Q.Two charges of magnitudes -4Q and +2Q are located at points (2a, 0) and (5a, 0) respectively. What is the electric flux due to these charges through a sphere of radius 4a with its centre at the origin?

[!TLDR] Only the -4Q charge (at 2a) lies inside the radius-4a sphere; +2Q at 5a is outside it and …

A sphere of radius $4a$ centred at the origin encloses any charge whose distance from the origin is less than $4a$. The charge $-4Q$ sits at $(2a,0)$, distance $2a<4a$ from the origin -- INSIDE the sphere. The charge $+2Q$ sits at $(5a,0)$, distance $5a>4a$ -- OUTSIDE the sphere. By Gauss' law, only the ENCLOSED charge contributes to the net flux through the sphere; a charge outside contributes exactly zero net flux (any field line from it that enters the sphere on one side must als …

Determine which of the two charges lies within radius 4a of the origin, …

A common mistake is to include BOTH charges in the flux calculation, forgetting that Gauss' law depends only on the charge strictly ENC …