Two particles having charges of 0.580 nC and 20.9 nC are separated by a distance of 1.10 m .
Where would the net electric field be zero if one of the charges were negative?
Enter your answer as a distance from the charge initially equal 0.580 nC .
keq1/r^2 = keq2/(1.1-r)^2 (the ke's cancel out)
-.580 x 10^-9/r^2 = 20.9 x 10^-9/(1.1+r)^2
(20.9 x 10^-9)r^2 = (-.580 x 10^-9)(1.1+r)^2
(-36.034)r^2 = (r+1.1)^2
-7.003r = 1.1
r = .157
I added that to the 1.1 to get the total distance of 1.257 m, but that answer is wrong. Could someone please help me figure out what I'm doing wrong?
To solve this problem, you've correctly set up the equation based on the principle that the net electric field is zero at a specific distance from the charge. However, there seems to be a mistake in your calculation. Let's go through the steps again to find out what might have gone wrong.
Let's assume the initial charge of 0.580 nC is positive, and the other charge of 20.9 nC is negative. The equation you've written is:
(0.580 x 10^-9)/r^2 = (-20.9 x 10^-9)/(1.1 - r)^2
Now, let's solve this equation step by step:
1. Cross-multiply to remove the denominators:
(0.580 x 10^-9)(1.1 - r)^2 = (-20.9 x 10^-9)(r^2)
2. Expand both sides of the equation:
(0.580 x 10^-9)(1.21 - 2.2r + r^2) = (-20.9 x 10^-9)(r^2)
3. Distribute and simplify:
0.7028 x 10^-9 - 1.2776 x 10^-9r + 0.580 x 10^-9r^2 = -20.9 x 10^-9r^2
4. Move all terms to one side of the equation:
0.7028 x 10^-9 + (1.2776 x 10^-9 - 0.580 x 10^-9)r^2 + 20.9 x 10^-9r^2 = 0
5. Combine like terms:
21.8776 x 10^-9r^2 + (0.6974 x 10^-9)r + 0.7028 x 10^-9 = 0
Now, you can solve this quadratic equation using the quadratic formula or factoring. The quadratic formula is often the simplest method:
r = (-b ± √(b^2 - 4ac)) / (2a)
Let's identify the coefficients in our quadratic equation:
a = 21.8776 x 10^-9
b = 0.6974 x 10^-9
c = 0.7028 x 10^-9
Now, we can substitute these values into the quadratic formula to find the value of r.
r = (-0.6974 x 10^-9 ± √((0.6974 x 10^-9)^2 - 4 * 21.8776 x 10^-9 * 0.7028 x 10^-9)) / (2 * 21.8776 x 10^-9)
r = (-0.6974 x 10^-9 ± √(0.48720076 x 10^-18 - 6.092523776 x 10^-18)) / (43.7552 x 10^-9)
r = (-0.6974 x 10^-9 ± √(-5.605323016 x 10^-18)) / (43.7552 x 10^-9)
At this point, we have a problem because the value inside the square root is negative. This indicates that there is no real solution for r in this case. Therefore, the net electric field will never be zero if one of the charges is negative.