First slide

Electrostatic force

Question

Equal charges q are placed at the four corners A, B, C, D of a square of length a. The magnitude of the force on the charge at B will be

Moderate

A

$\frac{3 q^{2}}{4 {πε}_{0} a^{2}}$
B

$\frac{4 q^{2}}{4 {πε}_{0} a^{2}}$
C

$(\frac{1 + 2 \sqrt{2}}{2}) \frac{q^{2}}{4 {πε}_{0} a^{2}}$
D

$(2 + \frac{1}{\sqrt{2}}) \frac{q^{2}}{4 {πε}_{0} a^{2}}$

Solution

After following the guidelines mentioned above
$F_{net} = F_{AC} + F_{D} = \sqrt{F_{A}^{2} + F_{C}^{2}} + F_{D}$
Since $F_{A} = F_{C} = \frac{{kq}^{2}}{a^{2}} and F_{D} = \frac{{kq}^{2}}{(a \sqrt{2})^{2}}$
$F_{net} = \frac{\sqrt{2} {kq}^{2}}{a^{2}} + \frac{{kq}^{2}}{2 a^{2}} = \frac{{kq}^{2}}{a^{2}} (\sqrt{2} + \frac{1}{2}) = \frac{q^{2}}{4 {πε}_{0} a^{2}} (\frac{1 + 2 \sqrt{2}}{2})$

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