## Problem Restatement Given: - Three types of capacitors: **parallel plate**, **cylindrical**, and **spherical** - All have: - The **same small gap size** between the plates - **Equal surface areas** on their plates **Task:** Order the capacitors from **greatest to least capacitance** and explain your reasoning. --- ## Capacitance Formulas Let's recall the capacitance formulas for each type: ### 1. Parallel Plate Capacitor \[ C = \epsilon_ \frac{A}{d} \] - \( A \): plate area - \( d \): distance between plates - \( \epsilon_ \): permittivity of free space ###2. Cylindrical Capacitor (Coaxial) \[ C = \frac{2\pi \epsilon_ L}{\ln(b/a)} \] - \( L \): length of cylinder (assume large compared to radius) - \( a, b \): inner and outer radii - Surface area constraint: \( 2\pi a L \) and \( 2\pi b L \) should each equal the given area ### 3. Spherical Capacitor (Concentric Spheres) \[ C = 4\pi \epsilon_ \frac{ab}{b - a} \] - \( a, b \): inner and outer radii - Surface area constraint: \( 4\pi a^2 \) and \( 4\pi b^2 \) should each equal the given area --- ## Assumptions - The **gap \( d \) is very small** (\( d \ll a, b \)), so \( b \approx a + d \). - **Surface area** for each plate is the same: let it be \( S \). --- ## Analysis and Ordering ### **Parallel Plate** \[ C_{pp} = \epsilon_ \frac{S}{d} \] - Directly proportional to area, inversely to gap. ### **Cylindrical** - For small gap, \( b = a + d \), so \[ \ln(b/a) = \ln\left(1 + \frac{d}{a}\right) \approx \frac{d}{a} \quad (\text{for } d \ll a) \] - For plate area \( S = 2\pi a L \Rightarrow L = \frac{S}{2\pi a} \): \[ C_{cyl} = \frac{2\pi \epsilon_ L}{d/a} = 2\pi \epsilon_ L \frac{a}{d} = 2\pi \epsilon_ \frac{S}{2\pi a} \frac{a}{d} = \epsilon_ \frac{S}{d} \] - **Same as parallel plate!** ### **Spherical** - For small gap, \( b = a + d \), so \[ C_{sph} = 4\pi \epsilon_ \frac{a(a + d)}{d} \approx 4\pi \epsilon_ \frac{a^2}{d} \] - For plate area \( S = 4\pi a^2 \Rightarrow a = \sqrt{\frac{S}{4\pi}} \): \[ C_{sph} \approx 4\pi \epsilon_ \frac{S}{4\pi d} = \epsilon_ \frac{S}{d} \] - **Same as parallel plate and cylindrical!** --- ## **Conclusion & Ordering** ### **Order (from greatest to least capacitance):** \[ \boxed{ \text{Parallel Plate} = \text{Cylindrical} = \text{Spherical} } \] ### **Reasoning:** When the **gap is very small** and **surface areas are equal**, all three geometries give the **same capacitance**: \[ C = \epsilon_ \frac{\text{(surface area)}}{(\text{gap size})} \] The differences due to geometry vanish in the small-gap limit, so the capacitance depends only on the area and the gap. --- ### **Summary Table** | Capacitor Type | Capacitance (\(C\)) (small \(d\)) | |------------------|------------------------------------| | Parallel Plate | \( \epsilon_ \frac{S}{d} \) | | Cylindrical | \( \epsilon_ \frac{S}{d} \) | | Spherical | \( \epsilon_ \frac{S}{d} \) |