An ideal parallel-plate capacitor consists of a set of two parallel plates of area A separated by a very small distance d. When the capacitor plates carry charges +Q and -Q, the capacitor stores energy U0. If the separation between the plates is doubled, how much electrical energy is stored in the capacitor?
a.4U0
b. U0/4
c. U0
d.2U0
e. U0/2

d.2U0

Explanation:

The capacitance of a capacitor is given by the following formula

And the electric potential energy stored in a capacitor is given by

Applying these formula to the given variables, the initial potential energy becomes

If d becomes 2d, then potential energy becomes

d. 2U₀

The electrical energy, U, stored in a capacitor is given by;

U =            ------------------(i)

Where;

Q = Charge on the plates of the capacitor

C = capacitance of the capacitor = Aε₀ / d

A = Area of the plates of the capacitor

d = distance of separation between the plates

ε₀ = permittivity of free space.

Substitute C = Aε₀ / d into equation (i) as follows;

U = Q²d / Aε₀           -----------(ii)

From the question;

Electrical energy U₀, is stored when the charges on the plates are +Q and -Q.

Substitute these into equation (ii) as follows;

U₀ = Q²d / Aε₀                ----------------(iii)

Now, when the distance is doubled (d = 2d), the electrical energy stored becomes U₁ which is given by;

U₁ = Q²(2d) / Aε₀

U₁ = 2 x Q²d / Aε₀          [ Q²d / Aε₀ = U₀ -------------- from equation (iii)]

U₁ = 2 x U₀

U₁ = 2U₀

Therefore, the electrical energy stored when the separation between the plates is doubled is twice the initial electrical energy stored.

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