Two point charges are fixed on the y axis: a negative point charge q1 = -26 μC at y1 = +0.17 m and a positive point charge q2 at y2 = +0.35 m. A third point charge q = +9.5 μC is fixed at the origin. The net electrostatic force exerted on the charge q by the other two charges has a magnitude of 26 N and points in the +y direction. Determine the magnitude of q2.
To determine the magnitude of q2, we can use Coulomb's Law and the principle of vector addition.
Coulomb's Law states that the magnitude of the electrostatic force between two point charges q1 and q2 is given by the equation:
F = k * |q1| * |q2| / r^2
Where F is the force, k is Coulomb's constant (8.99 x 10^9 N·m^2/C^2), |q1| and |q2| are the magnitudes of the charges, and r is the distance between the two charges.
In this problem, the net electrostatic force exerted on charge q by the other two charges (q1 and q2) has a magnitude of 26 N and points in the +y direction. We need to use this information to solve for q2.
Let's break down the problem step by step:
Step 1: Calculate the distance between q and q1 (r1)
The distance between q and q1 is the distance between their y-coordinates. In this case, it is the absolute difference between y1 and the origin:
r1 = |y1 - 0| = 0.17 m
Step 2: Calculate the distance between q and q2 (r2)
Similarly, the distance between q and q2 is the absolute difference between y2 and the origin:
r2 = |y2 - 0| = 0.35 m
Step 3: Use Coulomb's Law to find the magnitude of q2
Now we have all the information we need to apply Coulomb's Law. We have the magnitudes of charges q1 and q2, the net electrostatic force (F = 26 N), and the distances r1 and r2.
We'll focus on the net force between charge q and charge q2, as the question asks for the magnitude of q2. The net force, F, is the vector sum of the individual forces between q and q1, and q and q2. Since the net force points in the +y direction, the vector sum of these forces is in the same direction.
Mathematically, we can represent this vector addition as:
F = F1 + F2
Since the magnitudes of q1 and q2 are given (|q1| = 26 μC and |q2| = unknown), we can write Coulomb's Law equation for each force:
F1 = k * |q1| * |q| / r1^2 (Force between q and q1)
F2 = k * |q2| * |q| / r2^2 (Force between q and q2)
Substituting these equations into the vector sum equation, we get:
26 N = k * |q1| * |q| / r1^2 + k * |q2| * |q| / r2^2
Now we can plug in the known values:
26 N = (8.99 x 10^9 N·m^2/C^2) * (26 μC) * (9.5 μC) / (0.17 m)^2 + (8.99 x 10^9 N·m^2/C^2) * |q2| * (9.5 μC) / (0.35 m)^2
Simplifying and solving this equation will give us the magnitude of q2.