Capacitance with a Dielectric Slab Partially Filled in between the Plates:

When a dielectric slab is partially filled between the plates of a capacitor, the capacitance of the capacitor changes. The capacitance with a dielectric slab partially filled in between the plates is given by:

C = (epsilon * A) / (d1 + k * d2)

where epsilon is the permittivity of free space, A is the area of the plates, d1 is the distance between the plates without the dielectric slab, d2 is the distance between the dielectric slab and one of the plates, and k is the dielectric constant of the material of the slab.

Energy Stored inside a Capacitor:

The energy stored inside a capacitor is given by:

U = (1/2) * C * V^2

where U is the energy stored, C is the capacitance, and V is the potential difference across the plates of the capacitor.

Substituting the value of capacitance in the above equation, we get:

U = (1/2) * (epsilon * A) / (d1 + k * d2) * V^2

Since the electric field E is given by E = V / d, we can rewrite the above equation as:

U = (1/2) * (epsilon * A) * E^2 * (d1 + k * d2)

Finally, we can simplify the above equation as:

U = (1/2) * epsilon * E^2 * (A * d1 + k * A * d2)

Therefore, the energy stored inside a capacitor is proportional to the square of the electric field and the volume of the dielectric material between the plates. This equation is useful in calculating the energy stored in various types of capacitors for different applications.

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