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A source of potential difference V is connected to the combination of two identical capacitors as shown in the figure. When key 'K’ is closed, the total energy stored across the combination is E1. Now key ‘K’ is opened and dielectric of dielectric constant 5 is introduced between the plates of the capacitors. The total energy stored across the combination is now E2. The ratio E1/E2 will be :
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Detailed Solution
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