How to prove an electrostatics concept mathematically

[channel1]

"The third point charge should be placed at a location at which the forces on the third charge due to each of the other two point charges cancel. There can be no such place except on the line between the two point charges."

i need to be able to prove a similar statement mathematically (meaning i need to prove that the third point charge cannot be on the right or left of the system but that it must be in the middle). i have no idea how to go about this, please show me how to prove the above statement mathematically so that i can work out my homework assignment problem on my own. (the link below is to the problem containing the statement above---scroll to the top of the second page.)

i tried solving this using coulombs law, but it requires that i make an initial assumption for my R---which i can't do without first proving on which area within the system i need to place q_3

http://www.physics.wisc.edu/undergrads/courses/spring09/248/HWSolutions/HW5Solutions.pdf

 

[Delphi51]

It would sure help if you could prove the 3rd charge must be on the line through Q1 and Q2. This could be done by taking that line as the x-axis, and then considering the forces in the y-direction. What happens to the y forces on Q3 if Q3 is above the line? Below?

Once you have that Q3 lies on the line, you could take its position to be x distance from Q1 or Q2. Then you should be able to use Coulomb's law to get an expression for the force on Q3. It will have an x in it. You know the force is zero, so you can calculate x, proving that it is less than the Q1 to Q2 distance, hence Q3 lies between them.

 

[PeterO]

"The third point charge should be placed at a location at which the forces on the third charge due to each of the other two point charges cancel. There can be no such place except on the line between the two point charges."

i need to be able to prove a similar statement mathematically (meaning i need to prove that the third point charge cannot be on the right or left of the system but that it must be in the middle). i have no idea how to go about this, please show me how to prove the above statement mathematically so that i can work out my homework assignment problem on my own. (the link below is to the problem containing the statement above---scroll to the top of the second page.)

i tried solving this using coulombs law, but it requires that i make an initial assumption for my R---which i can't do without first proving on which area within the system i need to place q_3

http://www.physics.wisc.edu/undergrads/courses/spring09/248/HWSolutions/HW5Solutions.pdf

The two charges in the system are both positive. If a third positive charge is introduced, it will experience a repulsive force from each of the two originals.

To the left of the two charges, that means both forces would be left - so they could never cancel.
To the right of the two charges, that means both forces would be right - so they could never cancel.

only in between could you get one force to the left and one to the right.

If the third charge introduced was negative, the situation is little different, especially from the cancelling out point of view.

If the third charge was introduced above or below the line joining the two original charges [rather than ON the line joining them] then the resulting forces would both have an up or down component, which could never cancel.

 

[channel1]

thanks a bunch :-) i managed to draw diagrams to represent all the hypothetical situations so that made everything much clearer