The capacitance of a parallel plate capacitor is 2.5 µF. When it is half filled with dielectric, its capacitance becomes 5µF. What will be the dielectric constant of dielectric.

To determine the dielectric constant of the material used in the parallel plate capacitor, we can use the relationship between capacitance, dielectric constant, and the configuration of the capacitor. Let's break this down step by step.

Understanding Capacitance

The capacitance \( C \) of a parallel plate capacitor is given by the formula:

C = \frac{εA}{d}

Where:

• C is the capacitance.
• ε is the permittivity of the dielectric material (which is ε = ε₀ * κ, where ε₀ is the permittivity of free space and κ is the dielectric constant).
• A is the area of the plates.
• d is the separation between the plates.

Initial Capacitance

Initially, the capacitance of the capacitor is given as 2.5 µF. When the capacitor is half-filled with a dielectric, the effective capacitance increases to 5 µF. This change occurs because the dielectric material increases the capacitance due to its dielectric constant.

Capacitance with Dielectric

When half of the capacitor is filled with a dielectric, we can think of the capacitor as two capacitors in series: one with the dielectric and one without. The capacitance of the capacitor with the dielectric can be expressed as:

C_d = \frac{κC_0}{2}

Where:

• C_d is the capacitance with the dielectric.
• C_0 is the original capacitance without the dielectric (2.5 µF).

Calculating the Dielectric Constant

Given that the new capacitance \( C_d \) is 5 µF, we can set up the equation:

C_d = C_0 + C_d

Substituting the values:

5 µF = 2.5 µF + \frac{κ(2.5 µF)}{2}

Rearranging gives:

5 µF - 2.5 µF = \frac{κ(2.5 µF)}{2}

2.5 µF = \frac{κ(2.5 µF)}{2}

Multiplying both sides by 2:

5 µF = κ(2.5 µF)

Now, dividing both sides by 2.5 µF:

κ = \frac{5 µF}{2.5 µF} = 2

Final Result

The dielectric constant of the material used in the capacitor is 2. This means that the dielectric material doubles the capacitance compared to the air-filled capacitor, which is a significant enhancement in its ability to store electrical energy.