Alternating Current

The quality factor of an LCR circuit is given by: When the resistance is decreased by 50% and the inductance is increased by 100%, the new quality factor will change as follows: Since resistance decreases by 50%, the new resistance . The inductance is increased by 100%, so the new inductance . Thus, the new quality factor is:

Step 1: Analyzing the Condition:
Given , which means . The net reactance is positive. This implies the circuit is Inductive.
Step 2: Evaluating Options:

• (C) The circuit is predominantly inductive: True, since .
• (D) Current in the circuit lags source voltage: True. In an inductive circuit, voltage leads current (or current lags voltage).
• (A) Phase angle is negative: The phase angle is usually given by . Since , is positive in the impedance triangle convention. However, if the phase of current is expressed as relative to voltage, it is often referred to as a "lagging" phase angle. Whether this specific statement is strictly "true" or "false" depends on sign convention, but it describes the lag.
• (B) Power factor is zero: The power factor is . In an LCR series circuit, a resistor is present (unless specified otherwise). Therefore, and , so the power factor cannot be zero. A power factor of zero implies a purely reactive circuit (pure inductor or capacitor), which contradicts the "LCR series circuit" description. Thus, this statement is definitely incorrect.

Step 3: Conclusion:
Since the question asks for the incorrect option, (B) is the answer.

Step 1: Understand the cause of eddy current losses.
Eddy currents are loops of electric current induced within conductors by a changing magnetic field, according to Faraday's Law of Induction.
In a transformer, the alternating magnetic flux induces these currents in the iron core. These currents circulate through the core material, dissipating energy as heat ( loss), which is an unwanted energy loss.

Step 2: Analyze the function of a laminated core.
A laminated core is constructed from thin sheets (laminations) of soft iron, insulated from each other by a thin layer of varnish or oxide.
The laminations are arranged parallel to the magnetic flux. The insulation between them dramatically increases the resistance in the path of the circulating eddy currents.
Since the power loss due to eddy currents is proportional to , increasing the resistance ( ) significantly reduces the magnitude of the induced eddy currents ( ) and thus reduces the power loss.

Step 3: Evaluate the options.
(A) Winding the coils one over the other improves magnetic coupling but does not affect eddy currents.
(B) Using thick wire reduces copper losses (Joule heating in the windings) but does not affect the core's eddy current losses.
(C) Using a laminated core breaks up the path of the eddy currents, greatly increasing resistance and reducing the current magnitude, thereby effectively reducing eddy current losses. This is the correct method.
(D) Using magnetic material with low hysteresis loss reduces hysteresis losses, not eddy current losses.

In a series RC circuit, the phase difference ( ) between the voltage and current is given by:
.
Where is the capacitive reactance and is the resistance.
We are given:
Resistance, .
Phase difference, .
We can find the capacitive reactance :
.
.
.
The capacitive reactance is also given by the formula , where is the angular frequency and is the capacitance.
The voltage source is given by . Comparing this with the standard form , we can identify the angular frequency:
rad/s.
Now we can solve for the capacitance :
.
F.
F = F.
To express this in microfarads ( F), we use F.
F = 50 F.

Step 1: Understanding the Concept:
The brightness of the bulb depends on the current flowing through it. The circuit is an AC series circuit containing a resistor (the bulb) and an inductor. The inductor offers opposition to the flow of alternating current, known as inductive reactance. This reactance depends on the inductor's self-inductance.
Step 2: Analyzing the Circuit and the Effect of the Iron Rod:
1. Initial State: The circuit consists of an AC source, a bulb (resistance ), and an air-cored inductor (inductance ). The total opposition to the current, called impedance ( ), is given by , where is the inductive reactance ( is the angular frequency of the AC source). The current in the circuit is .
2. Inserting the Iron Rod: An iron rod is a ferromagnetic material with a high relative magnetic permeability ( ). The self-inductance of a coil is directly proportional to the permeability of its core material ( ). When the iron rod is inserted into the inductor, its self-inductance increases significantly: .
3. Consequences:
- The inductance increases.
- The inductive reactance increases.
- The total impedance of the circuit increases.
- According to Ohm's law for AC circuits ( ), if the impedance increases while the source voltage remains constant, the current in the circuit must decrease.
- The power dissipated by the bulb, which determines its glow, is . Since the current decreases, the power dissipated also decreases.
Step 3: Final Answer:
As the current through the bulb decreases, its glow diminishes. Therefore, the glow of the bulb decreases. Option (C) is correct.

• At resonance, the inductive reactance = capacitive reactance , so across R is , and current .
• Voltage across capacitor: .
• Series resonance voltage across L or C can exceed source voltage (series resonance phenomenon).
• Using .
• After calculating and plugging in, V.
• Hence, voltage across capacitor = 88 V.

1. In a pure inductive circuit, the current lags the voltage by 90 degrees, and the opposition to current is given by the inductive reactance .
2. The inductive reactance is calculated as , where is the frequency and is the inductance.
3. Given Hz and mH = 0.14 H, substitute: .
4. The RMS current in the circuit is , where V is the RMS voltage.
5. Calculate A. The value is exactly 3 A when using .
6. Therefore, the correct option is (3) 3 A.