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 = 6m.

To find the electric flux density ( D ) at ( R = 6 , text{m} ) due to the three cylindrical sheets, we can use Gauss’s law, which states that the electric flux through a closed surface is proportional to the charge enclosed by that surface.

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

[

D_1 = sigma_1 = 5 , text{C/m}^2

]

2. For the sheet at ( R = 4 , text{m} ) with ( sigma = -2 , text{C/m}^2 ):

[

D_2 = sigma_2 = -2 , text{C/m}^2

]

3. For the sheet at ( R = 5 , text{m} ) with ( sigma = -3 , text{C/m}^2 ):

[

D_3 = sigma_3 = -3 , text{C/m}^2

]

At ( R = 6 , text{m} ), all three sheets contribute to the electric flux density, therefore:

[

D_{total}