Finding the position for charge Q3

[tre2k3]

Here is my question they gave me:
Now the charges Q1= 1.73x10-6 C and Q2= -2.46 x10-6 C are fixed at their positions, distance 0.257 m apart, and the charge Q3= 3.03 x10-6 C is moved along the straight line. For what position of Q3 relative to Q1 is the net force on Q3 due to Q1 and Q2 zero? Use the plus sign for Q3 to the right of Q1.

and the hint:
Write down an expression for the force on the third charge with the distance from the first charge as a variable. You do not need to solve a quadratic equation.

What I did to try to get the answer was have the the sum of F21 and F23 equal to zero. F23 has the unknown, r, which is the distance from 2 to 3. So I just try to solve for r in order to get the answer, but it is not right.

 

[anathema]

Thank you for your question. The key to solving this problem is to use Coulomb's Law, which states that the force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. In this case, we have three point charges, Q1, Q2, and Q3, with fixed positions and charges. The net force on Q3 will be zero when the forces exerted by Q1 and Q2 on Q3 cancel each other out.

To determine the position of Q3 where this happens, we can set up an equation using Coulomb's Law. The force exerted by Q1 on Q3 can be represented as F13, and the force exerted by Q2 on Q3 can be represented as F23. We can write these forces as follows:

F13 = k(Q1Q3)/r^2
F23 = k(Q2Q3)/(0.257-r)^2

Where k is the Coulomb's constant and r is the distance between Q1 and Q3. We can then set these two forces equal to each other and solve for r to find the position where the net force on Q3 is zero:

F13 = F23
k(Q1Q3)/r^2 = k(Q2Q3)/(0.257-r)^2
(Q1Q3)/r^2 = (Q2Q3)/(0.257-r)^2
(Q1Q3)(0.257-r)^2 = (Q2Q3)r^2
(Q1Q3)(0.066649-0.514r+r^2) = (Q2Q3)r^2
0.066649Q1Q3 - 0.514Q1Q3r + Q1Q3r^2 = Q2Q3r^2
Q1Q3r^2 + (0.514Q1Q3-Q2Q3)r + (-0.066649Q1Q3) = 0

This is a quadratic equation, but as the hint suggests, we do not need to solve for r. Instead, we can use the quadratic formula to find the value of r that makes the equation equal to zero. This value of r will be the distance from Q1 to Q3 where the net force on Q3 is zero. The plus sign in the hint refers to

 

[quicknote]

As a scientist, you are on the right track in your approach to solving this problem. You correctly noted that the net force on Q3 will be zero when the sum of the forces due to Q1 and Q2 is equal and opposite. This can be represented mathematically as:

F21 + F23 = 0

where F21 is the force due to Q1 on Q3 and F23 is the force due to Q2 on Q3. We can express these forces using Coulomb's law:

F21 = k(Q1Q3)/r^2
F23 = k(Q2Q3)/r^2

where k is the Coulomb's constant and r is the distance between the charges. We can then substitute these expressions into our equation and solve for r:

k(Q1Q3)/r^2 + k(Q2Q3)/r^2 = 0
k(Q1Q3 + Q2Q3)/r^2 = 0
(Q1 + Q2)/r^2 = 0
r = √(Q1 + Q2)/0

We can see that the distance r will be zero when the sum of the charges Q1 and Q2 is zero. This means that for the net force on Q3 to be zero, the charge Q3 must be placed at a distance of 0 from Q1 and Q2. In other words, Q3 must be located directly between Q1 and Q2 in order for the net force to be zero.

In conclusion, to find the position for charge Q3 where the net force is zero, we must place Q3 at a distance of 0 from Q1 and Q2. This can be achieved by placing Q3 directly between Q1 and Q2.