Charges q1 and q2 lie on x axis at points x= -a and x= 5, respectively.

(a) how must q1 and q2 be related for the net force on charge +Q at x = a/2 to be zero ?
(b) and if +Q is placed at x = 3a/2.

Q at a/2

d1 = 1.5 a
d2 = 5 -.5 a

for same E field from each at a/2 (both + charges)
q1/d1^2 = q2/d2^2
d1/2.25a^2 =q2/(25 - 5a+.25a^2)

by the way, unless you mean 5a and not 5, the problem is not well defined. q1 and q2 must be same sign if Q is between them and of opposite signs if Q is to the right of 5

Ops I meant the other x is (a)

My bad

To determine how q1 and q2 must be related for the net force on charge +Q at x = a/2 to be zero, we need to consider the concept of electric forces and their properties.

(a) To find the relationship between q1 and q2, we need to understand the forces acting on charge +Q at x = a/2.

1. Electric force between q1 and +Q:
The electric force between q1 and +Q follows Coulomb's Law, given by:
F1 = (k * q1 * Q) / r1^2
where F1 is the force between q1 and +Q, k is the Coulomb constant, q1 and Q are the charges, and r1 is the distance between them.

2. Electric force between q2 and +Q:
Similarly, the electric force between q2 and +Q is given by:
F2 = (k * q2 * Q) / r2^2
where F2 is the force between q2 and +Q, k is the Coulomb constant, q2 and Q are the charges, and r2 is the distance between them.

Now, let's consider the forces on +Q at x = a/2:

- The force F1 between q1 and +Q is directed towards the left since q1 is located to the left of +Q.
- The force F2 between q2 and +Q is directed towards the right since q2 is located to the right of +Q.

For the net force on +Q to be zero, F1 and F2 must be equal in magnitude but opposite in direction:

F1 = F2
(k * q1 * Q) / r1^2 = (k * q2 * Q) / r2^2
q1 / r1^2 = q2 / r2^2

Since q1 is located at x = -a, r1 = a/2.
Similarly, since q2 is located at x = 5, r2 = 3a/2.

Substituting these values into the equation:

q1 / (a/2)^2 = q2 / (3a/2)^2
q1 / (a^2/4) = q2 / (9a^2/4)
q1 / a^2 = q2 / 9a^2
q1 = q2 / 9

Therefore, q1 must be equal to one-ninth of q2 for the net force on charge +Q at x = a/2 to be zero.

(b) If +Q is placed at x = 3a/2, we can use the same approach to find the relationship between q1 and q2.

The distance between q1 and +Q (r1) will now be a/2 + 3a/2 = 2a, as +Q is located 2a units to the right of q1.

Similarly, the distance between q2 and +Q (r2) will be 3a/2 - 5 = 3a/2 - 10/2 = 3a/2 - 5 = a/2 - 5.

Applying the condition for the net force to be zero:
q1 / r1^2 = q2 / r2^2

Substituting the values:
q1 / (2a)^2 = q2 / (a/2 - 5)^2

Simplifying the equation will yield the relationship between q1 and q2 for the net force on +Q at x = 3a/2 to be zero.