Two charges q1 = 2e and q2 = -
4e are placed on the x-axis at points (0; 0) and
(a; 0) respectively.
(a) What is the magnitude and direction of the electrostatic force on q2 due to
q1?
(b) Suppose a third charge q3 = -e is introduced at the location (b; 0) such that
b > a. What is the net force on q3 due to q1 and q2?
(c) If b < a, what should be the value of b as function of a so that the net
electrostatic force on q3 due to q1 and q2 exactly cancels out.
To calculate the electrostatic force between two charges, we can use Coulomb's law. Coulomb's law states that the magnitude of the electrostatic force between two point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them.
(a) To find the electrostatic force on q2 due to q1, we need to calculate the magnitude of the force and its direction.
Let's consider the force exerted by q1 on q2. According to Coulomb's law, the magnitude of the force F between two charges is given by:
F = k * (|q1| * |q2|) / r^2
where k is the electrostatic constant (approximately equal to 9 × 10^9 Nm^2/C^2), q1 and q2 are the magnitudes of the charges, and r is the distance between the charges.
Given:
q1 = 2e
q2 = -4e
Since the distance between them is a, the magnitude of the force exerted by q1 on q2 is:
F = k * (|q1| * |q2|) / a^2
Substituting the values, we have:
F = (9 × 10^9) * (2e) * (4e) / a^2
Now, we need to determine the direction of the force. Since q1 is positive and q2 is negative, the force between them will be attractive. Hence, the direction of the force on q2 due to q1 is towards q1.
(b) To find the net force on q3 due to q1 and q2, we need to consider the individual forces between q3 and q1, as well as q3 and q2. Then, we can simply add these forces vectorially to obtain the net force.
The force between q3 and q1 can be calculated using Coulomb's law in the same way we did in part (a):
F1 = k * (|q1| * |q3|) / b^2
The force between q3 and q2 can be calculated similarly:
F2 = k * (|q2| * |q3|) / (a - b)^2
Now, to find the net force on q3, we need to consider the direction as well. Since q1 and q3 have the same sign (both negative) and q2 and q3 have opposite signs, the forces F1 and F2 will have opposite directions. Hence, we should subtract the magnitudes of the forces:
Net force on q3 = F2 - F1
(c) If b < a, we need to find the value of b such that the net electrostatic force on q3 due to q1 and q2 exactly cancels out.
Setting the net force on q3 equal to zero, we have:
Net force on q3 = F2 - F1 = 0
Substituting the expressions for F1 and F2 from part (b), we get:
k * (|q1| * |q3|) / b^2 - k * (|q2| * |q3|) / (a - b)^2 = 0
Now, we can manipulate this equation to solve for b. Simplifying the equation, we have:
(|q1| * |q3|) / b^2 = (|q2| * |q3|) / (a - b)^2
Cross-multiplying and simplifying further, we get:
(|q1| * |q3|) * (a - b)^2 = (|q2| * |q3|) * b^2
Now, we can solve this equation algebraically to find the value of b in terms of a.