Where Should a 30nC Charge Be Placed to Nullify Electric Field at the Origin?

[LeeroyJenkins]

Homework Statement

In free space, Q1 = 10nc at P1(0,-4,0) and Q2 = 20nC at P2(0,0,4).
a. Find E at the origin (answered)
E(r) = 5.617ay - 11.235az
b. Where should a 30nC point charge be located so E=0 at the origin? (need help)

Homework Equations

E(r) = (1/4(pi)epsilon) sum ((Qm)am/(|r-rm|^2))
formula for electric field intensity

The Attempt at a Solution

E = 0, Q3 = 30nC
E = 0 = (1/4(pi)epsilon) ( (Q1)a1/(|r-r1|^2))+ ...)
a3 = (0-x)ax + (0-y)ay + (0-z)az/ |r-r3|
x = 0
P3 = (0, y, z)

5.617ay - 11.235 az = ((Q3)a3/(|r-r3|^2))

*5.617ay - 11.235 az = Q3((0-y)ay + (0-z)az/ |r-r3|^3)
**|r-r3| = sqrt(0^2 + (0-y)^2 + (0-z)^2)
* and ** aren't enough to solve I'm missing one thing any help would be appreciated.

 

[trav]

(0, -5.62, +11.23) = (3x10^-8+12)/(4pi8.854) (0, -y, z)\(y ²+z ²) 1.5 ; -5.62 = 270 -y/(y ²+z ²) 1.5 and 11.23 = 270 -z/ (y ² + z ²); -0.5 = y/z, (y ² + 4y ²)1.5=48y, y=2.07, z = -4.14, The location is (0, 2.07, -4.14). The trick is dividing y by z!

 

[nlightner]

To find the position of Q3 where E=0 at the origin, we can set up the equation as follows:

E = 0 = (1/4(pi)epsilon) ( (Q1)a1/(|r-r1|^2))+ ...) + ((Q2)a2/(|r-r2|^2)) + ((Q3)a3/(|r-r3|^2))

Since we know the values of Q1, Q2, and E(r) at the origin, we can plug these in and solve for the position of Q3:

0 = (1/4(pi)epsilon) ( (10nC)(-4ay)/(|r-(0,-4,0)|^2))+ ...) + ((20nC)(4az)/(|r-(0,0,4)|^2)) + ((Q3)a3/(|r-r3|^2))

Simplifying, we get:

0 = (-40ay)/(16) + (80az)/(16) + ((Q3)a3)/(|r-r3|^2)

0 = (-2.5ay) + (5az) + ((Q3)a3)/(|r-r3|^2)

Since we want E=0 at the origin, we can set the coefficients of ay and az to 0, and solve for the position of Q3:

0 = (Q3)a3/(|r-r3|^2)

0 = (30nC)a3/(|r-r3|^2)

Solving for |r-r3|, we get:

|r-r3| = (30nC)/(a3)

Since a3 is a unit vector, we can assume that it has a magnitude of 1. Therefore, |r-r3| = 30nC.

This means that the distance between Q3 and the origin should be 30nC. We can also see from the equation that the position of Q3 should be in the direction of a3 (0, y, z), where y and z can take any value as long as the magnitude of |r-r3| is 30nC.

Therefore, the position of Q3 can be any point on a circle with a radius of 30nC centered at the origin.