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I will make a generalized form of my question. There are two point charges $q$, $x$ distance apart. And there is a dielectric slab of thickness $t$ and of dielectric constant $K$. Should the force be sum of forces on them separately. i.e. $F=\frac{k\cdot q^2}{(x-t)^2} + \frac{k\cdot q^2}{t^2*K}.$ Is this right or am I doing something wrong ? |
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The correct equation, to represent the physical system, must give an answer that is bounded by the physical extremes of t = 0 and t = x, i.e.: $\frac{kq^2}{Kx^2} < F < \frac{kq^2}{x^2}$ The lower value is when t = x and the upper value is when t = 0. I'm thinking that we need to find an effective relative dielectric constant to use over the distance x. $F = \dfrac{kq^2}{K_{eff}x^2}$ Given the bounds above, I'm thinking a kind of weighted harmonic mean might work. $K_{eff} = \dfrac{x}{(x-t) + \dfrac{t}{K}}$ Note that when: $t = 0, K_{eff} = 1$ $t = x, K_{eff} = K$ Anyhow, if you stare at this a bit, you'll see that the force, as desired, decreases as t increases and that the formula reduces to the bounds above for t = 0 and t = x. |
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