A point charge of –3.00 μC is located at the origin; a point charge of 4.00 μC is located on the x axis at x = 0.200 m; a third point charge Q is located on the x axis at x = 0.320 m. The electric force on the 4.00 μC charge is 240 N in the +x direction. (a) Determine the charge Q. (b) With this configuration of three charges, at what location(s) is the electric field zero?
a) Q = -2.00 μC
b) The electric field is zero at the origin and at x = 0.320 m.
To find the charge Q and the location(s) where the electric field is zero, we can use Coulomb's Law and the principle that the electric field is zero when the net force on a charge is zero.
(a) To determine the charge Q, we need to find the net force on the 4.00 μC charge. The distance between the -3.00 μC charge at the origin and the 4.00 μC charge is the same as the distance between the 4.00 μC charge and the unknown charge Q. We'll call this distance d.
The electric force between two charges is given by Coulomb's Law:
F = k * |q1| * |q2| / r^2
where F is the force, k is the electrostatic constant (9.0 x 10^9 N.m^2/C^2), q1 and q2 are the magnitudes of the charges, and r is the distance between the charges.
The electric force on the 4.00 μC charge is given as 240 N in the +x direction. This means the force due to the -3.00 μC charge must balance out the force due to the unknown charge Q.
Thus, we set up the equation:
|-3.00 μC| * |4.00 μC| / (0.200 m)^2 = |3.00 μC| * |Q| / d^2
Simplifying the equation, we get:
(3.00 μC * 4.00 μC) / (0.200 m)^2 = (3.00 μC * |Q|) / d^2
Solving for Q, we find:
|Q| = (d^2 * (3.00 μC * 4.00 μC)) / (0.200 m)^2
Substituting the given values, we find:
|Q| = (0.320 m)^2 * (3.00 μC * 4.00 μC) / (0.200 m)^2
|Q| = 9.60 μC
Therefore, the charge Q is 9.60 μC.
(b) To find the location(s) where the electric field is zero, we need to consider the net forces on charge Q from the -3.00 μC charge and the 4.00 μC charge.
For the electric field to be zero at a particular location, the net force on charge Q must be zero. This means the force due to the -3.00 μC charge must balance out the force due to the 4.00 μC charge.
Let's consider the following equation:
F1 + F2 = 0
where F1 is the force due to the -3.00 μC charge and F2 is the force due to the 4.00 μC charge.
Using Coulomb's Law, we know that the force between two charges is inversely proportional to the square of the distance between them. Therefore, the forces from both charges will balance out when they are equidistant from charge Q.
The location where the electric field is zero is between the -3.00 μC charge and the 4.00 μC charge, at a distance d/2 from each charge.
Thus, at a location d/2 from the -3.00 μC charge and d/2 from the 4.00 μC charge, the electric field is zero.
Given the values of d = 0.320 m, the location(s) where the electric field is zero would be at 0.160 m from both charges.
Therefore, at x = 0.160 m, the electric field is zero.
To determine the charge Q, we can use Coulomb's law, which states that the electric force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
Let's denote the charge at the origin as Q1 (Q1 = -3.00 μC), the charge at x = 0.200 m as Q2 (Q2 = 4.00 μC), and the charge at x = 0.320 m as Q.
(a) To find the charge Q, we know that the electric force on the charge Q2 is 240 N in the +x direction. We can use this information to set up the equation:
F = k * |Q1| * |Q2| / r^2
Where F is the force between Q1 and Q2, k is the electrostatic constant (k = 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 them.
Substituting the known values, we get:
240 N = (8.99 x 10^9 N m^2/C^2) * |Q1| * |Q2| / (0.200 m)^2
Solving for |Q2|:
|Q2| = (0.200 m)^2 * 240 N * (1 / (8.99 x 10^9 N m^2/C^2)) / |Q1|
|Q2| ≈ 4.492 x 10^-5 C
Since Q2 is given as 4.00 μC (or 4 x 10^-6 C), we take its magnitude:
|Q2| = 4.00 x 10^-6 C
Thus, |Q2| = |Q| = 4.492 x 10^-5 C.
Therefore, the charge Q is approximately 4.492 x 10^-5 C.
(b) To find where the electric field is zero, we need to consider the superposition principle. The electric field at any point is the vector sum of the electric fields due to each individual charge.
The electric field due to a point charge is given by:
E = k * |Q| / r^2
Where E is the electric field, k is the electrostatic constant, |Q| is the magnitude of the charge, and r is the distance from the charge.
To find the location(s) where the electric field is zero, we need to investigate the points on the x-axis where the electric fields due to each charge cancel out.
For simplicity, let's consider the electric fields due to Q1 and Q2 at point x.
The electric field due to Q1 at point x is given by:
E1 = k * |Q1| / x^2
The electric field due to Q2 at point x is given by:
E2 = k * |Q2| / (0.200 m - x)^2
When the electric fields E1 and E2 cancel out, their magnitudes should be equal:
|E1| = |E2|
k * |Q1| / x^2 = k * |Q2| / (0.200 m - x)^2
Simplifying:
|Q1| / x^2 = |Q2| / (0.200 m - x)^2
Substituting the given values:
|-3.00 μC| / x^2 = |Q2| / (0.200 m - x)^2
By plugging in the values for |Q1|, |Q2|, and solving the equation, you can find the location(s) where the electric field is zero. As an AI text-based model, I don't have the capacity to solve this equation symbolically. However, you can use numerical techniques such as graphing or numerical approximation methods to find the location(s) where the electric field is zero.