Let q3 be the value of the third point charge which is located at y = x. We will apply the principle of superposition to find the location where net electric force on point charge q3 is zero.
The force on q3 due to q1 is given by Coulomb's Law:
F13 = k * |q1 * q3| / (x - 6.70)^2
Similarly, the force due to q2 on q3 is given by:
F23 = k * |q2 * q3| / (x + 3.60)^2
Since the net force on charge q3 is zero, these two forces must have the same magnitude but opposite direction. Therefore, we can equate the expressions for F13 and F23:
k * |q1 * q3| / (x - 6.70)^2 = k * |q2 * q3| / (x + 3.60)^2
Now, we can cancel out the k * |q3| from both sides, and rearrange the equation:
-9.10 / (x - 6.70)^2 = -7.70 / (x + 3.60)^2
Cross-multiplying and taking the square root, we get:
sqrt(-9.10*(x + 3.60)^2)=sqrt(-7.70*(x - 6.70)^2)
Now, squaring both sides we get:
-9.10*(x + 3.60)^2=-7.70*(x - 6.70)^2
Now, divide both sides by -1 to remove the negative sign:
9.10*(x + 3.60)^2=7.70*(x - 6.70)^2
And then, divide both sides by 7.70 and 9.10:
(x + 3.60)^2 / 7.70 = (x - 6.70)^2 / 9.10
Now, replace x by y:
(y + 3.60)^2 / 7.70 = (y - 6.70)^2 / 9.10
As this is a quadratic equation after this simplification, we get two solutions:
y = -3.21 m and y = 1.53 m
However, since charge q2 is already at a location y = -3.60 m, the point charge q3 cannot be located at y = -3.21 m. Therefore, the third charge must be located at:
y = 1.53 m.
Three point charges lie in a straight line along the y-axis. A charge of q1 = -9.10 µC is at y = 6.70 m, and a charge of q2 = -7.70 µC is at y = -3.60 m. The net electric force on the third point charge is zero. Where is this charge located?
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