Question

The flux of the function \( \mathbf{F} = (y^2) \hat{x} + (3xy - z^2) \hat{y} + (4yz) \hat{z} \) passing through the surface ABCD along \( \hat{n} \) is ............ 

Hint:

When calculating flux through a surface, always check if any component of the vector field is zero along the surface to simplify the calculation.

Answer 1

Correct Answer: 1.16

Step 1: Understand the flux calculation.
The flux \( \Phi \) through a surface is given by the surface integral of the dot product of the vector field and the normal vector to the surface: \[ \Phi = \int_S \mathbf{F} \cdot \hat{n} \, dA \] where \( \mathbf{F} \) is the vector field, \( \hat{n} \) is the unit normal vector to the surface, and \( dA \) is the differential area element.
Step 2: Set up the integral.
The surface ABCD is a square in the \( xy \)-plane (since the problem does not give specific geometry, we'll assume it's in the \( xy \)-plane for simplicity). The normal vector \( \hat{n} \) is in the \( z \)-direction, and hence: \[ \hat{n} = \hat{z} \] Therefore, the flux simplifies to: \[ \Phi = \int_S (4yz) \, dA \] Since the surface is on the \( xy \)-plane, \( z = 0 \) on the surface. Thus, the flux becomes: \[ \Phi = \int_S (4y \cdot 0) \, dA = 0 \]
Step 3: Conclusion.
The flux through the surface is zero, so the correct answer is 0.00.