Let's solve the problem step by step. --- ## **Given Values** - **Height of inner cylinder, \( h_1 \)**: 16.74 cm - **Height of outer cylinder, \( h_2 \)**: 19.63 cm - **Diameter of inner cylinder, \( d_1 \)**: 7.97 cm - **Diameter of outer cylinder, \( d_2 \)**: 13.83 cm - **Permittivity of free space, \( \epsilon_ \)**: \( 8.854 \times 10^{-12} \) F/m --- ## **Step 1: Convert All Units to Meters** - \( h_1 = .1674 \) m - \( h_2 = .1963 \) m - \( d_1 = .0797 \) m - \( d_2 = .1383 \) m --- ## **Step 2: Radii and Average Height** - Inner radius: \( r_1 = \frac{d_1}{2} = .03985 \) m - Outer radius: \( r_2 = \frac{d_2}{2} = .06915 \) m - Average height: \( h_\text{avg} = \frac{h_1 + h_2}{2} = \frac{.1674 + .1963}{2} = .18185 \) m --- ## **Step 3: Parallel Plate Model** ### **Area** - \( A_\text{avg} = 2\pi r_\text{avg} h_\text{avg} \) - Where \( r_\text{avg} = \frac{r_1 + r_2}{2} = \frac{.03985 + .06915}{2} = .0545 \) m \[ A_\text{avg} = 2 \pi \cdot .0545 \cdot .18185 \approx .0623 \text{ m}^2 \] ### **Distance** - \( d = r_2 - r_1 = .06915 - .03985 = .0293 \) m ### **Capacitance** \[ C_\text{parallel} = \frac{\epsilon_ A_\text{avg}}{d} = \frac{8.854 \times 10^{-12} \times .0623}{.0293} \] \[ C_\text{parallel} \approx \frac{5.515 \times 10^{-13}}{.0293} \approx 1.88 \times 10^{-11} \text{ F} \] --- ## **Step 4: Cylindrical Capacitor Model** \[ C_\text{cylinder} = \frac{2\pi \epsilon_ h_\text{avg}}{\ln(r_2/r_1)} \] - \( \ln(r_2/r_1) = \ln(.06915/.03985) = \ln(1.735) \approx .551 \) \[ C_\text{cylinder} = \frac{2\pi \cdot 8.854 \times 10^{-12} \cdot .18185}{.551} \] \[ = \frac{1.011 \times 10^{-11}}{.551} \approx 1.83 \times 10^{-11} \text{ F} \] --- ## **Step 5: Percent Difference** \[ \text{Percent Difference} = \frac{|C_\text{parallel} - C_\text{cylinder}|}{C_\text{cylinder}} \times 100\% \] \[ = \frac{|1.88 \times 10^{-11} - 1.83 \times 10^{-11}|}{1.83 \times 10^{-11}} \times 100\% \] \[ = \frac{.05 \times 10^{-11}}{1.83 \times 10^{-11}} \times 100\% \] \[ \approx 2.73\% \] --- ## **Final Answers** - \( C_\text{parallel} \approx 1.88 \times 10^{-11} \) F - \( C_\text{cylinder} \approx 1.83 \times 10^{-11} \) F - **Percent Difference**: **2.73%** --- Let me know if you'd like to see the calculations in more detail!

# Step-by-Step Solution to the Capacitor Problem ## Given Values - **Height of inner cylinder, \( h_1 \)**: 16.74 cm - **Height of outer cylinder, \( h_2 \)**: 19.63 cm - **Diameter of inner cylinder, \( d_1 \)**: 7.97 cm - **Diameter of outer cylinder, \( d_2 \)**: 13.83 cm - **Permittivity of free space, \( \epsilon_0 \)**: \( 8.854 \times 10^{-12} \text{ F/m} \) ## Step 1: Convert All Units to Meters - \( h_1 = 0.1674 \) m - \( h_2 = 0.1963 \) m - \( d_1 = 0.0797 \) m - \( d_2 = 0.1383 \) m ## Step 2: Calculate Radii and Average Height - **Inner radius**: \( r_1 = \frac{d_1}{2} = 0.03985 \) m - **Outer radius**: \( r_2 = \frac{d_2}{2} = 0.06915 \) m - **Average height**: \( h_{\text{avg}} = \frac{h_1 + h_2}{2} = \frac{0.1674 + 0.1963}{2} = 0.18185 \) m ## Step 3: Calculate Capacitance Using Parallel Plate Model ### Area Calculation - **Average radius**: \( r_{\text{avg}} = \frac{r_1 + r_2}{2} = \frac{0.03985 + 0.06915}{2} = 0.0545 \) m - **Area**: \[ A_{\text{avg}} = 2\pi r_{\text{avg}} h_{\text{avg}} \] \[ A_{\text{avg}} = 2 \pi (0.0545)(0.18185) \approx 0.0623 \text{ m}^2 \] ### Distance Calculation - **Distance between cylinders**: \[ d = r_2 - r_1 = 0.06915 - 0.03985 = 0.0293 \text{ m} \] ### Capacitance Calculation \[ C_{\text{parallel}} = \frac{\epsilon_0 A_{\text{avg}}}{d} = \frac{8.854 \times 10^{-12} \times 0.0623}{0.0293} \] Calculating: \[ C_{\text{parallel}} \approx \frac{5.515 \times 10^{-13}}{0.0293} \approx 1.88 \times 10^{-11} \text{ F} \] ## Step 4: Calculate Capacitance Using Cylindrical Capacitor Model ### Capacitance Formula \[ C_{\text{cylinder}} = \frac{2\pi \epsilon_0 h_{\text{avg}}}{\ln(r_2/r_1)} \] ### Natural Logarithm Calculation \[ \ln\left(\frac{r_2}{r_1}\right) = \ln\left(\frac{0.06915}{0.03985}\right) \approx \ln(1.735) \approx 0.551 \] ### Capacitance Calculation \[ C_{\text{cylinder}} = \frac{2\pi (8.854 \times 10^{-12}) (0.18185)}{0.551} \] Calculating: \[ C_{\text{cylinder}} \approx \frac{1.011 \times 10^{-11}}{0.551} \approx 1.83 \times 10^{-11} \text{ F} \] ## Step 5: Calculate Percent Difference ### Percent Difference Formula \[ \text{Percent Difference} = \frac{|C_{\text{parallel}} - C_{\text{cylinder}}|}{C_{\text{cylinder}}} \times 100\% \] ### Calculation \[ = \frac{|1.88 \times 10^{-11} - 1.83 \times 10^{-11}|}{1.83 \times 10^{-11}} \times 100\% \] \[ = \frac{0.05 \times 10^{-11}}{1.83 \times 10^{-11}} \times 100\% \] \[ \approx 2.73\% \] ## Final Answers - **Capacitance (Parallel Plate Model)**: \( C_{\text{parallel}} \approx 1.88 \times 10^{-11} \text{ F} \) - **Capacitance (Cylindrical Model)**: \( C_{\text{cylinder}} \approx 1.83 \times 10^{-11} \text{ F} \) - **Percent Difference**: **2.73%** Feel free to ask if you need further clarification or additional calculations!