Question

The value of alternating emf \(E\) in the given ckt will be. (a) \(100 \mathrm{~V}\) (b) \(20 \mathrm{~V}\) (c) \(220 \mathrm{~V}\) (d) \(140 \mathrm{~V}\)

Short answer

Since no specific information about the circuit is provided, we cannot determine the exact value of the alternating emf "E". However, once the details of the circuit are given, following the steps outlined above will help to find the correct answer from the given options.

Step-by-step solution

Understand the circuit components and parameters

First, we need to gather information about the different components of the circuit and their parameters (such as resistances, inductances, capacitances, etc.) from the given problem statement. Since no information is provided about the circuit, we'll have to assume that there is a series RL or RC circuit with known values for the resistance, inductor, and capacitor.

Determine the phasor representation of the voltage and current

Next, we need to represent the voltage and current phasor quantities using complex numbers. For a series RL or RC circuit, we can use the impedance (Z) for relating voltage and current. Impedance can be found by the formula: For RL circuit: \(Z = R + j{\omega}{L}\) For RC circuit: \(Z = R - \frac{j}{\omega}{C}\) Where R is the resistance, L is the inductance, C is the capacitance, j is the imaginary unit, and ω is the angular frequency of the alternating current.

Find the total voltage across the circuit

To find the total voltage across an RL or RC circuit, we can use Ohm's law: \(V = I \times Z\) where V is the total voltage, I is the current, and Z is the impedance.

Calculate the magnitude of the total voltage

To find the alternating emf "E," we need to calculate the magnitude of the total voltage. We can do this using the following equation for a complex number: \( \mathrm{Magnitude~of~Voltage} = |V| = \sqrt{\mathrm{Real~Part}^2 + \mathrm{Imaginary~Part}^2}\) Where Real Part and Imaginary Part are the real and imaginary components of the voltage complex number, respectively.

Compare the calculated magnitude to the given options

Finally, compare the calculated magnitude of the total voltage (|V|) to the four given options. The value that matches the calculated result will be the correct answer for the given problem. Without any specific information about the circuit, we can't provide a step-by-step solution for this problem. However, these steps can be followed once the necessary details are provided to arrive at the correct answer.

Key concepts

Impedance

Impedance plays a crucial role in alternating current (AC) circuits. It combines resistance, inductive reactance, and capacitive reactance into a single complex quantity. Impedance is used to describe how much a circuit resists the flow of electric current. It is symbolized as \( Z \) and is measured in ohms (\( \Omega \)).

When you encounter a circuit element, it might not just resist the flow of current, but react to it. This behavior depends on the element’s type:

• Resistive (R): Simply resists current. Impedance is equivalent to resistance \( Z = R \).
• Inductive (L): Reacts to current changes by creating a magnetic field, expressed as \( Z = j \omega L \), where \( j \) is the imaginary unit and \( \omega \) is the angular frequency.
• Capacitive (C): Stores and releases electrical energy, expressed as \( Z = -\frac{j}{\omega C} \).

By combining these into one formula, impedance helps us understand complex circuit behavior in AC circuits.

Phasor Representation

In alternating current circuits, phasors are a way to simplify the analysis of AC signals by transforming them into complex numbers that make calculations easier. Phasors represent sinusoidal voltages and currents by extending them into the complex plane, focusing on amplitude and phase rather than time.

A phasor lets us take a voltage or current waveform, typically expressed as \( V(t) = V_m \cos(\omega t + \phi) \), and represent it as \( V_m \angle \phi \), where:

• \( V_m \) is the maximum value (magnitude).
• \( \omega \) is the angular frequency.
• \( \phi \) is the phase angle.

This representation simplifies the calculation of circuit properties using algebraic methods, especially when combined with impedance to determine the overall circuit behavior. It essentially translates real-world oscillations into figures that are easier to manipulate mathematically.

Ohm's Law

Ohm's Law is fundamental in both direct and alternating current circuits. It determines the relationship between voltage, current, and impedance in a circuit. In an AC circuit, Ohm's Law takes the form of\( V = I \times Z \):

• \( V \) is the voltage across the circuit.
• \( I \) is the current flowing through the circuit.
• \( Z \) is the circuit's impedance.

Ohm’s Law for AC emphasizes that voltage across an element is proportional to the product of the current through it and the impedance. The formula becomes especially handy when dealing with phasors, allowing us to express voltage and current as sleek complex numbers for easier manipulation.

Using this law is crucial for solving AC circuit problems as it allows one to find unknown quantities once two others are known, making it a powerful tool in electrical engineering.

RL and RC Circuits

RL and RC circuits each combine different types of components and play unique roles in AC applications. They are widely used in various electronic and electrical applications.

**RL Circuits:**

• Combines resistors and inductors (coils).
• Inductor creates a magnetic field when current flows through it, making current lag behind voltage.
• This lag can be quantified by the expression \( \theta = \tan^{-1}\left(\frac{\omega L}{R}\right) \).

**RC Circuits:**

• Consists of resistors and capacitors.
• Capacitor stores electrical energy when voltage is applied, leading to voltage lagging behind current.
• The phase difference is described by \( \theta = \tan^{-1}\left(-\frac{1}{\omega RC}\right) \).

Both RL and RC circuits use impedance to describe their behavior. The presence of inductors and capacitors introduces reactance, impacting how current builds and decays in response to changes in voltage. This characteristic is essential for the design of filters, oscillators, and many other types of electronic equipment.