Problem 60 - College Physics

Question

In the text, it was shown that the energy stored in a capacitor \(C\) charged to a potential \(V\) is \(U=\frac{1}{2} Q V\) . Show that this energy can also be expressed as (a) \(U=Q^{2} / 2 C\) and (b) \(U=\frac{1}{2} C V^{2}\)

Step-by-Step Solution

Answer

The energy can be expressed as \( U = \frac{Q^2}{2C} \) and \( U = \frac{1}{2} C V^2 \).

Step 1: Understanding the Given Expression

We start from the given expression for the energy stored in a capacitor, which is \( U = \frac{1}{2} Q V \). Here, \( Q \) is the charge on the capacitor, and \( V \) is the potential across the capacitor.

Step 2: Using the Relation for Capacitance and Charge

The capacitance \( C \) is defined as \( C = \frac{Q}{V} \). Rearranging this formula gives \( Q = C V \). We'll use this in our calculations to derive the new expressions for energy.

Step 3: Deriving Expression for \( U = \frac{Q^2}{2C} \)

We begin with the expression for \( Q \) in terms of \( C \) and \( V \): \( Q = C V \). Substitute \( V = \frac{Q}{C} \) into the original energy expression \( U = \frac{1}{2} Q V \). This gives \( U = \frac{1}{2} Q \left( \frac{Q}{C} \right) = \frac{Q^2}{2C} \).

Step 4: Deriving Expression for \( U = \frac{1}{2} C V^2 \)

Substituting \( Q = C V \) into the original formula \( U = \frac{1}{2} Q V \), we have \( U = \frac{1}{2} (C V) V \). Simplify this to get \( U = \frac{1}{2} C V^2 \).

Step 5: Verification of Both Forms

Both derived expressions, \( U = \frac{Q^2}{2C} \) and \( U = \frac{1}{2} C V^2 \), are consistent with the original form \( U = \frac{1}{2} Q V \) when using the relation \( C = \frac{Q}{V} \). This confirms that the energy stored in a capacitor can be represented in these forms.

Key Concepts

energy storage

Capacitors are fascinating components in the world of electronics because they can store energy. Think of them like a small reservoir of electrical charge. When a capacitor is connected to a circuit, it can "hold" electric energy until it is needed. This stored energy can then be released to power devices or help with electrical signals.
That's why understanding how a capacitor stores energy is so important.

• Capacitors are used in various everyday electronics like cameras and computers to store and release energy quickly.
• The stored energy in a capacitor is key for devices that require a burst of energy.
• This concept of energy storage makes capacitors versatile and essential components.

By storing energy, capacitors help stabilize voltage and power flow, ensuring devices operate smoothly and efficiently.

capacitance formula

The capacitance formula is the heart of understanding how capacitors work. Capacitance \(C\) refers to a capacitor's ability to store an electric charge. It's defined as the amount of charge \(Q\) stored per unit voltage \(V\) across the capacitor plates. The basic formula is \(C = \frac{Q}{V}\), where:

• \(C\) is capacitance measured in farads (F).
• \(Q\) represents charge measured in coulombs (C).
• \(V\) is potential difference measured in volts (V).

This formula reveals a simple yet profound relationship: the greater the capacitance, the more charge can be stored at a given voltage. Furthermore, rearranging this formula can help us derive different expressions for the energy stored in a capacitor, such as \(U = \frac{1}{2} C V^2\) and \(U = \frac{Q^2}{2C}\). These expressions highlight how capacitance, charge, and potential are interlinked.

charge and potential

Charge and potential are fundamental to how capacitors function. The charge \(Q\) on a capacitor is the amount of electric charge stored on its plates. Potential, on the other hand, is the electrical pressure, or voltage \(V\), across the capacitor. These two aspects are closely related to the capacitance of the capacitor.
When a capacitor is connected to a power source, these charges accumulate, creating an electric field between the plates.

• This electric field gives rise to the potential difference.
• The stored charge equates to the capacitance multiplied by the potential difference, \(Q = C V\).

This relationship explains why altering the potential (voltage) changes the amount of charge a capacitor can store. The interplay of charge and potential allows capacitors to be flexible in handling various currents and voltages, making them integral components in electrical systems.

electrical potential energy

Electrical potential energy is the energy stored in a capacitor due to the charge on its plates. This energy can be calculated using different expressions based on the relationships among charge, capacitance, and voltage. One key formula is \(U = \frac{1}{2} Q V\), which represents the energy \(U\) as half of the product of charge and voltage. This formula highlights several factors:

• High charge \(Q\) stores more energy.
• High voltage \(V\) across the capacitor also means higher energy storage.

Other expressions like \(U = \frac{Q^2}{2C}\) and \(U = \frac{1}{2} C V^2\) can be derived from this. These demonstrate how energy storage depends on different parameters, such as charge and capacitance. Electrical potential energy is critical because it determines how much energy can be delivered to a circuit, influencing everything from small electronic devices to large power systems.