The dielectric constant of an insulator  in a transmission line is 2.25. Calculate velocity factor of the transmission line

Answer

To calculate the velocity factor of a transmission line given the dielectric constant, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship**: The velocity factor (VF) of a transmission line is defined as the ratio of the speed of light in the medium (v) to the speed of light in vacuum (c). Mathematically, it can be expressed as:
\[ VF = \frac{v}{c} \]

2. **Relate Velocity Factor to Dielectric Constant**: The speed of light in a medium is related to the dielectric constant (ε_r) of that medium. The relationship is given by:
\[ v = \frac{c}{\sqrt{\varepsilon_r}} \]
where ε_r is the relative permittivity (dielectric constant) of the material.

3. **Substitute into the Velocity Factor Formula**: By substituting the expression for v into the velocity factor formula, we get:
\[ VF = \frac{c/\sqrt{\varepsilon_r}}{c} = \frac{1}{\sqrt{\varepsilon_r}} \]

4. **Calculate the Velocity Factor**: Given that the dielectric constant (ε_r) is 2.25, we can substitute this value into the equation:
\[ VF = \frac{1}{\sqrt{2.25}} \]

5. **Simplify the Calculation**: Calculate the square root of 2.25:
\[ \sqrt{2.25} = 1.5 \]
Therefore, the velocity factor becomes:
\[ VF = \frac{1}{1.5} = \frac{2}{3} \]

6. **Final Result**: The velocity factor of the transmission line is:
\[ VF = \frac{2}{3} \approx 0.667 \]

### Summary:
The velocity factor of the transmission line with a dielectric constant of 2.25 is approximately 0.667 or \( \frac{2}{3} \).

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