Energy stored in a capacitor - Physics (9702) - Cambridge A-Level

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A-Level Physics (9702) Study Notes: Energy Stored in a Capacitor (19.2)

Hello future Physicist! This chapter is all about understanding the power hidden inside a capacitor. We know capacitors store charge, but that charge comes with a punch: stored energy! This energy is what makes a camera flash work or a defibrillator save a life. Ready to dive in and learn how to calculate this crucial quantity?

1. The Concept of Stored Energy: Why Work is Done

When a battery charges a capacitor, it must move charge (usually electrons) from one plate to the other, creating an electric field between them. This process doesn't happen for free—it requires energy.

Analogy: Pushing Charge Uphill

Imagine pushing boxes up a ramp (the battery doing work). At the start, the ramp is flat (zero voltage), so the first few boxes are easy to push. But as you push more boxes up, the ramp gets steeper (the voltage increases), making it harder and harder to push the new boxes against the weight of the ones already there.

The battery (the power source) has to do work to force incoming charge onto the plate, pushing against the electrostatic repulsion of the charge already accumulated.
This work done by the battery is converted into electric potential energy, which is stored in the electric field between the plates.

Crucial Point: Because the potential difference ($V$) across the capacitor increases as more charge ($Q$) is stored, the force needed to move subsequent charges is *not* constant. This changing potential difference is why we need to use graphical methods or calculus to find the total work done.

Key Takeaway: Energy is stored because work must be done against the increasing repulsive force (voltage) of the existing charge.

2. Calculating Energy Using the Potential-Charge (V-Q) Graph

The syllabus requires you to determine the energy stored from the area under the potential-charge graph. This is the most fundamental way to understand the derivation.

a) The V-Q Relationship

We start with the definition of capacitance ($C$):
$$ C = \frac{Q}{V} $$ Rearranging this gives us the voltage across the capacitor at any point during charging: $$ V = \frac{1}{C} Q $$ Since capacitance $C$ is a constant value for a given capacitor, the relationship between $V$ and $Q$ is linear.

b) Graphical Determination of Work Done (Energy)

Recall from AS level that Work Done (W) is generally calculated by integrating force over distance, or in electrical terms, voltage over charge:

$$ \Delta W = V \Delta Q $$

For a graph where $V$ is plotted on the y-axis and $Q$ is plotted on the x-axis, the work done (or energy stored) is represented by the area under the graph.

When the capacitor is fully charged to a total charge $Q$ and a maximum potential difference $V$, the graph forms a right-angled triangle.

The area of a triangle is given by:

$$ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} $$

Substituting the graph variables:

$$ \text{Energy Stored (W)} = \frac{1}{2} \times Q \times V $$

This is your first key formula for stored energy.

Quick Review: The slope of the V-Q graph is $1/C$. The area under the V-Q graph is the stored energy, $W = \frac{1}{2}QV$.

3. The Three Energy Formulas (and How to Use Them)

The formula $ W = \frac{1}{2} Q V $ is useful, but often you are given $C$ or need to calculate energy without knowing the final charge $Q$. We can derive two other equivalent forms using the capacitance definition $ Q = C V $.

Formula 1: Energy in terms of Q and V

$$ W = \frac{1}{2} Q V $$

Units: Energy $W$ in Joules (J), Charge $Q$ in Coulombs (C), Voltage $V$ in Volts (V).

Formula 2: Energy in terms of C and V (Most commonly used)

To get rid of $Q$, substitute $ Q = C V $ into Formula 1:

$$ W = \frac{1}{2} (C V) V $$ $$ W = \frac{1}{2} C V^2 $$

Units: Capacitance $C$ in Farads (F), Voltage $V$ in Volts (V), Energy $W$ in Joules (J).

Memory Aid: This formula looks like Kinetic Energy $E_k = \frac{1}{2}mv^2$. Just replace mass $m$ with capacitance $C$. Easy to remember!

Formula 3: Energy in terms of Q and C

To get rid of $V$, substitute $ V = Q/C $ into Formula 1:

$$ W = \frac{1}{2} Q \left(\frac{Q}{C}\right) $$ $$ W = \frac{Q^2}{2C} $$

Units: Charge $Q$ in Coulombs (C), Capacitance $C$ in Farads (F), Energy $W$ in Joules (J).

Note for Struggling Students: You only must recall and use $ W = \frac{1}{2}QV $ and $ W = \frac{1}{2}CV^2 $, as these are the ones explicitly listed in the syllabus. However, understanding all three gives you flexibility in problem-solving!

4. Application and Real-World Connection

The energy stored in a capacitor is potential energy. When the capacitor is discharged, this stored potential energy is rapidly converted into other forms, usually heat, light, or kinetic energy.

Did You Know? The Defibrillator

A medical defibrillator works by charging a large capacitor to a high voltage (often several kilovolts) and then rapidly discharging this stored energy (around 100 to 360 J) through the patient's chest. This rapid discharge requires the capacity to store a significant amount of energy, calculated using $ W = \frac{1}{2} C V^2 $.

If a defibrillator uses a 100 µF capacitor charged to 2000 V:
$ C = 100 \times 10^{-6} \text{ F} $
$ V = 2000 \text{ V} $
$ W = \frac{1}{2} (100 \times 10^{-6}) (2000)^2 $
$ W = \frac{1}{2} (100 \times 10^{-6}) (4,000,000) $
$ W = 200 \text{ J} $

Step-by-Step Problem Solving Guide

Identify Given Variables: Determine which two of $Q$, $V$, or $C$ you are given.
Choose the Best Formula: Select the formula that uses your given variables directly.
- Given $C$ and $V$? Use $ W = \frac{1}{2} C V^2 $.
- Given $Q$ and $V$? Use $ W = \frac{1}{2} Q V $.
- Given $Q$ and $C$? Use $ W = \frac{Q^2}{2C} $.
Convert Units: Ensure all units are SI base units (Farads, Coulombs, Volts) before calculating. Remember micro ($\mu$) means $10^{-6}$.

Key Takeaway: The three formulas for energy are mathematically identical. Choose the one that minimizes calculation steps based on the information provided in the question.

5. Common Mistakes and Quick Review

Avoid These Mistakes!

Forgetting the Half: This is the most common error. Students often use $ W = Q V $ or $ W = C V^2 $. Remember, the "half" is necessary because the voltage is building up from zero during charging. (Think back to the triangular area on the V-Q graph!)
Confusing Capacitance Formulas: Do not mix up the energy formula $ W = \frac{1}{2}CV^2 $ with the charge formula $ Q = C V $. The charge formula is linear, the energy formula is quadratic (due to the factor of $\frac{1}{2}$ and $V^2$).
Unit Errors: Always check your prefixes! If you use microfarads ($\mu\text{F}$) without converting to Farads ($\text{F}$), your energy answer will be wrong by a factor of a million.

Quick Review Box: Energy Stored (19.2)

Definition: The work done by the source to transfer charge between the capacitor plates, stored as electric potential energy.

Graphical Method: The area under the Potential-Charge (V-Q) graph.

Key Formulas: (You must know these two)
1. $ W = \frac{1}{2} Q V $
2. $ W = \frac{1}{2} C V^2 $

Derivation Principle: The factor of $\frac{1}{2}$ accounts for the fact that the charging potential difference starts at zero and only reaches its maximum value $V$ at the end.