Compute the Gauss law for D= 10ρ3 /4 i, in cylindrical coordinates with ρ= 4m, z=0 and z=5.

To compute Gauss’s Law using the displacement field D, we first write down Gauss’s Law for electric displacement fields:

[

Phi_D = oint_S mathbf{D} cdot dmathbf{A} = Q_{free}

]

Where Q_free is the free charge enclosed by the surface S. Here, the displacement field is given as:

[

mathbf{D} = frac{10rho^3}{4} mathbf{i}

]

In cylindrical coordinates, D can be expressed as a function of ρ, and given constants indicate that D has only an i component (along the x-axis).

Given:

– ρ = 4 m

– z = 0 (which we could consider as the bottom face of a cylindrical volume)

– z = 5 (top face of the volume)

1. Calculate D at ρ = 4 m:

[

mathbf{D} = frac{10(4)^3}{4} mathbf{i} = frac{10 times 64}{4} mathbf{i} = 160 mathbf{i}

]

2. Define the Gaussian surface: A cylinder with radius 4 m and height 5 m.

3. Calculate the area vector: The enclosed area of the Gaussian surface includes the curved surface and two flat circular areas.

– Curved surface area