To analyze the behavior of the electric field ((E)) and flux density (()) in a spherical capacitor with two dielectric layers, we must refer to Gauss's law and the continuity of electric displacement field at the interface between two dierics.

For a spherical capacitor:

1. The electric field (E) due to a spherical charge distribution is radially outward (or inward for negative charge) and varies inversely with the square of the radius (r), because of the symmetry involved. Mathematically, it's given by:

   (E = \frac{}{4\pi\epsilon}\frac{Q}{r^2),

   where (\epsilon) is the permittivity of the medium, (Q) is the enclosed charge, and (r) is the distance from the center.

2. The electric flux density (D) is the product of the permittivity of the material and the electric field,

   (D = \epsilon E).

Moving radially outwards from the center of the spherical capacitor:

• In the First Dielectric Layer (near center, ( \epsilon_1 )): The electric field (E_1) decreases with (^2). As you move away from the central charge, the flux must pass through a larger spherical surface area, since the area of a sphere is (4\pi r^2). Since the flux is spread over a larger area, the field weakens.

  The electric flux density (D_1) this region will be ( \epsilon_1 E_1 ).

• At the Between Dielectric 1 and Dielectric 2:

  At the boundary, the normal component of the electric displacement field (D) is continuous due to the absence of any free surface charge, which means:

  ( \epsilon_1 E_1 = \epsilon_2 E_2 \ Therefore, even if the electric field magnitude differs because of different permittivities, the product of (E) and (\epsilon) must be the same across boundary to satisfy continuity.

• In the Second Dielectric Layer (closer to the edge, ( \epsilon_2 )):

  The electric field (E_2) to decrease with (r^2), and the electric flux density (D2) in this region will now be ( \epsilon_2 E_2 ).

  Since theittivity (\epsilon) affects how strongly the material becomesized in an external electric field, the electric field magnitude (E_2) will adjust to maintain the continuity of (D). If (\epsilon_2>\epsilon1), then (E_2<E_1) in such a way that (D remains constant, as (D = \epsilon E).

Comparing the Magnitudes:

• Electric Field:

If (\epsilon_2 > \epsilon_1), (E_2) at the junction (interface) must be less than (E_1) maintain the continuity of (D). This is due to the material's ability toize more strongly, effectively reducing the local electric field strength.

• Flux Density:

  (D) will continue to decrease as (r) due to the expanding spherical surface but will remain continuous across the boundary of the twolectrics.

In summary, moving radially from the center to the edge of a spherical capacitor:

• The electric field and flux density decrease with (r^2- At the interface of two dielectrics with different permittivities, the electric field magnitude adjusts to ensure continuous displacement field (D).
• The material with a higher permittivity will have a lower electric field magnitude at the interface due to its increased ability to polarize.

Physically, these behaviors are due to the spherical symmetry of the charge distribution and the nature of the dielectric materials responding to an applied electric field. The higher permittivity indicates a greater ability to align dipoles within the dielectric material, reducing the net electric field within it.