A co-axial cable used in transmission line has inner radius of `0.1 mm` and outer radius of `0.6 mm`. Calculate capacitnace per meter of the cable.

Explanation

To calculate the capacitance per meter of a co-axial cable with an inner radius of \( r_A = 0.1 \, \text{mm} \) and an outer radius of \( r_B = 0.6 \, \text{mm} \), we will follow these steps:

### Step 1: Convert the radii to meters
The inner radius \( r_A \) and outer radius \( r_B \) need to be converted from millimeters to meters.

\[ r_A = 0.1 \, \text{mm} = 0.1 \times 10^{-3} \, \text{m} = 1.0 \times 10^{-4} \, \text{m} \]
\[ r_B = 0.6 \, \text{mm} = 0.6 \times 10^{-3} \, \text{m} = 6.0 \times 10^{-4} \, \text{m} \]

### Step 2: Use the formula for capacitance per meter
The formula for the capacitance per unit length \( C/L \) of a co-axial cable is given by:

\[ \frac{C}{L} = \frac{2 \pi \epsilon_0}{\ln\left(\frac{r_B}{r_A}\right)} \]

Where:
- \( \epsilon_0 \) is the permittivity of free space, approximately \( 8.85 \times 10^{-12} \, \text{F/m} \).
- \( \ln \) is the natural logarithm.

### Step 3: Calculate the ratio \( \frac{r_B}{r_A} \)
Now we will calculate the ratio of the outer radius to the inner radius:

\[ \frac{r_B}{r_A} = \frac{6.0 \times 10^{-4}}{1.0 \times 10^{-4}} = 6.0 \]

### Step 4: Calculate the natural logarithm
Next, we find the natural logarithm of the ratio:

\[ \ln\left(\frac{r_B}{r_A}\right) = \ln(6.0) \approx 1.7918 \]

### Step 5: Substitute values into the capacitance formula
Now we substitute the values into the capacitance formula:

\[ \frac{C}{L} = \frac{2 \pi (8.85 \times 10^{-12})}{1.7918} \]

Calculating \( 2 \pi \):

\[ 2 \pi \approx 6.2832 \]

Now substituting this value:

\[ \frac{C}{L} = \frac{6.2832 \times 8.85 \times 10^{-12}}{1.7918} \approx \frac{5.564 \times 10^{-11}}{1.7918} \approx 3.102 \times 10^{-11} \, \text{F/m} \]

### Step 6: Convert to picofarads per meter
To convert from Farads to picofarads (1 F = \( 10^{12} \) pF):

\[ \frac{C}{L} \approx 31.02 \, \text{pF/m} \]

### Final Result
The capacitance per meter of the co-axial cable is approximately:

\[ \frac{C}{L} \approx 31.02 \, \text{pF/m} \]

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