Question

a) Four identical particles with charge 3 μμC are located at the corners of a square. The square's sides are 2 cm in length. A fifth particle of charge 6 μμC is placed directly at the center of the square. What is the magnitude of the total electric force on any one of the four particles at the corners?

b) If you could adjust the charge of the particle at the center, what should you choose so that the net electric force on any of the corner particles is zero?

Answer 1

From Coulomb's Law

$F=kq_{1}q_{2}/r^{2}$ Where k is coulomb constant = 9x10⁹ and r is the distance between charges q₁ and q₂

Lets find Force on charge at corner B

Force due to charge 3uC at corner A , F_A = 9*10⁹ * 3*10^-6 *3*10^-6 / 0.02²

F_A = 202.5 N (+ve x direction)

Force due to charge 3uC at corner C , F_C =  9*10⁹ * 3*10^-6 *3*10^-6 / 0.02²

F_C = 202.5 N (+ve y direction)

Distance from corner D to B = $\sqrt{2}$ *0.02 = 0.0283 m

Force due to charge 3uC at corner D , F_D =  9*10⁹ * 3*10^-6 *3*10^-6 / 0.0283²

F_D = 101.137 N (at angle 45 deg to x axis)

center E is in the middle of diagonal DB ,so r = 0.0141 m

Force due to charge 6uC at centerE , F_E =  9*10⁹ * 6*10^-6 *3*10^-6 / 0.0141²

F_E = 814.85 N (at angle 45 deg to x axis)

Split F_E and F_D to its components and ADD

Force on x direction

Fx = F_A + F_Dcos45 + F_Ecos45 ---------------------(1)

= 202.5 + 101.137* cos45 +  814.85cos45

= 850.2 N

Fy = F_C + F_Dsin45 + F_Esin45

=202.5 + 101.137* sin45 +  814.85sin45

= 850.2 N

total force F = $\sqrt{Fx^{2}+Fy^{2}}$ = 1202.36 N

ANSWER : F = 1202.36 N

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(b)electric force on corner particles to zero

from equation (1)

if Fx = F_A + F_Dcos45 - F_Ecos45 =0 horizontal force is balanced

find F_E everythg else is unchanged

F_E = (F_A + F_Dcos45 )/cos45

= (202.5 + 101.137* cos45)/cos45

F_E=387.51 N

k*3*10^-6 * X /r² = 387.51 N here r =a/ $\sqrt{2}$ =0.0141 m

X*9*10⁹ *3*10^-6 /(0.0141)² =387.51 N

X = 387.51 *(0.0141)² /27000

X= -2.853uC

ANSWER : change 6uC to -2.853uC to reduce net electric force on all corner particles to ZERO