Energy Stored in Inductors

Overview of Inductor Energy

Inductors store energy in their magnetic fields when current flows through them. This energy can be released when the current changes, making inductors important components in energy storage and transfer systems. Understanding how energy is stored and released in inductors is crucial for analyzing circuits and electromagnetic devices.

The energy stored in an inductor is proportional to the square of the current and the inductance. This relationship is fundamental to understanding power electronics, energy conversion, and electromagnetic applications.

Energy Storage Formula

Energy Stored in Inductor

The energy stored in an inductor is:

\[ U_L = \frac{1}{2}LI^2 \]

Where:

\(U_L\) is the energy stored in joules (J)
\(L\) is the inductance in henrys (H)
\(I\) is the current in amperes (A)

Note: This formula is on the AP Physics C equation sheet.

Key Concepts

Energy Transfer

When current changes in an inductor, energy is transferred:

Increasing current: energy stored in magnetic field
Decreasing current: energy released from magnetic field
Constant current: no energy transfer
Energy flows between inductor and circuit

Energy Conservation

In an ideal inductor:

Energy stored = \(\frac{1}{2}LI^2\)
Energy released = \(\frac{1}{2}LI^2\)
No energy loss in storage
Energy can be recovered

Real inductors have resistance, causing energy loss as heat.

Physical Meaning

Energy is stored in the magnetic field created by the current. The field contains energy that can be released when the current changes.

Magnetic field stores energy
Energy proportional to I²
Energy proportional to L
Energy can be transferred

Formula Derivations

Derivation 1: Energy Storage Formula

Derive: \(U_L = \frac{1}{2}LI^2\)

Step-by-Step Derivation:

Power in inductor: \(P = \mathcal{E}I = -L\frac{dI}{dt}I\)
Energy stored: \(U = \int P \, dt = \int -LI\frac{dI}{dt} \, dt\)
Substitute: \(U = \int -LI \, dI\)
Integrate: \(U = -\frac{1}{2}LI^2 + C\)
At I = 0, U = 0, so C = 0
Therefore: \(U_L = \frac{1}{2}LI^2\)

Result: \(U_L = \frac{1}{2}LI^2\)

Example Problems

Example 1: Calculating Stored Energy

Problem: A 0.5 H inductor carries a current of 3 A. How much energy is stored?

Solution:

Use energy formula: \(U_L = \frac{1}{2}LI^2\)
Substitute values: \(U_L = \frac{1}{2}(0.5)(3)^2\)
Calculate: \(U_L = \frac{1}{2}(0.5)(9) = 2.25 \text{ J}\)

Answer: The stored energy is 2.25 J.

Applications of Inductor Energy

Energy Storage: Inductors store energy in magnetic fields
Power Supplies: Smooth current variations and store energy
Transformers: Transfer energy between circuits
LC Circuits: Energy oscillates between inductor and capacitor
Filters: Store and release energy in electronic filters
Circuit Protection: Absorb energy spikes and protect components

Quick Quiz: Inductor Energy

1. The energy stored in an inductor is proportional to:

The current

The square of the current

The inductance

The voltage

2. When current decreases in an inductor:

Energy is stored

Energy is released

No energy change

Energy is lost

4. What is the energy stored in a 1 H inductor with 2 A current?

1 J

2 J

4 J

8 J

3. When current is constant in an inductor:

Energy is stored

Energy is released

No energy transfer

Energy is lost

Learning Objectives

Calculate Stored Energy: Use \(U_L = \frac{1}{2}LI^2\) to find energy
Understand Energy Transfer: Know when energy is stored or released
Energy Conservation: Apply energy conservation in circuits
Real Applications: Connect theory to practical energy storage

Key Takeaways

Energy Storage: Inductors store energy in magnetic fields
Quadratic Dependence: Energy proportional to I²
Energy Transfer: Energy stored when current increases, released when current decreases
Applications: Essential in circuits and energy systems