Two capacitors are identical, except
that one is filled with a dielectric (� = 4.50). The empty capacitor is connected to a 12.0 Volt
battery. What must be the potential difference across the plates of the capacitor filled with a
dielectric so that it stores the same amount of electrical energy as the empty capacitor.
The capacitor with the dielectric will have 4.5 times the C of an empty capacitor with same area and gap size.
Since E = (1/2)CV^2, the dielectric-filled capacitor will require only 1/4.5 = 22.2% of the value of V^2 of the empty one, to have the same stored energy.
(Vfilled)^2 = 12^2/4.5 = 32 Volt^2
Vfilled = 5.66 Volts
The method in the previous comment is correct ^^^
To find the potential difference across the plates of the capacitor filled with a dielectric, we need to set the electrical energy stored by both capacitors equal to each other.
The electrical energy stored by a capacitor is given by the formula:
E = (1/2) * C * V^2
Where:
E = Electrical energy stored
C = Capacitance of the capacitor
V = Potential difference across the plates
Since the capacitors are identical except for the dielectric, the capacitance remains the same for both capacitors.
Therefore, we can write the equation for the electrical energy stored in the empty capacitor as:
E(empty) = (1/2) * C * V(empty)^2
And the equation for the electrical energy stored in the capacitor filled with a dielectric as:
E(dielectric) = (1/2) * C * V(dielectric)^2
To make the two capacitors store the same amount of electrical energy, we can equate the two equations:
(1/2) * C * V(empty)^2 = (1/2) * C * V(dielectric)^2
Since the capacitance is the same for both capacitors, we can cancel it out from both sides of the equation:
V(empty)^2 = V(dielectric)^2
To find the potential difference across the plates of the capacitor filled with a dielectric, we take the square root of both sides of the equation:
V(dielectric) = sqrt(V(empty)^2)
Plugging in the given value V(empty) = 12.0 Volts, we get:
V(dielectric) = sqrt((12.0 V)^2)
V(dielectric) = sqrt(144 V^2)
V(dielectric) = 12.0 V
Therefore, the potential difference across the plates of the capacitor filled with a dielectric must also be 12.0 Volts in order to store the same amount of electrical energy as the empty capacitor.
To find the potential difference across the plates of the capacitor filled with a dielectric so that it stores the same amount of electrical energy as the empty capacitor, we need to use the formula for the capacitance of a capacitor with a dielectric:
C = k * ε₀ * A / d
Where:
C is the capacitance (in Farads)
k is the dielectric constant
ε₀ is the vacuum permittivity (8.85 x 10^(-12) F/m)
A is the area of the plates (in square meters)
d is the separation distance between the plates (in meters)
Since the two capacitors are identical, we can assume they have the same area and separation distance. Let's call the capacitance of the empty capacitor C1 and the potential difference V1. The capacitance of the capacitor with the dielectric is C2 and the potential difference across its plates is V2.
Since the two capacitors store the same amount of electrical energy, we can equate their energies:
1/2 * C1 * V1^2 = 1/2 * C2 * V2^2
Now we need to express C1 and C2 in terms of the given information. The capacitance of the empty capacitor, C1, is only determined by the vacuum permittivity and the physical characteristics of the capacitor (area and separation distance):
C1 = ε₀ * A / d
The capacitance of the capacitor filled with the dielectric, C2, is given by:
C2 = k * ε₀ * A / d
Now we can substitute these expressions into the energy equation:
1/2 * (ε₀ * A / d) * V1^2 = 1/2 * (k * ε₀ * A / d) * V2^2
Since the area and separation distance are the same for both capacitors, they cancel out:
ε₀ * V1^2 = k * ε₀ * V2^2
Now we can solve for V2:
V2^2 = V1^2 / k
Taking the square root of both sides gives us:
V2 = V1 / √(k)
Thus, to find the potential difference across the plates of the capacitor filled with the dielectric so that it stores the same amount of electrical energy as the empty capacitor, we can divide the potential difference of the empty capacitor by the square root of the dielectric constant (k).