GR8677.054 [tagged]

This question might be difficult if you forgot the definition of dielectric constant, or how the dielectric works. Still, you can answer this question just by choice (and a bit of knowledge).

First, you should at least know that dielectric constant is "ratio" of something. One of possible (and normal) value is such a number is 1. Then, you can eliminate choice (A) since if $K=1$ , then the bound charge density goes to infinity, which is unlikely. Now, what else do you know about 1? It is actually the state of nothing, that is, there is no change (that's why the ratio is 1). Hence, plugging in $K=1$ to each choice, the only choice (E) gives answer zero. Which means that there is no change to the system. The correct answer is (E). You can guess this answer without knowing what K is.

Actually, $K$ , the dielectric constant, is the permittivity in the dielectric relative to the permittivity in vacuum. In other words, $K = \epsilon / \epsilon_0$ , where $\epsilon$ is the permittivity in the dielectric. Also, the dielectric constant determines the strength of the electric field as $E = E_0 /K$ , where $E_0$ is the electric field in the case that there is no dielectric there.

With this fact, we can construct "Gauss's box", to both bound charge in dielectric and free charge in conductor.

Then, by Gauss's Law, we got
For Gauss's box in dielectric $E - 0 = (\sigma_p + \sigma)/\epsilon_0$ .
Note that the electric field inside the conductor is zero (static case). Also, we can use $\epsilon_0$ here since, in this notion, we treat dielectric effect as bound charge already.

Rearranging term, we got $\sigma_p = \epsilon_0 E - \sigma$ (*)
We know that $E = E_0 /K$ .
Also, we know by Gauss's Law (try to prove by yourself) that $E_0 = \sigma/\epsilon_0$ .
Plug all this into (*), we got $\sigma_p = \frac{1}{K} \sigma - \sigma = \sigma \frac{1-K}{K}$

The correct answer is therefore (E). Only 16 of 100 people answered correctly.

Now you can see power of choice (and wise guess!)