Let's solve each part step by step: --- ## **Given Data** - Inner conductor radius, \( a = 2.9 \) mm \( = 2.9 \times 10^{-3} \) m - Outer conductor radius, \( b = 4. \) mm \( = 4. \times 10^{-3} \) m - Length of capacitor, \( L = 2.1 \) m - Vacuum between conductors - Potential difference (\( V \)) for parts B & C: \( 380 \) mV \( = .380 \) V --- ## **Part A: Capacitance per unit length** ### **Formula for Cylindrical Capacitor Capacitance per unit length:** \[ C' = \frac{2\pi \varepsilon_}{\ln(b/a)} \] Where: - \( C' \) = capacitance per unit length (F/m) - \( \varepsilon_ = 8.854 \times 10^{-12} \) F/m (vacuum permittivity) - \( a \) = inner radius (m) - \( b \) = outer radius (m) ### **Calculation** 1. Calculate \( \ln(b/a) \): \[ \ln\left(\frac{b}{a}\right) = \ln\left(\frac{4. \times 10^{-3}}{2.9 \times 10^{-3}}\right) = \ln(1.37931) \approx .321 \] 2. Substitute values: \[ C' = \frac{2 \pi (8.854 \times 10^{-12})}{.321} \] \[ = \frac{2 \times 3.1416 \times 8.854 \times 10^{-12}}{.321} \] \[ = \frac{55.614 \times 10^{-12}}{.321} \] \[ = 173.25 \times 10^{-12} \text{ F/m} \] \[ = 173.25 \text{ pF/m} \] **Final Answer Part A:** \[ \boxed{173 \ \text{pF/m}} \] --- ## **Part B: Charge on the Inner Conductor** ### **Formula:** \[ Q = C \cdot V \] But for the total capacitor: \[ C = C' \cdot L \] So: \[ Q = (C' \cdot L) \cdot V \] ### **Calculation:** \[ C' = 173.25 \times 10^{-12} \ \text{F/m} \] \[ L = 2.1 \ \text{m} \] \[ V = .380 \ \text{V} \] \[ C = 173.25 \times 10^{-12} \times 2.1 = 363.825 \times 10^{-12} \ \text{F} \] \[ Q = 363.825 \times 10^{-12} \times .380 = 138.25 \times 10^{-12} \ \text{C} \] \[ Q \approx 1.38 \times 10^{-10} \ \text{C} \] **Final Answer Part B:** \[ \boxed{1.38 \times 10^{-10} \ \text{C}} \] (The charge is **positive** on the inner conductor.) --- ## **Part C: Charge on the Outer Conductor** - By conservation of charge, if inner is \( +Q \), outer is \( -Q \). **Final Answer Part C:** \[ \boxed{-1.38 \times 10^{-10} \ \text{C}} \] (The charge is **negative** on the outer conductor.) --- ### **Summary Table** | Part | Quantity | Value | |------|------------------------------|--------------------------------| | A | Capacitance per unit length | \( 173 \) pF/m | | B | Charge on inner conductor | \( +1.38 \times 10^{-10} \) C | | C | Charge on outer conductor | \( -1.38 \times 10^{-10} \) C | Let me know if you need any clarifications!