A point charge of 10 microcoulombs is at an unspecified location inside a cube of side length 2.0 centimeters. Find the net electric flux through the surfaces of the cube.
We can call the charge of our point charge, 10 microcoulombs, 𝑄. We want to solve for the net electric flux through the sides of the cube that the point charge is inside. We can call this flux 𝜙 sub 𝐸. Let’s start by recalling the mathematical equation for electric flux.
The electric flux, 𝜙 sub 𝐸, through a closed surface is equal to the charge that surface encloses, 𝑄, divided by 𝜖 naught, the permittivity of free space, which we can take to be exactly 8.85 times 10 to the negative 12 farads per meter.
If we draw a sketch of our point charge and our cube, we can see that even though we don’t know exactly where in the cube charge is, that’s not important because the cube entirely encloses the charge.
That means that all the electric field lines emanating from this charge will pass through some side of the cube and be included in our calculation for 𝜙 sub 𝐸 in the flux.
So 𝜙 sub 𝐸 is 𝑄 over 𝜖 naught or 10 times 10 to the negative sixth coulombs, our charge, divided by 𝜖 naught, 8.85 times 10 to the negative 12 farads per meter.
When we calculate this fraction, we find 𝜙 sub 𝐸 is 1.1 times 10 to the sixth newton meter squared per coulomb. That’s the total electric flux that passes through the surfaces of this cube.