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 = 4.5m.

To find the electric flux density (D) at R = 4.5 m due to the three charged cylindrical sheets, we can use Gauss’s law for electricity, which relates the electric flux density to the surface charge density.

The electric flux density D due to a uniformly charged infinite cylindrical sheet is given by the equation:

[

D = frac{sigma}{2} hat{n}

]

where σ is the surface charge density and (hat{n}) is the unit normal vector pointing away from the sheet.

Given:
1. At R = 2 m, σ = 5 C/m²
2. At R = 4 m, σ = -2 C/m²
3. At R = 5 m, σ = -3 C/m²

Now, we calculate the contributions to D at R = 4.5 m from all three sheets.

1. Contribution from the sheet at R = 2 m:

[

D_1 = frac{5}{2} = 2.5 , text{(outward)}

]

2. Contribution from the sheet at R = 4 m:

[

D_2 = frac{-2}{2} = -1 , text{(inward)}

]

3. Contribution from the sheet at R = 5 m:

[

D_