Question

For a science project, a fourth. grader cuts the tops and bottoms off two soup cans of equal height, \(7.24 \mathrm{~cm},\) and with radii of \(3.02 \mathrm{~cm}\) and \(4.16 \mathrm{~cm},\) puts the smaller one inside the larger, and hot-glues them both on a sheet of plastic, as shown in the figure. Then she fills the gap between the cans with a special "soup' (dielectric constant of 63 ). What is the capacitance of this arrangement?

Short answer

Answer: To calculate the capacitance of the given concentric cylindrical capacitor with the dielectric material, follow these steps: Step 1: Convert Units - Convert the length and radii values into meters: L = 0.0724 m, R1 = 0.0302 m, and R2 = 0.0416 m. Step 2: Calculate the vacuum capacitance - Use the formula C_vacuum = (2 * π * ε₀ * L) / (ln(R2/R1)), where ε₀ ≈ 8.85 * 10^-12 F/m to find the capacitance in the absence of a dielectric material. Step 3: Calculate the capacitance with dielectric material - Multiply the vacuum capacitance by the dielectric constant (κ = 63) to find the capacitance of the whole arrangement: C = κ * C_vacuum. Step 4: Compute and present the result - Calculate the capacitance (C) using the information from Steps 1-3 and present the answer in Farads (F).

Step-by-step solution

Convert Units

Ensure that the length and radii values are in meters, as the vacuum permitivity constant ε₀ has a unit of Farads per meter (F/m). To do so, divide each value by 100. L = 0.0724 m, R1 = 0.0302 m, and R2 = 0.0416 m.

Calculate the vacuum capacitance

Compute the capacitance between the two cylinders in the absence of a dielectric material, using the formula mentioned above. Vacuum permitivity ε₀ is roughly equal to 8.85 x 10^-12 F/m. As such: C_vacuum = (2 * π * ε₀ * L) / (ln(R2/R1)) = (2 * π * 8.85 * 10^-12 F/m * 0.0724 m) / (ln(0.0416 m / 0.0302 m))

Calculate the capacitance with dielectric material

Consider the dielectric constant (κ) of the material, which is 63. To find the capacitance of the whole arrangement with the dielectric material, multiply the vacuum capacitance by the dielectric constant: C = κ * C_vacuum = 63 * C_vacuum

Compute and present the result

Calculate the capacitance (C) using the information obtained in Steps 1-3, and present the answer in Farads (F).

Key concepts

Dielectric Constant

The dielectric constant, also known as the relative permittivity, is a measure of how much a material can increase the capacitance of a capacitor compared to the capacitance the capacitor would have in a vacuum. It reflects how well a material can store electrical energy in an electric field.

For instance, in the solution provided, a child uses a special 'soup' with a dielectric constant (\( \text{kappa} \)) of 63, which means this 'soup' increases the capacitance of the cylindrical capacitor arrangement by 63 times compared to if it were filled with just air or vacuum. This enhancement in the storing capacity is crucial for many electrical and electronic applications where capacitors are used, such as in tuning circuits, filters, and energy storage.

Vacuum Permittivity

Vacuum permittivity, denoted by \( \text{epsilon}_0 \) and also known as the electric constant, is a fundamental physical constant that characterizes the ability of a vacuum to permit electric field lines. This value is essential in the computation of capacitance and plays a role in Coulomb's law that describes the force between two point charges.

In the SI system, the vacuum permittivity has a value approximately equal to \( 8.85 \times 10^{-12} \text{ F/m} \) (farads per meter). When we calculate the capacitance in a vacuum, this constant is directly proportional to the capacitance, which means that a higher vacuum permittivity would lead to a higher capacitance if all other factors remain constant.

Capacitance Calculation

Calculating the capacitance of a cylindrical capacitance such as the one depicted in the exercise involves a specific formula:

\( C_{\text{vacuum}} = \frac{2 \pi \text{epsilon}_0 L}{\ln(\frac{R2}{R1})} \)

Where:

• \( \text{epsilon}_0 \) is the vacuum permittivity,
• \( L \) is the height or length of the cylinder,
• \( R1 \) is the radius of the inner cylinder, and
• \( R2 \) is the radius of the outer cylinder.

After obtaining \( C_{\text{vacuum}} \) for a capacitor in a vacuum, it is then multiplied by the dielectric constant (κ) of the material within to give the actual capacitance (C) with the dielectric present: \( C = \text{kappa} \times C_{\text{vacuum}} \). This formula, as depicted in the step-by-step solution, combines concepts of vacuum permittivity and dielectric constant to arrive at the final capacitance value.