A capacitor is composed of two plates separated by a sheet of insulating material 13 mm thick and or relative permittivity 4. The distance between the plates is increased to allow the insertion of second sheet 5 mm thick and of relative permittivity `in_(r)`. If the capacitance of the capacitor so formed is 1/3 of the original capacitance , find `in_(r)`.

Explanation

To solve the problem, we need to find the relative permittivity \( \epsilon_r \) of the second insulating sheet inserted into the capacitor. We will follow these steps:

### Step 1: Understand the Original Capacitance
The original capacitor has a thickness of 13 mm and a relative permittivity \( \epsilon_r = 4 \). The formula for the capacitance \( C \) of a capacitor filled with a dielectric is given by:

\[ C_1 = \frac{\epsilon_r \epsilon_0 A}{d_1} \]
...