Five identical charges Q = 2μC are placed equidistant on a semicircle as shown in Figure 9. Another…
Five identical charges $Q = 2\mu\text{C}$ are placed equidistant on a semicircle as shown in Figure 9. Another point charge $q = 1\mu\text{C}$ is kept at the center of the circle of radius 2cm. Calculate the electrostatic force experienced by the charge q.
Let $R = 2\text{ cm} = 0.02\text{ m}$, $Q = 2\ \mu\text{C}$, $q = 1\ \mu\text{C}$. The charges are at $90^\circ$, $135^\circ$, $180^\circ$, $225^\circ$, $270^\circ$ about q.
Forces due to $Q_1$ and $Q_5$ (diametrically opposite) are equal and opposite and cancel.
$F_3 = \dfrac{qQ}{4\pi\epsilon_0R^2} = \dfrac{9\times10^9\times 1\times10^{-6}\times 2\times10^{-6}}{(0.02)^2} = 45\text{ N}$
$Q_2$ and $Q_4$: the components perpendicular to the axis cancel; the components along it add: $F_{24} = 2F\cos45^\circ = \sqrt2\times 45\text{ N}$.
$F = \dfrac{1}{4\pi\epsilon_0}\dfrac{qQ}{R^2}(1+\sqrt2) = 45(1+\sqrt2)\approx 108.6\text{ N}$
