Problem 16 - University Physics with Modern Physics

Question

An inductor used in a dc power supply has an inductance of 12.0 \(\mathrm{H}\) and a resistance of 180\(\Omega .\) It carries a current of 0.300 \(\mathrm{A}\) . (a) What is the energy stored in the magnetic field? (b) At what rate is thermal energy developed in the inductor? (c) Does your answer to part (b) mean that the magnetic-field energy is decreasing with time? Explain.

Step-by-Step Solution

Answer

(a) 0.54 J; (b) 16.2 W; (c) Magnetic-field energy is constant if current is constant.

Step 1: Energy Stored in the Magnetic Field

To find the energy stored in the magnetic field of the inductor, we use the formula: \( E = \frac{1}{2} L I^2 \), where \( E \) is the energy, \( L \) is the inductance, and \( I \) is the current. Substitute \( L = 12.0 \, \text{H} \) and \( I = 0.300 \, \text{A} \) into the formula:\[ E = \frac{1}{2} \times 12.0 \times (0.300)^2 \]Calculate the energy:\[ E = \frac{1}{2} \times 12.0 \times 0.09 = 0.54 \, \text{J} \] The energy stored in the magnetic field is 0.54 Joules.

Step 2: Rate of Thermal Energy Development

The rate at which thermal energy is developed in the inductor can be found using the power formula for resistive heating: \( P = I^2 R \), where \( P \) is the power, \( I \) is the current, and \( R \) is the resistance.Substitute \( I = 0.300 \, \text{A} \) and \( R = 180 \, \Omega \) into the formula:\[ P = (0.300)^2 \times 180 \]Calculate the power:\[ P = 0.09 \times 180 = 16.2 \, \text{W} \]The rate at which thermal energy is developed is 16.2 Watts.

Step 3: Analysis of Magnetic-Field Energy Decrease

The answer to part (b) does not imply that the magnetic-field energy is decreasing. The power calculated in part (b) represents the rate at which electrical energy is converted to thermal energy in the resistor. The energy of the magnetic field remains constant as long as the current is constant. If current doesn't change, the magnetic-field energy will not decrease, since energy loss to heat happens through the resistance.

Key Concepts

Inductance

Inductance is a fundamental characteristic of an inductor, often denoted by the symbol \( L \). It refers to the ability of the inductor to store energy in the form of a magnetic field when a current passes through it. The larger the inductance, the more energy the inductor can store for a given current. This concept is central to many applications in electronics, such as in power supplies, filters, and transformers. The formula for the energy stored in an inductor’s magnetic field is:

• \( E = \frac{1}{2} L I^2 \)

where:

• \( E \) is the energy in joules (J)
• \( L \) is the inductance in henries (H)
• \( I \) is the current in amperes (A)

This equation tells us how changes in the current affect the energy stored within the magnetic field of an inductor. The energy stored is directly proportional to the square of the current, highlighting the importance of managing current levels in inductive circuits.

Resistance

Resistance is an inherent property of materials to oppose the flow of electric current, measured in ohms (\( \Omega \)). In an inductor, resistance originates from the wire coil used to create the magnetic field. This resistance contributes to power losses in the form of heat, commonly known as resistive heating.The formula that describes the power loss due to resistance in inductors is:

• \( P = I^2 R \)

where:

• \( P \) is the power loss in watts (W)
• \( I \) is the current in amperes (A)
• \( R \) is the resistance in ohms (\( \Omega \))

This equation shows that power losses grow with the square of the current. Thus, in designing circuits, it’s critical to minimize resistance to enhance efficiency. For constant current, the power lost to resistance becomes an important factor in energy management.

Magnetic Field Energy

Magnetic field energy in an inductor is the energy stored within its magnetic field when an electric current flows through it. The calculation of magnetic energy is crucial for designing and analyzing circuits that use inductors, like transformers and motors.The energy stored can be calculated using the formula:

• \( E = \frac{1}{2} L I^2 \)

This formula indicates that the energy is dependent on both the inductance and the square of the current it carries. In practical applications, energy stored in the magnetic field can be released back into the circuit when needed, such as to maintain a current flow when the power supply is interrupted. This capability is especially valuable in power electronic applications, ensuring uninterrupted operation and efficient energy usage.

Thermal Energy Development

Thermal energy development in an inductor is the energy converted to heat due to the resistance of the wire coil. When current flows through an inductor’s resistive elements, some of the electrical energy is lost as heat.The rate of thermal energy development can be calculated using:

• \( P = I^2 R \)

This formula emphasizes the direct relationship between current flow and thermal loss. As current flows through the inductor, it encounters resistance, causing some of the electrical energy to convert to thermal energy (heat).Although thermal energy development is often seen as a loss, understanding this aspect is crucial for thermal management in electronic circuits. Properly designed systems will allow for heat dissipation to reduce the risk of overheating and ensure reliable operation of the components.