Three charged cylindrical sheets are present in three spaces with σ = 5 at R = 2m, σ = -2 at R = 4m and σ = -3 at R = 5m. Find the flux density at R = 3m.

To find the electric flux density ( mathbf{D} ) at ( R = 3 , text{m} ) due to the three cylindrical sheets, we can use Gauss’s law for electric displacement. The displacement field ( mathbf{D} ) is given by:

[

mathbf{D} = varepsilon_0 mathbf{E} + mathbf{P}

]

Where:

– ( sigma ) is the surface charge density,

– ( varepsilon_0 ) is the permittivity of free space (( approx 8.854 times 10^{-12} , text{C}^2/text{N m}^2 )),

– In this case, we can consider only the contribution from the surface charge density since there are no free charges outside of the cylindrical sheets.

### Electric Flux Density Calculation:

1. Cylindrical Sheet at ( R = 2 ) m with ( sigma = 5 , text{C/m}^2 ):

– For ( R < 2 , text{m} ), ( mathbf{D} = 0 ) (inside the cylindrical sheet).

– For ( R > 2 , text{m} ), the flux density due to this charge:

[

D = frac{sigma}{2} = frac{