Consider point charges + Q and +9Q, which are separated by 12 meters. At what point, on a line between the two charges, would it be possible that the electric field would be zero.
A. There is no such point possible
B. 9 meters from the +Q charge
C. 9 meters from the +9 Q charge
D. 1 meter from the +Q charge
E. None of the above
Show work
Let x be the distance from the Q charge.
E=0=kQ/x^2 -k9Q/(12-x^2
or
9x^2=144-24x+x^2
or
8x^2+24x-144=0
Use the quadratic equation.
A. There is no such point possible
B. 9 meters from the +Q charge
C. 9 meters from the +9 Q charge
D. 1 meter from the +Q charge
E. None of the above
I don't understand how you use the quadratic formula to solve for this problem
getting x = 1.5 + -3.9686i...
To find the point at which the electric field is zero, we can use the concept of superposition of electric fields.
According to Coulomb's law, the electric field produced by a point charge Q at a distance r from it is given by:
E = k * Q / r^2
Similarly, the electric field produced by a point charge 9Q at a distance 12 - r from it is:
E' = k * (9Q) / (12 - r)^2
At the point where the electric field is zero, the two electric fields must cancel each other out. Therefore, we have:
E + E' = 0
Substituting the values of E and E', we get:
k * Q / r^2 + k * (9Q) / (12 - r)^2 = 0
Simplifying the equation, we get:
Q / r^2 + 9Q / (12 - r)^2 = 0
Now, multiplying both sides by (12 - r)^2, we get:
Q * (12 - r)^2 + 9Q * r^2 = 0
Expanding and rearranging the equation, we get:
Q * (144 - 24r + r^2) + 9Q * r^2 = 0
144Q - 24Qr + Qr^2 + 9Qr^2 = 0
Simplifying further, we have:
144Q - 15Qr + 10Qr^2 = 0
Dividing both sides by 5Q, we get:
28.8 - 3r + 2r^2 = 0
Rearranging the equation, we have:
2r^2 - 3r + 28.8 = 0
To solve this quadratic equation, we can use the quadratic formula:
r = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 2, b = -3, and c = 28.8. Substituting these values into the formula, we get:
r = (-(-3) ± √((-3)^2 - 4 * 2 * 28.8)) / (2 * 2)
Simplifying further, we have:
r = (3 ± √(9 - 230.4)) / 4
r = (3 ± √(-221.4)) / 4
Since the value inside the square root is negative, it means there are no real solutions. Therefore, it is not possible to find a point on the line between the two charges at which the electric field is zero.
Therefore, the correct answer is A. There is no such point possible.