Energy Storage in Capacitors and Inductors

Hello!

In our previous lessons, we established the fundamental behaviors of capacitors and inductors. We saw that a capacitor stores energy in an electric field to oppose changes in voltage ($i=C\frac{dv}{dt}$), while an inductor stores energy in a magnetic field to oppose changes in current ($v=L\frac{di}{dt}$).

Today, we will quantify exactly how much energy these components store. This lesson directly addresses the learning outcome: Calculate the energy stored in a capacitor and an inductor. In line with your preference for building deep physical intuition, we won't just learn the formulas. We'll derive them from first principles, focusing on the concepts of work and power to understand why the formulas take the shape they do.

1. Energy Stored in a Capacitor: The Work of Building a Field

To store energy in a capacitor, a power source must do work to move charge from one plate to the other against the repulsive force of the growing electric field. Think of it as compressing an "electric spring." The total energy stored is equal to the total work done in this process.

A common first guess for the stored energy is simply the final charge $Q$ multiplied by the final voltage $V$. However, this is incorrect. Let's watch a video that builds the intuition for why this is wrong and then derives the correct formula.

As the video masterfully explains, because the voltage across the capacitor increases from 0 to its final value $V$ during the charging process, the average voltage is only $\frac{1}{2}V$. This leads to the correct formulas for the energy $U_C$ stored in a capacitor:

You can use whichever form is most convenient depending on whether you know the voltage, charge, or capacitance.

A Deeper Look: The Case of the "Missing" Energy

Your interest in energy flow brings up a fascinating puzzle. If a battery with a constant voltage $V$ charges an empty capacitor, the total work done by the battery is $W_{batt}=QV$. But we just found that the capacitor only stores $U_C=\frac{1}{2}QV$. Where did the other half of the energy go?

This is a profound insight: charging a capacitor from a constant voltage source is an inherently lossy process. Exactly 50% of the energy drawn from the source is converted to heat in the circuit's resistance. Even in an "ideal" circuit with zero resistance, that energy would be lost, radiated away as an electromagnetic wave.

2. Energy Stored in an Inductor: The Energy of a Moving Current

In the last lesson, we developed an analogy of an inductor having "electrical inertia." Now, we'll see how this connects to energy. Just as a moving mass has kinetic energy ($\frac{1}{2}m{v}^2$), a flowing current in an inductor has magnetic energy. To get the current flowing, the source must do work to build the magnetic field. That work is then stored as energy in the field.

The analogy to kinetic energy is striking and provides a strong conceptual hook:

• Inductance ($L$) is analogous to mass ($m$), representing inertia.
• Current ($i$) is analogous to velocity ($v$), representing motion.
• Magnetic Energy ($\frac{1}{2}L{I}^2$) is analogous to Kinetic Energy ($\frac{1}{2}m{v}^2$).

The derivation confirms this intuition. By integrating the power ($P=vi$) delivered to the inductor over time, we find that the total energy $U_L$ stored in the magnetic field is:

Unlike the capacitor, there is only one common form for this formula.

3. Practice: Tying It All Together

Let's solve a problem that requires you to apply these new formulas. This circuit problem also serves as a review, as you'll first need to determine the steady-state conditions of the circuit—a concept from our previous lessons.

Consider the following circuit, which has been connected for a long time. Your task is to calculate the energy stored in the capacitor and the inductor.

Your approach should be:

1. Analyze the circuit in DC steady-state. How does an inductor behave after a long time in a DC circuit? How does a capacitor behave? Redraw the circuit based on this.
2. Solve for the key quantities. Using your simplified circuit, find the voltage across the capacitor ($V_C$) and the current through the inductor ($I_L$).
3. Calculate the stored energy. Apply the formulas $U_C=\frac{1}{2}C{V}^2$ and $U_L=\frac{1}{2}L{I}^2$.

Give it a try! Once you have your answers, you can check your work against the solution below.

Conclusion

In this lesson, we quantified the energy storage capabilities of capacitors and inductors, deriving their respective formulas from the physical concepts of work and power.

Key Takeaways:

• Energy is stored in a capacitor's electric field. The amount is given by $U_C=\frac{1}{2}C{V}^2$.
• Energy is stored in an inductor's magnetic field. The amount is given by $U_L=\frac{1}{2}L{I}^2$.
• The factor of $\frac{1}{2}$ in the capacitor formula arises because the voltage increases as it is charged.
• The inductor energy formula is conceptually analogous to the formula for kinetic energy, with inductance acting as inertia and current as motion.
• In DC steady-state, you can calculate stored energy by first simplifying the circuit (capacitors become opens, inductors become shorts) to find the relevant voltages and currents.

Preview of the Next Lesson:
Now that we understand these components individually, our next step is to analyze how they behave when combined. We will derive the formulas for equivalent capacitance and inductance for series and parallel combinations. You might be surprised to find that the rules are "swapped" compared to resistors—parallel capacitors add, while series inductors add. We will explore the physical reasoning for this difference.