Two point charges are fixed on the y axis: a negative point charge q1 = -26 µC at y1 = +0.21 m and a positive point charge q2 at y2 = +0.38 m. A third point charge q = +8.5 µC is fixed at the origin. The net electrostatic force exerted on the charge q by the other two charges has a magnitude of 24 N and points in the +y direction. Determine the magnitude of q2.
Set the sum of the Coulomb forces on q equal to 24 N (in +y direction) , and solve for the only unknown, q2.
The force due to q1 is up (+y direction) and the force due to q2 is down.
k(8.5*10^-6)*[26*10^-6/(0.21)^2 -q2/(0.38)^2] = 24
k is the Coulomb constant.
To determine the magnitude of q2, we need to use Coulomb's law to calculate the individual forces exerted by q1 and q2 on q and then find the magnitude of q2 that satisfies the given conditions.
1. Coulomb's law states that the magnitude of the electrostatic force between two point charges is given by:
F = k * |q1 * q2| / r^2
Where:
F is the magnitude of the force,
k is the electrostatic constant which is approximately 8.99 x 10^9 Nm^2/C^2,
q1 and q2 are the magnitudes of the charges,
r is the distance between the charges.
2. Calculate the distance between q1 and q as follows:
r1 = y1 - 0
r1 = 0.21 m
3. Calculate the distance between q2 and q as follows:
r2 = y2 - 0
r2 = 0.38 m
4. Determine the forces exerted by q1 and q2 on q.
For q1:
F1 = k * |q1 * q| / r1^2
For q2:
F2 = k * |q2 * q| / r2^2
5. Given that the net force on q has a magnitude of 24 N and points in the +y direction, we can set up the following equation:
F2 - F1 = 24 N
6. Substitute the expressions for F1 and F2 into the equation and solve for q2:
k * |q2 * q| / r2^2 - k * |q1 * q| / r1^2 = 24 N
q2 / r2^2 - q1 / r1^2 = 24 N / (k * q)
7. Substitute the given values:
q1 = -26 µC
q = +8.5 µC
r1 = 0.21 m
r2 = 0.38 m
q2 / (0.38 m)^2 - (-26 µC) / (0.21 m)^2 = 24 N / (8.99 x 10^9 Nm^2/C^2 * 8.5 µC)
8. Solve the equation for q2:
q2 = [(24 N) + (-26 µC / (0.21 m)^2) * (0.38 m)^2] * (8.99 x 10^9 Nm^2/C^2 * 8.5 µC)
q2 = 2.109 x 10^-5 C
Therefore, the magnitude of q2 is approximately 2.109 x 10^-5 C.
To determine the magnitude of q2, let's break down the problem into steps:
Step 1: Calculate the electrostatic force exerted by q1 on q.
The electrostatic force between two charged particles can be calculated using Coulomb's law:
F = k * |q1 * q |/ r1^2,
where F is the force, k is Coulomb's constant (8.99 x 10^9 Nm^2/C^2), q1 and q are the charges, and r1 is the distance between the charges.
In this case, q1 = -26 µC, q = +8.5 µC, and r1 = 0.21 m. Plugging in these values:
F(q1 on q) = (8.99 x 10^9 Nm^2/C^2) * (|-26 x 10^-6 C * 8.5 x 10^-6 C| / (0.21 m)^2) = x N (where x is the calculated value).
Step 2: Calculate the electrostatic force exerted by q2 on q.
Since the net electrostatic force exerted by q1 and q2 on q is in the +y direction, the force exerted by q2 on q must also be in the +y direction.
So, the force exerted by q2 on q can be calculated as:
F(q2 on q) = 24 N - F(q1 on q) = 24 N - x N = y N (where y is the calculated value).
Step 3: Calculate the distance between q2 and q.
The distance between q2 and q is given as r2 = 0.38 m.
Step 4: Calculate the magnitude of q2.
Using Coulomb's law again, we can now find the value of q2:
y N = (8.99 x 10^9 Nm^2/C^2) * (|q2 * 8.5 x 10^-6 C| / (0.38 m)^2) = (8.99 x 10^9 Nm^2/C^2) * (q2 * 8.5 x 10^-6 C / (0.38 m)^2).
Solving this equation for q2 will give us the magnitude of q2.