Three-point charges are arranged on a line. Charge q3= +5.00 nC is at origin. Charge q2= -3.00 nC is at x= +4.00 cm. Charge q1 is at x = +2.00 cm. What is the magnitude and sign of q1 if the net force on q3 is zero?
To find the magnitude and sign of q1, we need to consider the forces exerted on q3 by q1 and q2. Since the net force on q3 is zero, the magnitudes of these forces must be equal.
The force between two charges is given by Coulomb's Law:
F = k * (|q1| * |q3|) / r^2
Where:
- F is the force
- k is the electrostatic constant (k ≈ 8.99 x 10^9 N m^2 / C^2)
- |q1| and |q3| are the magnitudes of the charges
- r is the distance between the charges
Since the forces on q3 due to q1 and q2 are equal, we can set up the following equation:
k * (|q1| * |q3|) / (2^2) = k * (|q2| * |q3|) / (4^2)
Simplifying the equation:
|q1| / 4 = |q2| / 16
Now we substitute the given charge values:
|q1| / 4 = 3.00 nC / 16
To find the magnitude of q1, we cross multiply and solve for |q1|:
|q1| = (3.00 nC) * 4 / 16
|q1| = 0.75 nC
So the magnitude of q1 is 0.75 nC.
To determine the sign of q1, we observe that q2 is negative. Since the forces on q3 due to q1 and q2 must be equal, and the force due to q2 is towards q3, the force due to q1 must also be towards q3. Thus, q1 must be positive.
Therefore, the magnitude of q1 is 0.75 nC, and its sign is positive.