Show that energy stored in an inductor L, when a current I is established through it is 1/2LI2.

To demonstrate that the energy stored in an inductor $L$ when a current $I$ flows through it is given by the formula $\frac{1}{2}L{I}^2$, we can break this down step by step, using some fundamental concepts from electromagnetism and circuit theory. Let’s walk through the reasoning together.

The Basics of Inductance

An inductor is a passive electrical component that stores energy in a magnetic field when an electric current passes through it. The key characteristic of an inductor is its inductance $L$, which is measured in henries (H). Inductance quantifies how effectively an inductor can store energy.

The Formula for Inductance

The voltage $V$ across an inductor is related to the change in current $I$ through it by the equation:

• V = L \frac{dI}{dt}

Here, $\frac{dI}{dt}$ represents the rate of change of current with respect to time. If we want to find the energy stored in the inductor, we need to consider how much work is done to establish the current $I$ in the inductor from rest (when $I=0$).

Calculating the Work Done

When a current increases through the inductor, the work $W$ done to increase the current is equal to the integral of the voltage over time. Since the voltage across the inductor changes as the current changes, we can express the work done in terms of the current:

• W = ∫ V \, dt = ∫ L \frac{dI}{dt} \, dt

Rearranging gives us:

• W = ∫ L \, dI

Now, if the current increases from $0$ to $I$, the work done can be expressed as:

• W = L \int_0^I dI

Carrying out this integral, we have:

• W = L \left[ \frac{I^2}{2} \right]_0^I = L \cdot \frac{I^2}{2} - L \cdot \frac{0^2}{2} = \frac{1}{2}LI^2

Understanding Energy Storage

The result $W=\frac{1}{2}L{I}^2$ tells us that the energy $W$ stored in the inductor when a current $I$ is established is half of the product of the inductance and the square of the current. This is crucial because it shows that the energy stored increases with the square of the current, meaning that even small increases in current can lead to significant increases in stored energy.

Real-World Analogy

Think of an inductor like a water tank. The amount of water in the tank represents the energy stored in the inductor. The current flowing into the tank can be likened to the flow of water being pumped in. As you increase the flow rate (current), the water level (energy) rises, but the relationship isn't linear. If you double the flow, the water level increases by a factor of four—illustrating the quadratic relationship of energy storage.

Final Thoughts

In summary, the formula $\frac{1}{2}L{I}^2$ encapsulates how inductors store energy based on their inductance and the current flowing through them. This relationship is fundamental in circuit design, especially in applications involving energy storage, filtering, and in various electronic devices. Understanding this concept is essential for grasping more complex electrical engineering principles.