An uncharged parallel plate capacitor having a dielectric of dielectric constant K is connected to a similar air-filled capacitor charged to a potential V. The two share the charge, and the common potential is V’. The dielectric constant K is:

To find the dielectric constant \( K \) of the uncharged capacitor after it is connected to the charged air-filled capacitor, we can use the principles of capacitance and charge sharing.

Capacitance Basics

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

• C = εA/d

Where:

• ε is the permittivity of the dielectric material (ε = Kε₀ for the dielectric capacitor and ε = ε₀ for the air-filled capacitor).
• A is the area of the plates.
• d is the distance between the plates.

Charge and Potential Relationship

When the two capacitors are connected, they share charge. The initial charge \( Q \) on the air-filled capacitor can be expressed as:

• Q = C₁V

Where \( C₁ \) is the capacitance of the air-filled capacitor and \( V \) is its initial voltage.

Common Potential After Connection

The common potential \( V' \) after connection can be expressed as:

• V' = (Q + 0) / (C₁ + C₂)

Here, \( C₂ \) is the capacitance of the dielectric-filled capacitor.

Finding the Dielectric Constant

Substituting the expressions for charge and capacitance, we can derive the formula for the dielectric constant \( K \):

• K = (C₂ / C₁) * (V / V')

By rearranging and substituting the values of capacitance, you can calculate \( K \) based on the known values of \( V \) and \( V' \).

In summary, the dielectric constant \( K \) can be determined by the ratio of the capacitances and the voltages before and after connecting the capacitors.