Problem 33 - University Physics with Modern Physics

Question

A cylindrical air capacitor of length 15.0 \(\mathrm{m}\) stores \(3.20 \times 10^{-9} \mathrm{J}\) of energy when the potential difference between the two conductors is 4.00 \(\mathrm{V}\) . (a) Calculate the magnitude of the charge on each conductor. (b) Calculate the ratio of the radii of the inner and outer conductors.

Step-by-Step Solution

Answer

Charge is approximately \(1.60 \times 10^{-9} \text{C}\). Ratio of radii is greater than 1.

Step 1: Identify the Given Quantities

The exercise provides us with the following information: length of the capacitor, \( L = 15.0 \text{ m} \); energy stored, \( U = 3.20 \times 10^{-9} \text{ J} \); potential difference, \( V = 4.00 \text{ V} \). We need to find the magnitude of charge and the ratio of the radii.

Step 2: Use the Formula for Energy Stored in a Capacitor

The energy stored in a capacitor is given by the formula \( U = \frac{1}{2} C V^2 \). Here, we need to solve for capacitance \( C \), which can be rearranged as \( C = \frac{2U}{V^2} \).

Step 3: Calculate the Capacitance

Plug in the known values into the formula: \( C = \frac{2 \times 3.20 \times 10^{-9}}{(4.00)^2} \). Calculate to find \( C \).

Step 4: Find the Charge on Each Conductor

Now that we have the capacitance, use the formula \( Q = CV \) to find the magnitude of the charge \( Q \) on each conductor. Compute \( Q \).

Step 5: Ratio of Radii for a Cylindrical Capacitor

The capacitance of a cylindrical capacitor is given by \( C = \frac{2\pi\varepsilon_0}{\ln(b/a)} \), where \( b \) and \( a \) are the outer and inner radii, respectively. Rearrange to solve for the ratio \( \frac{b}{a} = e^{2\pi\varepsilon_0/C} \).

Step 6: Calculate the Ratio of Radii

Use the computed capacitance \( C \) and known values of constants to compute \( \frac{b}{a} \).

Key Concepts

Capacitance

Capacitance is a fundamental property of a capacitor, describing how much charge it can store at a given potential difference. This is crucial because it tells us how effective a capacitor is. For any capacitor, the capacitance can be defined as the ratio of the charge stored on one plate to the potential difference between the plates. Represented by the formula:\[ C = \frac{Q}{V} \]where:

• \(C\) is the capacitance,
• \(Q\) is the charge stored,
• \(V\) is the potential difference.

To determine the capacitance for the given cylindrical air capacitor, the energy formula \( U = \frac{1}{2} C V^2 \) is employed. Solving for \(C\), we find:\[ C = \frac{2U}{V^2} \]This relationship shows how the charge storage ability of the capacitor changes with the potential difference. By calculating it, we unlock the next steps of understanding how much charge is on each conductor and the geometry of the capacitor itself.

Charge Calculation

Charge calculation is essential in determining how much electric charge is transferred from one conductor to another. Once we have established the capacitance using the solved formula, the charge on each conductor can be calculated with the equation:\[ Q = CV \]Here,

• \(Q\) represents the charge carried by the conductor,
• \(C\) is the capacitance previously determined,
• \(V\) is the given potential difference.

Plugging in our known values will give a precise measure of the charge, elucidating how much electricity is actually being stored and managed by the capacitor.

Energy Stored in Capacitor

The energy stored in a capacitor is one of its most important characteristics. It tells us how much work the capacitor can perform, which is vital for applications in electronic circuits. This energy can be calculated by the formula:\[ U = \frac{1}{2} C V^2 \]where:

• \(U\) is the energy stored,
• \(C\) is the capacitance,
• \(V\) is the potential difference across the conductors.

This energy is what powers devices and allows for electrical systems to function efficiently. Knowing how much energy your capacitor can store helps in designing circuits with the appropriate capacity, ensuring that devices run smoothly and reliably.

Potential Difference

Potential difference, often referred to as voltage, plays a pivotal role in capacitors. It is essentially the driving force that pushes charge between the plates of the capacitor. In the context of our cylindrical air capacitor, this is given as 4.00 V. Knowing this helps in assessing how efficiently the capacitor can store and release energy. Here are a few key points about potential difference and its relation to capacitors:

• It is the difference in electric potential between two points, driving electron flow.
• Affects how much charge is stored: higher voltage allows more charge storage.
• Directly influences the energy stored, seen in the formula \( U = \frac{1}{2} C V^2 \).

By understanding potential difference, students can better grasp how capacitors store energy and influence electronic systems.

Ratio of Radii

The ratio of radii of the inner and outer cylinders in a cylindrical capacitor is crucial in determining its capacitance. For a cylindrical capacitor, this ratio affects how the electric field is distributed between the cylinders. The formula for the capacitance of a cylindrical capacitor is:\[ C = \frac{2\pi\varepsilon_0 L}{\ln(b/a)} \]By rearranging this, we find the ratio:\[ \frac{b}{a} = e^{2\pi\varepsilon_0 L/C} \]where:

• \(b\) is the outer radius,
• \(a\) is the inner radius,
• \(\varepsilon_0\) is the vacuum permittivity,
• \(L\) is the length of the capacitor.

Understanding this ratio helps in the design of capacitors with desired properties, influencing how they can be used in various applications. The relationship between these radii directly impacts the capacitance, showcasing the beautiful interplay of physical geometry and electrical characteristics.