First slide

Electrostatic force

Question

Two charges Q₁and Q₂ are distance d apart. Two dielectrics of thickness t₁ and t₂ and dielectric constant k₁ and k₂ are introduced as shown. Find the force between the charges

Moderate

A

$\frac{Q_{1} Q_{2}}{4 {πε}_{0} {[d - (t_{1} + t_{2}) + k_{1} t_{1} + k_{2} t_{2}]}^{2}}$
B

zero
C

$\frac{Q_{1} Q_{2}}{4 {πε}_{0} {(d + \sqrt{k_{1}} t_{1} + \sqrt{k_{2}} t_{2})}^{2}}$
D

$\frac{Q_{1} Q_{2}}{4 {πε}_{0} {[(\sqrt{k_{1}} - 1) t_{1} + (\sqrt{k_{2}} - 1) t_{2} + d]}^{2}}$

Solution

Effective distance in vacuum is given by
$r = t_{1} \sqrt{K_{1}} + t_{2} \sqrt{K_{2}} + d - (t_{1} + t_{2}) so, F = \frac{Q_{1} Q_{2}}{4 {πε}_{0} {(t_{1} \sqrt{K_{1}} + t_{2} \sqrt{K_{2}} + d - t_{1} - t_{2})}^{2}} \Rightarrow F = \frac{Q_{1} Q_{2}}{4 {πε}_{0} {[(\sqrt{K_{1}} - 1) t_{1} + (\sqrt{K_{2}} - 1) t_{2} + d]}^{2}}$

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