Question

A 20 volts ac is applied to a circuit consisting of a resistance and a coil with negligible resistance. If the voltage across the resistance is \(12 \mathrm{~V}\), the voltage across the coil is, (a) 16 volts (b) 10 volts (c) 8 volts (d) 6 volts

Short answer

The voltage across the coil is 8 volts, which corresponds to option (c).

Step-by-step solution

Apply Kirchhoff's Voltage Law (KVL)

KVL states that the sum of the voltages around a closed loop in a circuit is zero. In this case, we have a simple series circuit with an AC voltage source, a resistor, and a coil. The sum of the voltages across the resistor (\(V_R\)) and the coil (\(V_C\)) must equal the voltage supplied by the AC source (\(V_S\)): \(V_S = V_R + V_C\) We are given \(V_S = 20V\) and \(V_R = 12V\). We can plug these values into the equation to solve for \(V_C\).

Solve for the Voltage Across the Coil (\(V_C\))

Using the equation from Step 1, we can now solve for the voltage across the coil: \(20V = 12V + V_C\) Subtracting 12V from both sides of the equation, we get: \(V_C = 20V - 12V = 8V\) So, the voltage across the coil is 8 volts.

Check for the Correct Answer

We can now look at the given options to see which one matches our calculated voltage across the coil: (a) 16 volts (b) 10 volts (c) 8 volts (d) 6 volts The correct answer is (c) 8 volts.

Key concepts

AC Circuit

An AC (Alternating Current) circuit consists of various electrical components powered by an AC voltage source. In AC circuits, the current changes direction periodically. Unlike DC (Direct Current) circuits, AC circuits have a sinusoidal current wave.
AC voltage and current have both magnitude and direction, meaning the flow alternates over time. This alternating behavior is crucial for power distribution as it allows efficient transmission over long distances.

• A typical AC circuit may include resistors, coils (inductors), and capacitors.
• AC circuits can be more complex due to the phase difference between voltage and current.
• Understanding the impedance in AC circuits is key, as it combines resistance, inductance, and capacitance effects.

AC circuits are widely used in homes and industries due to their flexibility and efficiency. When studying AC circuits, familiarity with terms like phase angle, frequency, and waveform is beneficial.

Series Circuit

A series circuit is a basic type of electrical circuit where components are connected end-to-end, forming a single loop for current to flow through each component sequentially.
In a series circuit, all components share the same current but have different voltage drops. This is because the entire current flows through each component one after another.

• Total resistance in a series circuit is the sum of individual resistances: \( R_{total} = R_1 + R_2 + R_3 + \ldots \)
• Voltage across the circuit is the sum of the voltages across each component due to Kirchhoff's Voltage Law.
• Failure of one component interrupts the entire circuit.

Series circuits are often seen in holiday lights or electrical strings, where if one bulb goes out, the whole string stops working.

Voltage Across Resistance

The voltage across a resistor in an electric circuit is established by Ohm's Law, which states that the voltage (\( V_R \)) across the resistance is the product of the current (\( I \)) flowing through it and its resistance (\( R \)): \( V_R = I \times R \).
In an AC series circuit, the resistor has a voltage drop that directly affects the overall circuit voltage. Resistors are passive elements, meaning they only consume power. Their main role is to limit current and divide voltage within the circuit.

• Resistance is measured in ohms (\( \Omega \)).
• This voltage drop is always in phase with the current.
• For practical purposes, resistors determine part of a circuit's total impedance.

Understanding how to calculate the voltage across resistance is essential in AC circuit analysis and effectively utilizing Kirchhoff’s Voltage Law.

Voltage Across Coil

The voltage across a coil, also known as an inductor, in an AC circuit, is influenced by its inductive reactance. Inductors are components that store energy in a magnetic field when current flows through them.
The voltage (\( V_C \)) across a coil is determined by the rate of change of current through it. In a series circuit, use Kirchhoff's law to find the voltage across a coil by subtracting the known resistor voltage from the total voltage.

• Inductive reactance (\( X_L \)) is given by: \( X_L = 2\pi f L \), where \( f \) is the frequency and \( L \) is the inductance.
• Voltage across the coil leads the current by 90 degrees due to phase difference.
• Inductors oppose changes in current, which contributes to the characteristic behavior of AC circuits.

Understanding the voltage across the coil allows for analyzing how inductors affect circuit dynamics, crucial in applications like transformers and radio frequency circuits.