Charge Density on a Conductive Slab

[MengMei]

Homework Statement

An infinite sheet of charge, oriented perpendicular to the x-axis, passes through x = 0. It has a surface charge density σ1 = -4.7 μC/m2. A thick, infinite conducting slab, also oriented perpendicular to the x-axis, occupies the region between a = 2.8 cm and b = 4.5 cm. The conducting slab has a net charge per unit area of σ2 = 87 μC/m2. (Recall that the surface charge densities σa and σb on the slab surfaces at a and b, respectively, sum to equal the net charge per unit area: σa + σb = σ2.)

What is σb, the charge per unit area on the surface of the slab located at x = 4.5 cm?

Relevant Equations

σa + σb = σ2

Okay, so I tried thinking of this as like a simple balancing of equations. There's an infinite sheet of charge on the left and a conductor on the right with some charge already on it. My thought process was that the side nearer to the charged sheet would have 4.7 more μC/m2 than the far side. Knowing this, I assumed that the near side would have 87-4.7 = 82.3 and the far side would have 87+4.7 = 91.7. That didn't work, so I took at step back, looked at the equation and then thought, "Oh, the charges on each side of the slab have to equal 87."

So I thought to half the total net charge and then find the difference between the two sides due to the infinite sheet with charge. That also didn't work. Now I'm just really confused and don't know what to do anymore.

 

[haruspex]

Instead of guessing, think about the field inside the slab. What do you know about that, and how can you calculate it from the surface charge densities?