What will be the total flux through the faces of the cube as given in the figure with a side of length 'a' if a charge q is placed at?


Match the following:

Column I Column II
(a) a corner of the cube (i) \(\phi=\dfrac{q}{2\epsilon_0}\)
(b) mid-point of an edge of the cube (ii) \(\phi=\dfrac{q}{8\epsilon_0}\)
(c) centre of a face of the cube (iii) \(\phi=\dfrac{q}{2\epsilon_0}\)
(d) mid-point of B and C (iv) \(\phi=\dfrac{q}{4\epsilon_0}\)

1. (A)\(\rightarrow \)(iv), (B)\(\rightarrow \)(ii), (C)\(\rightarrow \)(iii), (D)\(\rightarrow \)(i)
2. (A)\(\rightarrow \)(ii), (B)\(\rightarrow \)(iv), (C)\(\rightarrow \)(i), (D)\(\rightarrow \)(iii)
3. (A)\(\rightarrow \)(iii), (B)\(\rightarrow \)(ii), (C)\(\rightarrow \)(iv), (D)\(\rightarrow \)(i)
4. (A)\(\rightarrow \)(i), (B)\(\rightarrow \)(iii), (C)\(\rightarrow \)(ii), (D)\(\rightarrow \)(iv)

Hint: Use Gauss' law.
Step 1: Find the electric flux in each case passing through the cube.
(a) There are eight corners in a cube so, the total charge for the cube is q8. Thus, the electric flux at A=q8ε0.
(b) When the charge q is placed at B, the middle point of an edge of the cube, it is being shared equally by 4 cubes. Therefore, total flux through the faces of the given cube=q/4ε0.
(c) When the charge q is placed at C, the centre of a face of the cube, it is being shared equally by 2 cubes. Therefore, total flux through the faces of the given cube =q/2ε0.
(d) Similarly, when charge q is placed at D, the mid-point of B and C, it is being shared equally by 2 cubes. Therefore, total flux through the faces of the given cube =q/2ε0.