charge 1 is 3.94 µC and is located at x₁ = -4.7 m, and charge 2 is 6.14 µC and is at x2 = 12.2 m. What is the x-coordinate of the point at which the net force on a point charge of 0.300 μC is zero?
We can calculate the net force on the point charge by summing up the individual forces due to the two charges. The 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 (approximately equal to 8.99 x 10^9 N * m^2 / C^2), q1 and q2 are the charges, and r is the distance between the charges.
Let's denote the x-coordinate of the point where the net force is zero as x. The force exerted by charge 1 on the point charge is:
F1 = k * (q1 * q3) / (x - x1)^2
The force exerted by charge 2 on the point charge is:
F2 = k * (q2 * q3) / (x2 - x)^2
Since the net force is zero, we have:
F1 + F2 = 0
Substituting the expressions for F1 and F2, we get:
k * (q1 * q3) / (x - x1)^2 + k * (q2 * q3) / (x2 - x)^2 = 0
Now, we can substitute the given values for q1, q2, x1, and x2:
k * (3.94 * 10^-6 * 0.3) / (x + 4.7)^2 + k * (6.14 * 10^-6 * 0.3) / (12.2 - x)^2 = 0
Simplifying the equation, we get:
(3.94 * 10^-6 * 0.3) / (x + 4.7)^2 + (6.14 * 10^-6 * 0.3) / (12.2 - x)^2 = 0
Now, we can solve this equation for x.
To find the x-coordinate of the point where the net force on a point charge of 0.300 μC is zero, we can use Coulomb's Law.
Coulomb's Law states that the force between two charged particles is given by:
F = k * ((q1 * q2) / r²)
Where:
F is the force between the charges,
k is the electrostatic constant (9.0 x 10^9 N*m²/C²),
q1 and q2 are the magnitudes of the charges, and
r is the distance between the charges.
Since we want to find the x-coordinate where the net force is zero, we know that the forces exerted on the 0.300 μC charge by both charges 1 and 2 must be equal in magnitude but opposite in direction.
Let's first find the force between charge 1 and the 0.300 μC charge.
F₁ = k * ((q1 * q3) / r₁²)
Where:
F₁ is the force between charge 1 and the 0.300 μC charge,
q3 is the magnitude of the 0.300 μC charge, and
r₁ is the distance between charge 1 and the 0.300 μC charge.
Given that q1 = 3.94 µC (3.94 x 10^-6 C), q₃ = 0.300 μC (0.300 x 10^-6 C), and r₁ is the distance between charge 1 and the point we want to find, we can find F₁ by substituting these values into the equation.
Now, let's find the force between charge 2 and the 0.300 μC charge.
F₂ = k * ((q2 * q3) / r₂²)
Where:
F₂ is the force between charge 2 and the 0.300 μC charge,
q3 is the magnitude of the 0.300 μC charge, and
r₂ is the distance between charge 2 and the 0.300 μC charge.
Given that q2 = 6.14 µC (6.14 x 10^-6 C), q₃ = 0.300 μC (0.300 x 10^-6 C), and r₂ is the distance between charge 2 and the point we want to find, we can find F₂ by substituting these values into the equation.
Since the net force is zero, we know that F₁ = -F₂.
(k * ((q1 * q3) / r₁²)) = - (k * ((q2 * q3) / r₂²))
We can simplify this equation by canceling out the electrostatic constant.
(q1 * q3) / r₁² = - (q2 * q3) / r₂²
We are solving for r₁, so let's isolate it.
r₁² = - (q2 * q3) / ((q1 * q3) / r₂²)
r₁² = - (q2 / q1) * (r₂²)
Taking the square root of both sides:
r₁ = √ (- (q2 / q1) * (r₂²))
Now we can substitute the given values into the equation to find r₁.
Given that q1 = 3.94 µC (3.94 x 10^-6 C), q2 = 6.14 µC (6.14 x 10^-6 C), r₂ = 12.2 m, we can substitute these values to find r₁.
r₁ = √ (- ((6.14 x 10^-6 C) / (3.94 x 10^-6 C)) * (12.2 m)²)
r₁ = √ (- (6.14 / 3.94) * (12.2 m)²)
r₁ = √ (- 1.56 * (12.2 m)²)
r₁ = √ (- 1.56 * 149.6 m²)
r₁ = √ (- 233.376 m²)
Since the distance cannot be negative, we can conclude that there is no x-coordinate where the net force on a point charge of 0.300 μC is zero in this specific case.