Question

For a vector field \( \mathbf{D} = \rho \cos^2 \phi a_{\rho} + z^2 \sin^2 \phi a_{\phi} \) in a cylindrical coordinate system \( (\rho, \phi, z) \) with unit vectors \( a_{\rho}, a_{\phi}, a_z \), the net flux of \( \mathbf{D} \) leaving the closed surface of the cylinder \( (\rho = 3, 0 \leq z \leq 2) \) (rounded off to two decimal places) is .

Hint:

To compute the flux through a surface, apply Gauss's law and integrate the dot product of the field and the surface area.

Answer 1

Correct Answer: 56.5

To find the net flux, we use Gauss's law: \[ \Phi = \int \mathbf{D} \cdot dA \] For a cylinder with radius \( \rho = 3 \) and height \( 0 \leq z \leq 2 \), the flux through the surface is computed by integrating the component of \( \mathbf{D} \) along the normal to the surface. After performing the integration, we find the flux to be: \[ \Phi = 56.55 \] Thus, the net flux of \( \mathbf{D} \) is \( 56.55 \).