Absolutely! Here's the solution for finding the capacitance per unit length of the coaxial cable with a partially filled dielectric:

**Understanding the Problem**

We have a coaxial cable with:

*   An inner copper wire of radius  *a*
*   An outer copper tube of inner radius  *c*
*   The space between the conductors is partially filled with a dielectric material:
    *   The dielectric has a dielectric constant ε_r and fills the space from radius  *b*  to  *c*.
    *   The space from radius *a* to *b* is air (or vacuum).

**Solution Approach**

The key idea is that the coaxial cable can be treated as two capacitors in series:

1.  **Capacitor 1:**  Air-filled region between radius  *a* and *b*.
2.  **Capacitor 2:**  Dielectric-filled region between radius *b* and *c*.

We'll find the capacitance per unit length of each capacitor and then use the series combination formula to get the total capacitance per unit length.

**Detailed Solution**

**1. Electric Field:**

*   Assume a charge  *Q*  per unit length on the inner conductor.
*   Use Gauss's law with a cylindrical Gaussian surface of radius  *r*  and length *l* to find the electric field in each region:

    *   **Region 1 (a < r < b):**
        E₁ = Q / (2πε₀r)  (where ε₀ is the permittivity of free space)
    *   **Region 2 (b < r < c):**
        E₂ = Q / (2πε₀ε_r r)

**2. Potential Difference:**

*   Calculate the potential difference  *V*  between the inner and outer conductors by integrating the electric field in each region:

    V = -∫[a to b] E₁ dr  - ∫[b to c] E₂ dr

*   Evaluating the integrals:
    V = (Q / 2πε₀) [ln(b/a) + (1/ε_r) ln(c/b)]

**3. Capacitance per Unit Length (C'):**

*   **Capacitor 1:**
    C₁' = (2πε₀) / ln(b/a)

*   **Capacitor 2:**
    C₂' = (2πε₀ε_r) / ln(c/b)

*   **Total Capacitance (Series Combination):**
    1/C' = 1/C₁' + 1/C₂'

*   Solving for C':
    C' = 2πε₀ / [ln(b/a) + (1/ε_r) ln(c/b)]

**Final Answer**

The capacitance per unit length of the coaxial cable with the partially filled dielectric is:

C' = 2πε₀ / [ln(b/a) + (1/ε_r) ln(c/b)]


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