https://file.aiquickdraw.com/mathbot/file/1727054705718uw6va2lk.jpg

Let's solve this problem for the capacitance per unit length of a cylindrical capacitor.

### Given:
- Inner conductor radius \( r_1 = 2.2 \, \text{mm} = 2.2 \times 10^{-3} \, \text{m} \)
- Outer conductor radius \( r_2 = 3.1 \, \text{mm} = 3.1 \times 10^{-3} \, \text{m} \)
- Length of the capacitor \( L = 2.3 \, \text{m} \)
- The space between the conductors is vacuum, so the permittivity of free space \( \varepsilon_0 = 8.854 \times 10^{-12} \, \text{F/m} \).

### Formula for Capacitance per Unit Length \( \frac{C}{L} \) for a cylindrical capacitor:
\[
\frac{C}{L} = \frac{2\pi \varepsilon_0}{\ln \left( \frac{r_2}{r_1} \right)}
\]

Substituting the values:
\[
\frac{C}{L} = \frac{2\pi (8.854 \times 10^{-12})}{\ln \left( \frac{3.1 \times 10^{-3}}{2.2 \times 10^{-3}} \right)}
\]

Let me calculate this.The capacitance per unit length for the given cylindrical capacitor is approximately **162.22 pF/m**.

Would you like further clarification on any part of this calculation or more details about the process?

Here are 5 related questions you could explore:
1. How does the presence of a dielectric material between the conductors affect the capacitance?
2. What happens to the capacitance if the radii of the conductors are increased?
3. Can we derive the formula for capacitance per unit length from the general expression for a cylindrical capacitor?
4. How does capacitance change with the length of the cylindrical capacitor?
5. How does the permittivity of the material between the conductors influence the capacitance?

**Tip:** In cylindrical capacitors, capacitance increases with a smaller distance between the conductors (inner and outer radii).

What is the capacitance per unit length of a cylindrical capacitor with an inner conductor radius of 2.2 mm, an outer conductor radius of 3.1 mm, and a vacuum between the conductors? The total length of the capacitor is 2.3 m.

Solve the capacitance per unit length for a cylindrical capacitor with inner conductor radius 2.2 mm, outer conductor radius 3.1 mm, separated by vacuum. This example demonstrates key electrostatic concepts and logarithmic calculations, suitable for undergraduate physics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation