Question

In Solved Problem 30.3 , the voltage supplied by the source of timevarying emf is \(33.0 \mathrm{~V}\), the voltage across the resistor is \(V_{R}=I R=13.1 \mathrm{~V}\), and the voltage across the inductor is \(V_{L}=I X_{L}=30.3 \mathrm{~V}\). Does this circuit obey Kirchhoff's rules?

Short answer

Answer: No, the circuit does not obey Kirchhoff's voltage law because the sum of the voltage drops across the resistor and the inductor (43.4 V) is not equal to the voltage supplied by the source (33.0 V).

Step-by-step solution

Identify the given values

In the given problem, we are provided with the following values: - Voltage supplied by the source of time-varying emf: \(33.0 \mathrm{~V}\) - Voltage across the resistor: \(V_{R} = 13.1 \mathrm{~V}\) - Voltage across the inductor: \(V_{L} = 30.3 \mathrm{~V}\)

Apply Kirchhoff's Voltage Law

According to Kirchhoff's voltage law, in a closed loop, the sum of all voltage sources must be equal to the sum of voltage drops across all elements. To verify this equation, add the voltage drops across the resistor and the inductor and equate it with the given voltage supplied. \(V_{source} = V_{R} + V_{L}\)

Calculate and compare the voltage sums

Substitute the given values into the equation: \(33.0 \mathrm{~V} = 13.1 \mathrm{~V} + 30.3 \mathrm{~V}\) Now, sum the voltage drops across the resistor and the inductor: \(13.1 \mathrm{~V} + 30.3 \mathrm{~V} = 43.4 \mathrm{~V}\) Comparing the two values, we find that: \(33.0 \mathrm{~V} \neq 43.4 \mathrm{~V}\) As the sum of the voltage drops across the resistor and the inductor is not equal to the voltage supplied by the source, this circuit does not obey Kirchhoff's voltage law.

Key concepts

Kirchhoff's rules

Kirchhoff's rules are fundamental in understanding the flow of electric current through a circuit. These rules are comprised of two main laws: Kirchhoff's Current Law (KCL) and Kirchhoff's Voltage Law (KVL). KCL, also known as the junction rule, states that the total current entering a junction must equal the total current leaving the junction. This is based on the principle of conservation of charge.

KVL, on the other hand, focuses on the conservation of energy within a closed loop of a circuit. It asserts that the total sum of all electromotive forces (EMFs) and potential differences (voltages) around any closed loop is zero. In essence, it tells us that the energy supplied by the sources (like batteries or generators) is exactly balanced by the energy 'used up' or dropped across the various circuit components like resistors, capacitors, and inductors. If a circuit involves time-varying EMFs, dealing with KVL can be more complex, and additional factors such as inductive reactance and capacitive reactance may need to be considered.

Voltage across resistor

When current flows through a resistor in an electrical circuit, there is a voltage drop across that resistor. This is due to the resistance, which impedes the flow of current, causing energy to be dissipated in the form of heat. The relationship between the voltage across the resistor (\( V_R \text{)}), the current flowing through it (\( I \text{)}), and its resistance (\( R \text{)}), is described by Ohm's law as:\[ V_R = I \times R \]

This equation signifies that the voltage drop across a resistor is directly proportional to the current passing through it and the resistance of the resistor itself. For a fixed resistance, a higher current will result in a greater voltage drop and vice versa. In the context of Kirchhoff's rules, this voltage drop must be included when applying KVL to determine the total voltage in a circuit loop.

Voltage across inductor

Inductors are components in a circuit that oppose changes in the current passing through them, primarily due to the magnetic field created around their coiled wire. The voltage across an inductor (\( V_L \text{)}) is proportional to the rate of change of the current (\( \frac{dI}{dt} \text{)}) and the inductance (\( L \text{)}), which is expressed by the following equation:\[ V_L = L \times \frac{dI}{dt} \]

This phenomenon is also known as inductive reactance and it introduces a phase difference between the current and voltage waveforms in an AC circuit. For direct current (DC) circuits, after an initial transient period, the inductor will simply oppose any change in current, but once steady-state is reached, the voltage drop across an ideal inductor is zero. Understanding how an inductor's behavior affects the voltage within a loop is crucial when applying Kirchhoff's Voltage Law in circuits containing inductors.

Time-varying emf

Electromotive force (emf) is the energy per unit charge provided by a source, such as a battery or generator, which is needed to drive the charge around a complete circuit. When an emf changes over time, we refer to it as time-varying emf. This can happen in circuits with components like transformers or when the source itself is time-varying, such as in AC circuits.

A time-varying emf is characterized by its waveform, which could be sinusoidal, square, triangular, or another shape. In calculations involving Kirchhoff's Voltage Law, time-varying emfs require careful consideration because they can introduce capacitive or inductive elements into the circuit that affect the total voltage at any point in time. This makes the analysis of circuits more complex, as it's necessary to consider factors such as frequency, reactance, and phase shifts when summing voltages around a loop.