A parallel-plate capacitor has square plates that are 7.20 cm on each side and 3.40 mm apart. The space between the plates is completely filled with two square slabs of dielectric, each 7.20 cm on a side and 1.70 mm thick. One slab is Pyrex glass and the other slab is polystyrene. If the potential difference between the plates is 96.0 V, find how much electrical energy (in nJ) can be stored in this capacitor.

U = 218 nJ

Explanation:

We are given;

Spacing between the plates; d = 3.4 mm = 3.4 * 10^ (-3) m

Voltage across the capacitor; V = 96 V

Dimension of the square plates is 7.2cm x 7.2cm.

So, Area = 7.2 * 7.2 = 51.84 cm² = 51.84 * 10^ (-4) m²

Permittivity of free space; ε_o = 8.85 * 10^ (-12) C²/N. m²

From relative permeability table;

Dielectric constant of Pyrex; k1 = 5.6

Dielectric constant of polystyrene; k2 = 2.56

Now, formula for capacitance of a capacitor with Dielectric is;

C = kC_o

Where, C_o = ε_o (A/d)

Since there are 2 capacitors, d will now be d/2 = (3.4 * 10^ (-3)) / 2 m = 1.7 * 10^ (-3)

Since we have 2 capacitor, thus;

C1 = k1*ε_o * (A/d)

C1 = (5.6 * 8.85 * 10^ (-12) * (51.84 * 10^ (-4)) / (1.7 * 10^ (-3))

C1 = 1.51 * 10^ (-10) F

Similarly;

C2 = (2.56 * 8.85 * 10^ (-12) * (51.84 * 10^ (-4)) / (1.7 * 10^ (-3))

C2 = 0.691 * 10^ (-10) F

For capacitors in series, formula for total capacitance (Cs) is;

1/Cs = (1/C1) + (1/C2)

Simplifying this, we have;

Cs = (C1*C2) / (C1 + C2)

Plugging in the relevant values;

Cs = (1.51 * 10^ (-10) * 0.691 * 10^ (-10)) / ((1.51 * 10^ (-10)) + (0.691 * 10^ (-10)))

Cs = 0.474 * 10^ (-10) F

The formula for energy stored in a capacitor with 2 Dielectrics is given as;

U = ½Cs*V²

So,

U = ½ * 0.474 * 10^ (-10) * 96²

U = 2.18 * 10^ (-7) J = 218 * 10^ (-9) = 218 nJ