4a Three point charges lie along the x axis, The positive charge q115.0C is at x 2.00 m, the positive charge q2 6.00 C is at the origin, and the net force acting on q3 is zero. What is the x coordinate of q3?
b Find the net force on q2 and its direction
a) To find the x coordinate of q3, we need to find the point where the net force acting on q3 is zero.
Since the net force on q3 is zero, the forces exerted by q1 and q2 on q3 must cancel each other out.
The force exerted by q1 on q3 is given by Coulomb's law: F1 = k * (q1 * q3) / r1^2, where k is the electrostatic constant, q1 and q3 are the charges, and r1 is the distance between them.
Similarly, the force exerted by q2 on q3 is given by F2 = k * (q2 * q3) / r2^2, where q2 and r2 are the charge and distance between q2 and q3, respectively.
Since the net force is zero, we equate the magnitudes of F1 and F2:
k * (q1 * q3) / r1^2 = k * (q2 * q3) / r2^2
Simplifying this equation, we get:
(q1 * q3) / r1^2 = (q2 * q3) / r2^2
(q1 * q3 * r2^2) = (q2 * q3 * r1^2)
(q1 * r2^2) = (q2 * r1^2)
Substituting the given values, we have:
(q1 * (0 - x)^2) = (q2 * (x - 2)^2)
(115C * (0 - x)^2) = (6C * (x - 2)^2)
Simplifying the equation further, we get:
115 * x^2 = 6 * (x - 2)^2
115 * x^2 = 6 * (x^2 - 4x + 4)
115 * x^2 = 6 * x^2 - 24 * x + 24
109 * x^2 + 24 * x - 24 = 0
This is a quadratic equation in terms of x. Solving it using the quadratic formula or factoring, we get:
x ≈ 0.205 m
Therefore, the x coordinate of q3 is approximately 0.205 m.
b) To find the net force on q2 and its direction, we need to calculate the forces exerted by q1 and q3 on q2, and then add them vectorially.
The force exerted by q1 on q2 is given by Coulomb's law: F1 = k * (q1 * q2) / r1^2, where k is the electrostatic constant, q1 and q2 are the charges, and r1 is the distance between them.
Similarly, the force exerted by q3 on q2 is given by F3 = k * (q3 * q2) / r3^2, where q3 and r3 are the charge and distance between q3 and q2, respectively.
Since we already know the x coordinates of q1, q2, and q3, we can calculate the distances r1 and r3:
r1 = x - 2 m
r3 = x m
Substituting the given values, we have:
F1 = k * (115C * 6C) / (x - 2)^2
F3 = k * (115C * q3) / x^2
The net force on q2 is the vector sum of F1 and F3:
Net Force = F1 + F3
To find the direction of the net force, we look at the sign of the net force:
If the net force is positive, it means the net force acts in the positive x-direction.
If the net force is negative, it means the net force acts in the negative x-direction.
To determine the exact magnitude and direction of the net force, we need the value of x, which we solved for in part (a). Once we substitute the value of x, we can calculate the net force and its direction.
To find the x coordinate of q3, we can use the principle of electrostatic equilibrium, which states that the net force on a charge must be zero.
Given:
Charge q1 has a magnitude of 15.0 C and is located at x = 2.00 m.
Charge q2 has a magnitude of 6.00 C and is located at the origin.
The net force acting on q3 is zero.
Let's solve for the x coordinate of q3:
Step 1: Set up the equation using Coulomb's Law:
According to Coulomb's Law, the force between two charges can be calculated using the equation:
F = k * (|q1| * |q2|) / r^2
Where F is the force between the two charges, k is the electrostatic constant (k = 9.0 × 10^9 Nm^2/C^2), |q1| and |q2| are the magnitudes of the charges, and r is the distance between the charges.
We need to calculate the force between q1 and q3, and between q2 and q3.
Step 2: Calculate the force between q1 and q3:
Since the net force on q3 is zero, the forces between q1 and q3 and between q2 and q3 must cancel each other out.
F(q1 on q3) = F(q2 on q3)
Using Coulomb's Law, we have:
k * (|q1| * |q3|) / (r1^2) = k * (|q2| * |q3|) / (r2^2)
Plugging in the given values:
(9.0 × 10^9 Nm^2/C^2) * (15.0 C * |q3|) / (2.00 m)^2 = (9.0 × 10^9 Nm^2/C^2) * (6.00 C * |q3|) / (0 m)^2
Simplifying the equation:
(15.0 C * |q3|) / 4.00 = 6.00 C * |q3|
Step 3: Solve for |q3|:
Dividing both sides of the equation by 6.00 C * |q3|:
(15.0 C * |q3|) / (4.00) = 1
Simplifying the equation:
15.0 C * |q3| = 4.00 C
Dividing both sides of the equation by 15.0 C:
|q3| = 4.00 C / 15.0 C
|q3| = 0.27 C
Since q3 is assumed to have the same sign as q1, the value of q3 is positive.
Therefore, the x coordinate of q3 is 2.00 m.
For part b:
To find the net force on q2 and its direction, we need to consider the forces between q1 and q2, and between q3 and q2.
Step 1: Calculate the force between q1 and q2:
Using Coulomb's Law, we have:
F(q1 on q2) = (9.0 × 10^9 Nm^2/C^2) * (15.0 C * 6.00 C) / (2.00 m)^2
Simplifying the equation:
F(q1 on q2) = 202.5 N
Step 2: Calculate the force between q3 and q2:
Using Coulomb's Law, we have:
F(q3 on q2) = (9.0 × 10^9 Nm^2/C^2) * (0.27 C * 6.00 C) / (2.00 m)^2
Simplifying the equation:
F(q3 on q2) = 0.3645 N
Step 3: Calculate the net force on q2:
The net force, F(net) on q2 is the vector sum of the forces between q1 and q2 and between q3 and q2:
F(net) = F(q1 on q2) + F(q3 on q2)
F(net) = 202.5 N + 0.3645 N
F(net) = 202.86 N
The net force on q2 is approximately 202.86 N.
Since both forces are acting in the positive x-direction, the net force on q2 is also in the same direction, which is the positive x-direction.