Question

A parallel plate capacitor consists of square plates of edge length \(2.00 \mathrm{~cm}\) separated by a distance of \(1.00 \mathrm{~mm}\). The capacitor is charged with a 15.0 -V battery, and the battery is then removed. A 1.00 -mm- thick sheet of nylon (dielectric constant of 3.50 ) is slid between the plates. What is the average force (magnitude and direction) on the nylon sheet as it is inserted into the capacitor?

Short answer

Answer: The average force on the nylon sheet is 8.71 × 10^{-4} N in the direction perpendicular to the plates.

Step-by-step solution

Calculate the capacitance of the capacitor

The capacitance of the parallel plate capacitor can be calculated using the formula: \( C = \epsilon_0 \frac{A}{d} \) where \(C\) is the capacitance, \(\epsilon_0\) is the vacuum permittivity constant (8.85 × 10⁻¹² F/m), \(A\) is the area of one plate, and \(d\) is the distance between the plates. Let's calculate the area of one plate: \( A = L^2 = (0.02m)^2 = 4 \times 10^{-4} m^2 \) Now, calculate the capacitance: \( C = (8.85 × 10^{-12} F/m) \frac{4 \times 10^{-4} m^2}{0.001m} = 3.54 × 10^{-9} F \)

Calculate the charge on the capacitor

When the capacitor is charged with a 15.0 V battery, the charge on the capacitor can be calculated using the formula: \( Q = CV \) where \(Q\) is the charge and \(V\) is the voltage. Now, calculate the charge: \( Q = (3.54 × 10^{-9} F)(15.0 V) = 53.1 × 10^{-9} C \)

Calculate the electric field between the plates

We can calculate the electric field between the plates of the capacitor using the formula: \( E = \frac{Q}{A \epsilon_0} \) where \(E\) is the electric field. Now, calculate the electric field: \( E = \frac{53.1 \times 10^{-9} C}{(4 \times 10^{-4} m^2 )(8.85 × 10^{-12} F/m)} = 1.50 × 10^6 N/C \)

Calculate the potential difference across the nylon

Next, we need to calculate the potential difference across the nylon. The potential difference can be calculated using the formula: \( V_\text{nylon} = Ed_\text{nylon} \) where \(d_\text{nylon}\) is the thickness of the nylon sheet. Now, calculate the potential difference across the nylon: \( V_\text{nylon} = (1.50 × 10^6 N/C)(0.001m) = 1.50 × 10^3 V \)

Calculate the force on the nylon sheet

Finally, we can calculate the force on the nylon sheet using the formula: \( F = \frac{1}{2} \epsilon_\text{nylon} \epsilon_0 A V_\text{nylon}^2 \) where \(F\) is the force, \(\epsilon_\text{nylon}\) is the dielectric constant of nylon, and \(\epsilon_0\) is the vacuum permittivity constant. Now, calculate the force: \( F = \frac{1}{2} (3.50)(8.85 × 10^{-12} F/m)(4 \times 10^{-4} m^2)(1.50 × 10^3 V)^2 = 8.71 × 10^{-4} N \) The average force on the nylon sheet as it is inserted into the capacitor is \(8.71 \times 10^{-4} N\) in the direction perpendicular to the plates.

Key concepts

Capacitance Calculation

Understanding how to calculate the capacitance of a parallel plate capacitor is fundamental in grasping the principles of capacitor function. Capacitance, denoted by the symbol C, is a measure of a capacitor's ability to store charge per unit of voltage across its plates. The formula for calculating capacitance in a vacuum or air is C = \(\epsilon_0 \frac{A}{d}\), where \(\epsilon_0\) represents the vacuum permittivity constant, A is the area of one of the plates, and d is the distance between the two plates.

For a parallel plate capacitor with square plates, the area A is simply the square of the edge length of a plate. In our case, the edge length of 2.00 cm gives an area of 4 \times 10^{-4} m^2. With the distance d of 1.00 mm, or 0.001 m, and using the known value of \(\epsilon_0\), we can calculate a specific capacitance. It’s crucial to ensure consistency in units when performing these calculations to avoid errors.

Dielectric Constant

The dielectric constant, also referred to as the relative permittivity, is a number that describes how much a material can increase the capacitance of a capacitor compared to a vacuum. When a dielectric material, such as nylon, is inserted between the plates of a capacitor, it affects the electric field and thus the capacitance.

The dielectric constant, denoted as \(\epsilon_r\), is a dimensionless quantity and for nylon, it’s given as 3.50. This implies that the capacitance of the capacitor with the nylon inserted will be 3.50 times greater than the capacitance of the capacitor in a vacuum. However, it’s important to note that the dielectric constant also influences how the electric field interacts with the material, as it reduces the electric field strength within the dielectric by the same factor.

Electric Field

The electric field is a vector quantity that relates to the force which would be exerted on a positive charge at a point in space. For a parallel plate capacitor without any dielectric material, the electric field is calculated using the formula E = \(\frac{Q}{A \times \epsilon_0}\), where E is the electric field strength, Q is the charge stored on the capacitor, and A with \(\epsilon_0\) have been previously defined.

The direction of the electric field is from the positive to the negative plate and is uniform between them. In the problem at hand, inserting a dielectric does not alter the charge Q (since the battery was removed prior to insertion), but it will alter the potential difference across the capacitor and, as a consequence, the electric field strength is changed too. Understanding the interaction of electric fields with dielectric materials is essential when analyzing the force exerted on the material, which is a key aspect of the application of parallel plate capacitors.