To determine the potential difference across the plates of the capacitor filled with a dielectric, we can use the formula for the energy stored in a capacitor.
The energy stored in a capacitor is given by the equation:
E = (1/2) * C * V^2
Where:
E is the energy stored
C is the capacitance of the capacitor
V is the potential difference across the plates
The capacitance of a capacitor is given by the equation:
C = (k * Īµā * A) / d
Where:
k is the dielectric constant
Īµā is the permittivity of free space (8.85 x 10^-12 F/m)
A is the area of the plates
d is the distance between the plates
Since we have two identical capacitors, the area and distance between the plates are the same. Let's assume A and d are constant.
For the empty capacitor (Cā):
Cā = Īµā * A / d
For the capacitor filled with a dielectric (Cā):
Cā = (k * Īµā * A) / d
We want both capacitors to store the same amount of electrical energy. Therefore, we can equate the energies:
Eā = Eā
(1/2) * Cā * Vā^2 = (1/2) * Cā * Vā^2
Plugging in the expressions for Cā and Cā:
(1/2) * (Īµā * A / d) * Vā^2 = (1/2) * ((k * Īµā * A) / d) * Vā^2
The area and distance between the plates cancel out, giving us:
Īµā * Vā^2 = k * Īµā * Vā^2
Dividing both sides by Īµā:
Vā^2 = k * Vā^2
Taking the square root of both sides:
Vā = ā(k) * Vā
Substituting the given values:
Vā = ā(3.9) * Vā
Since the potential difference of the empty capacitor (Vā) is given as 17 V, we can solve for Vā:
17 V = ā(3.9) * Vā
Dividing both sides by ā(3.9):
Vā = 17 V / ā(3.9)
Calculating the value:
Vā ā 9.40 V
Therefore, the potential difference across the plates of the capacitor filled with a dielectric should be approximately 9.40 V in order to store the same amount of electrical energy as the empty capacitor.