Where is the electric field is zero?

[3.14159265358979]

two charges of 1.5X10^-6 c and 3.0X10^-6 c are 0.2 m apart. Where is the electric field between them equal to zero?

heres what i got...

the electric field will be 0 when the field strength of the first charge minus the field strength of the second charge equals 0.

therefore,

q(1)-----x------P-------(0.2 - x)-------q(2)

where...
P is where the electric field equals 0
q(1) is the first charge
q(2) is the second charge
x is the distance (in metres) from the charge

[ kq(1) / (x)^2 ] - [ kq(2)/ ((0.2-x)^2) ] = 0

from here,
i cancel out the k's
find the common denominator and cancel it out once my numerator is expanded
try and use the quadratic equation to solve for x. however, when i try to solve for x i get a complex number...what am i doing wrong? the book says the answer is 0.08m (approx.) if you know a faster and much easier way, please do tell...thanks a bunch...

 

[Janitor]

Let the neutral point be a distance r from the smaller charge Q and a distance R-r from the bigger charge 2Q. Note that

kQ/r^2 - 2kQ/(R-r)^2 = 0 expresses the neutrality of the electric force along the line between the charges, by Coulomb's law.

After some algebraic manipulation you get

r^2 + 2Rr - R^2 = 0.

Applying the quadratic formula,

r = -R +/- sqrt(2)R. (I am too lazy to figure out how to stack the plus or minus symbol, so I wrote it as +/-.)

Discard the root that does not lie between the two charges. This leaves you with

r = [sqrt(2) - 1]R.

Here R=0.2 m, so you have

r = [1.414 - 1] 0.2

which works out to about 0.08 meters, that being the distance from the smaller charge.

 

[M98Ranger]

Your approach is correct, but it seems like you may have made a mistake in your calculations when solving for x using the quadratic equation. Here is a step-by-step solution for finding the value of x:

1. Set up the equation:

[ kq(1) / (x)^2 ] - [ kq(2)/ ((0.2-x)^2) ] = 0

2. Cancel out the k's:

[ q(1) / (x)^2 ] - [ q(2)/ ((0.2-x)^2) ] = 0

3. Expand the denominators:

[ q(1) / x^2 ] - [ q(2) / (0.04 - 0.4x + x^2) ] = 0

4. Find the common denominator:

[ q(1)(0.04 - 0.4x + x^2) / x^2(0.04 - 0.4x + x^2) ] - [ q(2)(x^2) / x^2(0.04 - 0.4x + x^2) ] = 0

5. Simplify:

[ 0.04q(1) - 0.4q(1)x + q(1)x^2 - q(2)x^2 ] / [ x^2(0.04 - 0.4x + x^2) ] = 0

6. Combine like terms:

[ (q(1) - q(2))x^2 + 0.4q(1)x + 0.04q(1) ] / [ x^2(0.04 - 0.4x + x^2) ] = 0

7. Set the numerator equal to 0:

(q(1) - q(2))x^2 + 0.4q(1)x + 0.04q(1) = 0

8. Use the quadratic formula to solve for x:

x = [ -0.4q(1) ± √(0.4q(1))^2 - 4(q(1) - q(2))(0.04q(1)) ] / 2(q(1) - q(2))

9. Simplify:

x = [ -0.4q(1) ± √(