ierces the Gaussian cube of edge length 0.420 m and positioned as shown in the figure. (The magnitu?

Through the top and bottom faces, only the j component will result in flux.  Flux through the bottom face is 

(-8.7)(6.7 N/C)(0.420 m)^2 

= use calculator.

Flux through the top face is 

(-8.7)(0.420^2 + 6.7)N/C*(0.420m)^2 = use calculator.

Flux through the left face is 

(7.7 N/C)(0.420 m)^2 = use calculator.

Flux through the back face is zero.

Net flux into the cube is the (positive) difference between the fluxes through the top and bottom faces, as there is no difference between the left and right faces.

We usually use the sign convention: flux leaving a Gaussian volume is positive and flux entering the volume is negative.

(a) Top face:

Only the component of the field in the y -direction flows through the top face because the face is perpendicular to the y-direction (making it parallel to the xz plane).

Field component in y-direction is -8.7(y² + 6.7)  NC⁻¹.

The top face has area 0.420² m² and  y= 0.420m everywhere in the face.

Flux = EA = -8.7(0.420² + 6.7)*0.420² = -10.553 = -10.6 NC⁻¹m² rounded.

We need to check the sign is correct.  Since the y-component of the field is negative it is directed in the -y-direction.  This means the flux enters the cube through the top face and the negative sign is correct.

(b) Bottom face:

Field component in y direction is E_y = -8.7(y² + 6.7)  N/C.

The top face has area 0.420² m² and  y=0 everywhere in the face.

Flux = E_y*A = -8.7(0² + 6.7)*0.420² = -10.282 = -10.3 NC⁻¹m² rounded.

Since the y-component of the field is negative it is directed in the -y direction.

This means the flux leaves  the cube though the bottom face and the negative sign must be changed to positive.

Flux = 10.3 NC⁻¹m²

(c) Left face:

Only the x-component of field passes through this face.

Flux = E_x*A = 7.7*0.420² = 1.358 = 1.36 NC⁻¹m² rounded.

This flux is entering the cube through this face so we need to give it a negative sign

Flux = -1.36 NC⁻¹m²

(d) Back face:

The field has no z-component so zero flux passes through the back face.

(e) Total:

Note, fluxes through the right and left faces will be equal magnitude but opposite signs, one negative (entering cube) and one positive (leaving cube).  So they will cancel.

The  fluxes through the back and front faces are both zero.

The total flux is -10.553 +10.282 = -0.27  NC⁻¹m²

Obviously, check my arithmetic.