Where electric field is zero

[craigl34]

Homework Statement

Two particles with positive charges q_1 and q_2 are separated by a distance s.

Along the line connecting the two charges, at what distance from the charge q_1 is the total electric field from the two charges zero?

(Express your answer in terms of some or all of the variables s, q_1, q_2 and K =1/(4*pi*$\epsilon$. If your answer is difficult to enter, consider simplifying it, as it can be made relatively simple with some work.)

Homework Equations

E = K*(q/(d)^2)
E_net = E1 + E2

The Attempt at a Solution

Since both the charges are positive, my E_net = E1 - E2. So I can solve this by finding where E1 and E2 are equal.

Setting the two equations equal I get K(q_1/s^2) = K(q_2/s^2)

Since I'm just concerned with finding the distance from q_1 to the point where the e-field is zero, wouldn't my equation be:

s_1 = (q_1 - (q_2/(s_2)^2)

Since none of the variables are defined, I'm having a hard time figuring out how to choose my 's' (distance). Wouldn't the distance ('s') depend on the magnitude of the charge on q_1 and q_2? How can I show that algebraically without somehow renaming the distance variable something other than 's'?

 

[Tomer]

I'm not sure I understood you.
s is a given distance between the two charges.
You need to find a point between them in which the field is zero - in other words, like you wrote, that E1 = E2.
Let's assume this point has the distance "x" from q1.
What is then the distance of this point from q2? (draw it to yourself if you're having a hard time).

Then use the appropriate formulas to deduce what x should be - in a similar manner to what you've done, but right this time :-)

 

[craigl34]

See I understand that the distance from q_2 would be equal to (total separation 's' - distance from q_2), I just don't know how to represent that with only being able to use the variable 's' representing the total separation.

 

[craigl34]

Nevermind, I figured it out.

I didn't realize that if I put in another variable into the equation 'x = distance from q_1' that it would end up cancelling out during the simplification process.

The answer for me would be:

x = s/1 + sqrt(q_2/q_1)

 

[rwisz]

Your approach is correct, but your final equation is not quite correct. The distance s_1 should be the distance from q_1 where the electric field is zero, so it should be equal to the distance s from q_1 to the point where the electric field from q_2 is equal to the electric field from q_1. This can be written as:

s_1 = s

Substituting this into your equation, we get:

s = (q_1 - (q_2/s^2)

Now, to solve for s, we need to rearrange the equation:

s^3 = q_1/(q_1-q_2)

s = (q_1/(q_1-q_2))^(1/3)

This is the distance from q_1 to the point where the electric field is zero. As you can see, the distance does not depend on the magnitude of the charges q_1 and q_2, but only on their ratio (q_1/q_2). This makes sense because the electric field is a property of the charges, not the distance between them. Therefore, the distance where the electric field is zero will always be the same for two charges with the same ratio of charges, regardless of their individual magnitudes.