If the electric field inside a capacitor exceeds the dielectric strength of the dielectric between its plates, the dielectric will break down, discharging and ruining the capacitor. Thus, the dielectric strength is the maximum magnitude that the electric field can have without break-down occurring. The dielectric strength of air is 3.0 x 106 V/m, and that of neoprene rubber is 1.2 x 107 V/m. A certain air-gap, parallel-plate capacitor can store no more than 0.046 J of electrical energy before breaking down. How much energy can this capacitor store without breaking down after the gap between its plates is filled with neoprene rubber? Take the dielectric constant of air to be 1.0, and of neoprene rubber to be 6.7.

To solve this problem, we need to determine the maximum electric field that the neoprene rubber can withstand before breaking down.

First, we can calculate the maximum electric field that the air-gap capacitor can withstand by using the formula:

E_air = dielectric strength of air / dielectric constant of air

Substituting the given values, we have:

E_air = (3.0 x 10^6 V/m) / 1.0 = 3.0 x 10^6 V/m

Next, we can calculate the maximum electric field that the neoprene rubber can withstand by using the formula:

E_neoprene = dielectric strength of neoprene rubber / dielectric constant of neoprene rubber

Substituting the given values, we have:

E_neoprene = (1.2 x 10^7 V/m) / 6.7 = 1.79 x 10^6 V/m

Now, we can calculate the maximum electrical energy that the capacitor can store without breaking down after the gap between its plates is filled with neoprene rubber. Let's assume the area of the plates is A and the distance between them (gap) is d.

The electric field between the plates is given by:

E = V / d

where V is the voltage across the capacitor.

The electrical energy stored in the capacitor is given by:

U = (1/2) * ε * A * E^2

where ε is the permittivity, A is the area, and E is the electric field.

For the air-gap capacitor, the energy is given as 0.046 J. Therefore:

0.046 J = (1/2) * ε_air * A * E_air^2

Simplifying, we have:

ε_air * A * E_air^2 = 2 * 0.046 J

Similarly, for the neoprene-filled capacitor, we have:

U_neoprene = (1/2) * ε_neoprene * A * E_neoprene^2

To find U_neoprene, we need to determine the electric field E_neoprene. Assuming that the electric field is the same for both cases (air and neoprene-filled), we have:

E_neoprene = E_air

Therefore, we can write:

U_neoprene = (1/2) * ε_neoprene * A * E_air^2

To find the energy that the capacitor can store without breaking down after the gap between its plates is filled with neoprene rubber, we need to solve for U_neoprene.

Dividing the equation for U_neoprene by the equation for U_air, we have:

(U_neoprene / U_air) = (ε_neoprene / ε_air)

Substituting the given values, we have:

(U_neoprene / 0.046 J) = (6.7 / 1.0)

Simplifying, we get:

U_neoprene = (6.7 / 1.0) * 0.046 J

Therefore, the energy that the capacitor can store without breaking down after the gap between its plates is filled with neoprene rubber is:

U_neoprene = 6.7 * 0.046 J

U_neoprene = 0.3082 J

Hence, the capacitor can store a maximum of 0.3082 J of electrical energy without breaking down after the gap between its plates is filled with neoprene rubber.