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

 

[rwisz]

To prove this statement mathematically, we can use the principle of superposition. This states that the total force on a charge due to a collection of other charges is equal to the vector sum of the individual forces on that charge due to each individual charge. In other words, the total force on a charge is the sum of the forces exerted on it by each of the other charges.

In this case, we have two point charges, q1 and q2, and we want to find the location of a third point charge, q3, where the forces on q3 due to q1 and q2 cancel out. We can represent this mathematically as:

F_total = F_1 + F_2

Where F_total is the total force on q3, F_1 is the force on q3 due to q1, and F_2 is the force on q3 due to q2.

Using Coulomb's law, we can write the forces F_1 and F_2 as:

F_1 = (k*q_1*q_3)/r_1^2 and F_2 = (k*q_2*q_3)/r_2^2

Where k is the Coulomb constant, q_1 and q_2 are the charges of q1 and q2 respectively, q_3 is the charge of q3, and r_1 and r_2 are the distances between q1 and q3 and between q2 and q3 respectively.

Now, we need to find the value of q_3 and the distance r_3 (the distance between q3 and the line connecting q1 and q2) that will make F_total equal to zero. This means that we need to solve the equation:

F_total = F_1 + F_2 = 0

Substituting the expressions for F_1 and F_2, we get:

(k*q_1*q_3)/r_1^2 + (k*q_2*q_3)/r_2^2 = 0

Simplifying and rearranging, we get:

q_3 = -(r_1/r_2)*q_2

This means that the value of q_3 is dependent on the ratio of the distances r_1 and r_2. Now, we need to find the value of r_3 that makes F_total equal to zero. To do this, we