Consider the following situation where a source of potential difference V is connected to the combination of two identical capacitors. The total energy stored across the combination when the key 'K' is closed is denoted by E₁. When the key 'K' is opened and a dielectric with a dielectric constant of 5 is introduced between the plates of the capacitors, the total energy stored across the combination is now denoted by E₂. The task is to calculate the ratio of E₁ to E₂.

Step 1: When the Key 'K' is Closed

When the switch is closed, the two capacitors are connected in parallel. For two identical capacitors, the equivalent capacitance is:

Ceq = 2C
    

The energy stored in the capacitors is given by the formula:

E1 = 1/2 * Ceq * V2 = 1/2 * 2C * V2 = C * V2
    

Thus, the energy stored when the key 'K' is closed is:

E1 = C * V2
    

Step 2: When the Key 'K' is Opened and the Dielectric is Introduced

When the switch is opened, the charge on the right capacitor remains as C * V, while the potential across the left capacitor remains the same. Now, a dielectric with a dielectric constant K = 5 is introduced between the plates of the capacitors. This increases the capacitance of the capacitors as follows:

C' = K * C = 5C
    

The energy stored in the combination after introducing the dielectric can be calculated as the sum of the energies stored in both capacitors:

E2 = 1/2 * (5C) * V2 + 1/2 * C * V2
    

Expanding the terms:

E2 = 1/2 * 5C * V2 + 1/2 * C * V2 = 5/2 * C * V2 + 1/2 * C * V2
    

Combining the terms gives:

E2 = (5/2 * C * V2 + 1/2 * C * V2) = 13/2 * C * V2
    

Step 3: Ratio of E₁ to E₂

Finally, we can calculate the ratio of the energies:

E1 / E2 = (C * V2) / (13/2 * C * V2)
    

Simplifying the expression:

E1 / E2 = 2 / 13
    

Final Answer:

The ratio of E₁ to E₂ is:

E1 / E2 = 5 / 13