Problem 12 - University Physics with Modern Physics

Question

An inductor used in a de 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) No, the magnetic field energy is constant with steady current.

Step 1: Identify the Given Information

We have an inductor with an inductance of \( L = 12.0 \text{ H} \) and a resistance of \( R = 180 \Omega \). The current flowing through the inductor is \( I = 0.300 \text{ A} \). We need to calculate the energy stored in the magnetic field, the rate of thermal energy development in the resistor, and interpret the relation between these two energies.

Step 2: Calculate Magnetic Field Energy

The energy stored in the magnetic field of an inductor is given by the formula \( E_L = \frac{1}{2}LI^2 \). Here, substitute \( L = 12.0 \text{ H} \) and \( I = 0.300 \text{ A} \) to find \( E_L \):\[E_L = \frac{1}{2} \times 12.0 \times (0.300)^2\]\[E_L = \frac{1}{2} \times 12.0 \times 0.09 = 0.54 \text{ Joules}\]

Step 3: Calculate Rate of Thermal Energy Development

The rate of thermal energy (power) developed in an inductor is given by \( P = I^2R \). Substitute \( I = 0.300 \text{ A} \) and \(R = 180 \Omega \):\[P = (0.300)^2 \times 180\]\[P = 0.09 \times 180 = 16.2 \text{ Watts}\]

Step 4: Analyze Energy Changes

The calculation in Step 3 shows that thermal energy is being developed at a rate of \(16.2 \text{ watts}\), but this does not imply the magnetic field energy is decreasing. The magnetic field energy remains constant if the current is constant, while the thermal energy results from the resistive loss due to the inductor's resistance.

Key Concepts

Magnetic Field Energy

Inductors are fascinating components in electrical circuits, essential for storing energy in the form of a magnetic field. When current flows through an inductor, it creates a magnetic field around itself. The energy stored in this magnetic field can be calculated using the equation:

• \( E_L = \frac{1}{2}LI^2 \)

Where \( L \) is the inductance and \( I \) is the current flowing through the inductor. In our scenario, with an inductance of 12.0 Henrys and a current of 0.300 Amperes, the energy stored is 0.54 Joules. This energy is essentially "trapped" by the magnetic field created by the flowing current.
This energy plays a crucial role in many applications where magnetic fields are needed or beneficial, such as in transformers and motors. It can be released to perform work in the circuit, offering a temporary form of energy storage. Understanding how to calculate magnetic field energy helps in designing efficient circuits by predicting how much energy an inductor can temporarily store.

Thermal Energy

Thermal energy in a circuit occurs largely due to resistance. As current flows through a resistor— or through any resistive component, like an inductor with resistance—it generates heat. This resistive heating is often referred to as thermal energy loss.
The rate of this energy development, or the power of this heat, can be found using the formula:

• \( P = I^2R \)

In our example, the inductor has a resistance of 180 Ohms and carries a current of 0.300 Amperes. The power calculated is 16.2 Watts. This rate tells us how quickly the inductor's resistance dissipates energy as heat in the circuit.
While this heating is a necessary side-effect in circuits, it often needs managing to avoid damage or performance issues. Understanding thermal energy helps in designing circuits for better energy efficiency and safety, ensuring components can operate without overheating.

Inductance and Resistance

Inductance and resistance are vital properties that define how an inductor behaves in a circuit. Inductance represents an inductor's ability to store energy in a magnetic field when electrical current flows through it. It is a measure of the inductor's efficiency in creating this field.
Resistance, on the other hand, is the property that restricts the flow of current through the inductor, causing energy to be lost as heat. The combination of these two properties means that an inductor can store energy in its magnetic field, all while also experiencing some energy loss due to its inherent resistance.
In practical terms:

• High inductance levels enable more energy storage.
• High resistance levels increase thermal energy losses.

Designers and engineers must balance these properties when choosing inductors for specific applications, considering both the efficiency of energy storage and the potential energy loss through resistance.