Question

Two parallel plate capacitors have identical plate areas and identical plate separations. The maximum energy each can store is determined by the maximum potential difference that can be applied before dielectric breakdown occurs. One capacitor has air between its plates, and the other has Mylar. Find the ratio of the maximum energy the Mylar capacitor can store to the maximum energy the air capacitor can store.

Short answer

Answer: U_ratio = (3.1 * V_1^2) / (V_0^2), where V_0 and V_1 are the breakdown voltages for air and Mylar capacitors, respectively. To obtain the numerical value of this ratio, the actual breakdown voltages V_0 and V_1 need to be found through research or given data.

Step-by-step solution

Determine the dielectric constants and breakdown voltages

First, we need to find the dielectric constants (K) and breakdown voltages (V) for air and Mylar. The dielectric constants can be found in a reference table or online. The breakdown voltages can also be found using a reference table or can be calculated using the formula: V_breakdown = K * E_breakdown where E_breakdown is the electronegativity of the dielectric material. For simplicity, we will assume that the breakdown voltages are given, but you could find the values through research if desired. According to available resources, we find: - For air: K_air = 1, V_air = V_0 (let's assume its breakdown voltage is V_0) - For Mylar: K_mylar = 3.1, V_mylar = V_1 (let's assume its breakdown voltage is V_1)

Compute the capacitance for each capacitor

To calculate the capacitance, we need to use the formula for capacitance for parallel plate capacitors: C = (ε_0 * K * A) / d where ε_0 is the vacuum permittivity, K is the dielectric constant, A is the plate area, and d is the plate separation. Since both capacitors have identical plate areas and plate separations, we only need to adjust the dielectric constant for each capacitor: C_air = (ε_0 * K_air * A) / d = (ε_0 * 1 * A) /d = ε_0 * A/d C_mylar = (ε_0 * K_mylar * A) / d = (ε_0 * 3.1 * A) /d = 3.1 * ε_0 * A/d

Compute the maximum energy for each capacitor

To find the maximum energy each capacitor can store, we need to use the formula for the energy stored in a capacitor: U = 0.5 * C * V^2 Using the calculated capacitances and breakdown voltages for each capacitor: U_air = 0.5 * C_air * V_0^2 = 0.5 * (ε_0 * A/d) * V_0^2 U_mylar = 0.5 * C_mylar * V_1^2 = 0.5 * (3.1 * ε_0 * A/d) * V_1^2

Compute the ratio of maximum energy storage between the capacitors

We are asked to find the ratio of maximum energy stored in the Mylar capacitor to the maximum energy stored in the air capacitor. To do this, we will divide the energy stored in the Mylar capacitor by the energy stored in the air capacitor: U_ratio = U_mylar / U_air U_ratio = [(0.5 * (3.1 * ε_0 * A/d) * V_1^2)] / [(0.5 * (ε_0 * A/d) * V_0^2)] Notice that the factors 0.5, ε_0, A, and d all cancel out: U_ratio = (3.1 * V_1^2) / (V_0^2) This is our final answer for the ratio of maximum energy stored in the Mylar capacitor to the maximum energy stored in the air capacitor. To obtain the numerical value of this ratio, we would need the actual breakdown voltages V_0 and V_1 for air and Mylar capacitors, which can be found through research or from given data.