Answer:
a) The net electric force on the middle sphere due to spheres 1 and 3 is 404N. b) The direction of the net electric force is to the left.
Explanation:
In order to solve this problem, we must first draw our free body diagram (See attached picture).
By definition, two opposite charges attract each other, so the force due to sphere 1 will go to the left (being the left direction negative) and the force due to sphere 3 will go to the right (being the right direction positive).
Knowing this we can use the electric force formula to calculate each of the forces:
![F_{e}=k_{e}\frac{q_{1}q_{2} }{r^{2} }](https://tex.z-dn.net/?f=F_%7Be%7D%3Dk_%7Be%7D%5Cfrac%7Bq_%7B1%7Dq_%7B2%7D%20%7D%7Br%5E%7B2%7D%20%7D)
where
=![8.99x10^{9}N\frac{m^{2} }{C^{2} }](https://tex.z-dn.net/?f=8.99x10%5E%7B9%7DN%5Cfrac%7Bm%5E%7B2%7D%20%7D%7BC%5E%7B2%7D%20%7D)
So knowing this, we can now calculate each force. Let's start with the force exerted by sphere 1 over sphere 2:
![F_{12}=8.99x10^{9}N\frac{m^{2} }{C^{2} }\frac{(71x10^{-6}C)(52x10^{-6}C) }{(0.17m)^{2} }](https://tex.z-dn.net/?f=F_%7B12%7D%3D8.99x10%5E%7B9%7DN%5Cfrac%7Bm%5E%7B2%7D%20%7D%7BC%5E%7B2%7D%20%7D%5Cfrac%7B%2871x10%5E%7B-6%7DC%29%2852x10%5E%7B-6%7DC%29%20%7D%7B%280.17m%29%5E%7B2%7D%20%7D)
Which gives me a force of:
![F_{12}=-1148.48N](https://tex.z-dn.net/?f=F_%7B12%7D%3D-1148.48N)
In this case the force will be negative because it's directed towards sphere 1, this is to the left.
We can do the same with the force due to sphere 3:
![F_{23}=8.99x10^{9}N\frac{m^{2} }{C^{2} }\frac{(46x10^{-6}C)(52x10^{-6}C) }{(0.17m)^{2} }](https://tex.z-dn.net/?f=F_%7B23%7D%3D8.99x10%5E%7B9%7DN%5Cfrac%7Bm%5E%7B2%7D%20%7D%7BC%5E%7B2%7D%20%7D%5Cfrac%7B%2846x10%5E%7B-6%7DC%29%2852x10%5E%7B-6%7DC%29%20%7D%7B%280.17m%29%5E%7B2%7D%20%7D)
Which gives me a force of:
![F_{23}=744.09N](https://tex.z-dn.net/?f=F_%7B23%7D%3D744.09N)
In this case the force is positive because it's directed towards sphere 3, this is it goes to the right.
With these two values I can now find the net force electric force on the middle sphere due to spheres 1 and 3.
∑F
![F_{net}=F_{12}+F_{23}](https://tex.z-dn.net/?f=F_%7Bnet%7D%3DF_%7B12%7D%2BF_%7B23%7D)
![F_{net}=-1148.48N+744.09N](https://tex.z-dn.net/?f=F_%7Bnet%7D%3D-1148.48N%2B744.09N)
![F_{net}=-404.39N](https://tex.z-dn.net/?f=F_%7Bnet%7D%3D-404.39N)
a) Since part a of the problem only asks us for the magnitude, then the net electric force on the middle sphere due to spheres 1 and 3 is 404.39N
b) since the answer when solving this problem was negative, this means that the force will be directed towards sphere 1, this is to the left.