As shown in the figure, 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?
To find the x-coordinate of the point where the net force on a point charge of 0.300 μC is zero, we need to calculate the net force at different x-coordinates and find where it equals zero.
The net force between two charges can be calculated using Coulomb's Law:
F = k * (|q1| * |q2|) / r^2
Where:
F is the net force
k is the electrostatic constant (k = 8.99 x 10^9 Nm^2/C^2)
|q1| and |q2| are the magnitudes of the charges
r is the distance between the charges
Let's calculate the net force at different x-coordinates:
For x = -4.7 m:
F1 = k * (|q1| * |q3|) / r1^2
= (8.99 x 10^9 Nm^2/C^2) * (|3.94 µC| * |0.300 μC|) / (-4.7 m)^2
For x = 12.2 m:
F2 = k * (|q2| * |q3|) / r2^2
= (8.99 x 10^9 Nm^2/C^2) * (|6.14 µC| * |0.300 μC|) / (12.2 m)^2
We want to find the x-coordinate where the net force is zero, so we can set F1 + F2 = 0.
F1 + F2 = 0
k * (|q1| * |q3|) / r1^2 + k * (|q2| * |q3|) / r2^2 = 0
Let's solve this equation to find the x-coordinate.
To determine the x-coordinate at which the net force on a point charge of 0.300 μC is zero, we can use Coulomb's law and the principle of superposition.
Coulomb's law states that the magnitude of the force between two charges is given by:
F = k * |q1 * q2| / r^2
Where:
F is the electrostatic force
k is the electrostatic constant (k = 8.99 × 10^9 N m^2/C^2)
q1 and q2 are the magnitudes of the charges
r is the distance between the charges
The net force acting on a point charge due to multiple charges can be found by summing the individual forces.
In our case, we have three charges: q1 = 3.94 µC, q2 = 6.14 µC, and q3 = 0.300 μC.
Let's assume the x-coordinate of the point where the net force is zero is x.
The force between q1 and the point charge would be:
F1 = k * |q1 * q3| / (x - x1)^2
The force between q2 and the point charge would be:
F2 = k * |q2 * q3| / (x - x2)^2
For the net force to be zero, the magnitudes of F1 and F2 must be equal.
Setting F1 = F2, we have:
k * |q1 * q3| / (x - x1)^2 = k * |q2 * q3| / (x - x2)^2
Simplifying the equation, we have:
|q1 * q3| / (x - x1)^2 = |q2 * q3| / (x - x2)^2
Taking the square root of both sides, we get:
|q1 * q3| / (x - x1) = |q2 * q3| / (x - x2)
Now we can substitute the given values for the charges and solve for x.
Plugging in the values, we have:
(3.94 µC * 0.300 μC) / (x - (-4.7 m)) = (6.14 µC * 0.300 μC) / (x - 12.2 m)
Simplifying further:
(1.182 µC^2) / (x + 4.7 m) = (1.842 µC^2) / (x - 12.2 m)
Cross multiplying, we have:
(1.182 µC^2) * (x - 12.2 m) = (1.842 µC^2) * (x + 4.7 m)
Expanding the equation, we get:
1.182 µC^2 * x - 14.4284 µC * m = 1.842 µC^2 * x + 8.6154 µC * m
Rearranging the equation:
1.182 µC^2 * x - 1.842 µC^2 * x = 14.4284 µC * m + 8.6154 µC * m
-0.66 µC^2 * x = 23.0438 µC * m
Dividing both sides by -0.66 µC^2, we have:
x = (23.0438 µC * m) / -0.66 µC^2
Solving this equation, we find:
x ≈ -34.9288 m
Therefore, the x-coordinate of the point at which the net force on a point charge of 0.300 μC is zero is approximately -34.9288 m.