 degrees out of phase

[Solution Description]

Since $$V_L$$ leads the current by $$\pi/2$$ and $$V_C$$ lags the current by $$\pi/2$$, the angle between $$V_L$$ and $$V_C$$ is $$\pi$$. This means $$V_L$$ and $$V_C$$ are 180 degrees out of phase.

Key Concept: Phasor Diagram Interpretation

d) In the same direction

[Solution Description] In a phasor diagram for an ideal resistor, both the voltage and current phasors are aligned in the same direction since they are always in phase with each other.

Key Concept: Basic Concept

b) Decrease voltage
[Solution Description]
A step-down transformer decreases the voltage while increasing the current according to the relation $$\frac{V_s}{V_p} = \frac{N_s}{N_p}$$, where $$N_s < N_p.$$

Key Concept: Efficiency Calculation

c) 90%

[Solution Description]

Efficiency is calculated using the formula:

$$\text{Efficiency (%)} = \left( \frac{\text{Output Power}}{\text{Input Power}} \right) \times 100$$

Substituting the given values:

$$\text{Efficiency (%)} = \left( \frac{1350}{1500} \right) \times 100 = 90\%$$

Thus, the efficiency of the transformer is 90%.

Key Concept: Magnetic Flux Dynamics, Inductive Reactance

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

[Solution Description] According to Faraday’s law, the induced emf is proportional to the rate of change of magnetic flux. In an AC generator, a rapid change in magnetic flux indeed results in a higher induced emf. However, this does not directly cause a phase shift between voltage and current; rather, it is the nature of inductive reactance that introduces such a phase difference. Inductive reactance $$X_L$$ is given by $$X_L = 2\pi f L$$, where f is the frequency and L is the inductance. This shows that the inductive reactance affects how the current lags behind the voltage, creating a phase shift in an AC circuit. Therefore, while both statements are true individually, the reason provided is not the correct explanation for the assertion.

Key Concept: Complex Impedance, Real World Application

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that low power factor leads to increased power losses in transmission, which is true. According to the formula for average power $$P = VI\cos\phi$$, a low $$\cos\phi$$ means that more apparent power $$(VI)$$ is needed for the same real power (P), increasing the current and thus resistive losses $$(I^2 R)$$ losses in the lines.

The reason explains that power factor correction reduces the total current required, which is also true. By improving the power factor using capacitors or other methods, we decrease the phase angle between voltage and current, which increases $$\cos\phi$$. This allows for the same real power to be transmitted with less current, thereby decreasing line losses.

Both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Basic Formula

c) $$\omega_0 = \frac{1}{\sqrt{LC}}$$
[Solution Description] The resonant frequency $$\omega_0$$ of an LC circuit is given by the basic formula:
$$
\omega_0 = \frac{1}{\sqrt{LC}}
$$
where L is the inductance and C is the capacitance.

Key Concept: Phase Relationship

c) 90 degrees

[Solution Description]

In a purely capacitive AC circuit, the current leads the voltage by $$\frac{\pi}{2}$$ radians. Converting this into degrees:

$$\frac{\pi}{2} \times \frac{180}{\pi} = 90 \text{ degrees}$$

Hence, the current is 90 degrees ahead of the voltage.

Key Concept: Transformer Function

d) To change the AC voltage level.
[Solution Description]
A transformer’s main function in an AC circuit is to change the voltage level. It either steps up or steps down the input AC voltage by utilizing electromagnetic induction between its windings. This characteristic makes it essential for efficient power distribution, as stepping up voltage reduces currents in transmission lines, minimizing losses, while stepping down ensures safe utilization voltage at consumption points.

Key Concept: Inductive Reactance

a) Ohm
[Solution Description]
Inductive reactance is defined as $$X_L = ωL$$, where $$ω$$ is the angular frequency and L is the inductance. The unit of inductive reactance, being a type of resistance, is Ohm.

Key Concept: Complex Circuit Analysis, Power Calculation

d) 360 W

[Solution Description]

The impedance Z of the circuit can be calculated using:

$$Z = \sqrt{R^2 + (X_L – X_C)^2} \\
= \sqrt{20^2 + (30 – 10)^2} \\
= \sqrt{400 + 400} \\
= \sqrt{800} \\
= 28.28 \, \Omega$$

The RMS current I is given by:

$$I = \frac{V}{Z}$$

$$I = \frac{120}{28.28}$$

$$I = 4.24 \, \text{A}$$

The power factor $$\cos{\phi}$$ for a phase difference of $$45^\circ$$ is:

$$\cos{45^\circ} = \frac{1}{\sqrt{2}}$$

Therefore, the average power P consumed is:

$$P = V \cdot I \cdot \cos{\phi} = 120 \times 4.24 \times \frac{1}{\sqrt{2}} = 360 \, \text{W}$$

Key Concept: Real-World Application, Complex Impedance

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that the impedance is minimized at resonant frequency, which improves efficiency. In an LC circuit, at resonance, $$X_L = X_C$$, so their effects cancel each other. This leads to total impedance $$Z = \sqrt{R^2 + (X_L – X_C)^2}$$ simplifying to $$Z = R$$, which is minimal and purely resistive. Hence, both assertion and reason are true, and reason correctly explains why impedance is minimized at resonance.

Key Concept: Role of Components

c) Resonance requires both L and C components

[Solution Description]

Resonance occurs in a series RLC circuit when the inductive reactance and capacitive reactance are equal, i.e., $$X_L = X_C$$. This condition ensures that the reactive components cancel each other, allowing the circuit to resonate. Thus, both L and C must be present for resonance.

Key Concept: Resonance Frequency

b) $$\omega_0 = \frac{1}{\sqrt{LC}}$$

[Solution Description] The formula for the resonance frequency $$\omega_0$$ in a series LCR circuit is given by $$\omega_0 = \frac{1}{\sqrt{LC}}$$. Resonance occurs when the reactive components cancel each other out, resulting in maximum current amplitude at this frequency.

Key Concept: Real-World Application, Impedance Concept, Phase Difference Impact

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

[Solution Description]

The assertion states that in an AC circuit with a pure resistor, the power factor is unity. This is true because for purely resistive circuits, voltage and current are indeed in phase, meaning their phase difference is zero. The power factor, defined as $$\cos(\phi)$$ where $$\phi$$ is the phase difference between voltage and current, is 1 when $$\phi = 0$$. Hence, this makes the Assertion true.

The reason suggests that power dissipation in resistive AC circuits depends on the impedance angle affecting the in-phase alignment of voltage and current. For a pure resistor, the impedance is real, and there is no reactive component; thus, the impedance angle is zero, directly leading to in-phase voltage and current. However, the statement that power dissipation *depends* on the impedance angle is misleading in the context of resistive circuits alone because the angle is always zero, causing maximum power transfer without any phase shift consideration beyond ohmic behavior.

As both statements are separately correct but do not have a direct causal relationship in terms of explaining each other within a basic resistive scenario, Reason does not explain Assertion correctly.

Key Concept: Complex Circuit Analysis, Advanced Power Factor

c) Assertion is true, but Reason is false.

[Solution Description]

The assertion states that in an RLC circuit operating at its resonance frequency, the power factor becomes unity. This is true because, at resonance, the inductive and capacitive reactances are equal in magnitude but opposite in phase, canceling each other out. Hence, the circuit behaves as a pure resistor with no reactive component, making the power factor (cosine of the phase angle between voltage and current) equal to 1.

The reason claims that at resonance, the impedance of a series RLC circuit is purely inductive. This statement is false. At resonance, the impedance is actually purely resistive, not inductive, as the reactance components cancel each other out.

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

Key Concept: Simple Functionality

c) To maintain contact with slip rings
[Solution Description] Brushes are stationary components in an AC generator that maintain contact with slip rings. They help in transferring current from the rotating armature to the external circuit.