four  particles  form  a   square. the  charges  are  q1 = q4 = Q  &  q2 = q3  = q

(a) What  is  Q/q  if  the  net  electostatic  force  on  particles  1& 4  is  zero? (b) is  there  any  value  of  q  that  makes  the  net  electrostatic  force  on  each  of  the  four  particles  zero? Explain.

Look at teh above figure

(a)

the net x-component of force on particle 1 will be given as

F_1x = F₂ + F_4x

 there is not effect due to presence of 3 in x-direction

so,

F_1x= kq.Q / a² + [kQ.Q / (a√2)²].cos45

= kQ. [ q / a² + (Q / 2a² )cos45 ]

now, as per given

if the net electostatic force on particles 1& 4 is zero

then,

F_1x= 0

or

 kQ. [ q / a² + (Q / 2a² )cos45 ] = 0

by solving further, we get

q + Q / 2√2 = 0 

or

Q/q = -2√2

..

(b)

the net y-component of force on particle 2 will be given as

F_2y = F_3y + F_4y

 there is not effect due to presence of 1 in y-direction

so,

F_1x= [kq.q / (a√2)²].sin45 +  kq.Q / a² 

= kq. [ (q / 2a² )sin45  +  Q / a² ]

now, as per given

if the net electrostatic force on particles 1& 4 is zero

then,

F_2y= 0

or

 kq.[ (q / 2a² )sin45  +  Q / a² ]  = 0

by solving further, we get

q / 2√2 + Q  = 0 

or

Q/q = - 1 / (2√2)

.

so, as the ratios of  Q/q in both cases do not match it will be impossible for us to make a situation in which force on all charges will be equal. No, equilibrium situation will be made here.