Three charges are fixed in the x−y plane as follows: 1.7nC at the origin (0, 0); 2.3nC at (0.60m , 0); –2.0nC at (0, 1.20m ).
Find the force acting on the negative charge.
Express your answer using two significant figures. Enter the x and y components of the force separated by a comma
To find the net force acting on the negative charge, we'll first find the individual forces due to the other two charges and then add them vectorially.
Let's denote the three charges as q1 (1.7 nC), q2 (2.3 nC), and q3 (-2.0 nC), and their positions as r1 (0, 0), r2 (0.60 m, 0), and r3 (0, 1.20 m) respectively.
First, we'll find the force due to q1 on q3. The distance between q1 and q3 is the same as the y-coordinate of q3, which is 1.20 m. So, the magnitude of the force F13 is given by Coulomb's Law:
F13 = k * |q1*q3| / r13^2
where k is Coulomb's constant (8.99 × 10^9 N m^2/C^2), and r13 is the distance between q1 and q3.
F13 = (8.99 × 10^9) * (1.7 × 10^-9 * 2.0 × 10^-9) / (1.20)^2
F13 = 5.12 × 10^-8 N
The direction of this force is along the negative y-axis, as q1 is positively charged and q3 is negatively charged. So, F13 = (0, -5.12 × 10^-8) N.
Next, we'll find the force due to q2 on q3. The distance between q2 and q3 is given by:
r23 = sqrt((0.60)^2 + (1.20)^2) = sqrt(1.80) = 1.34 m
The magnitude of the force F23 is given by Coulomb's Law:
F23 = k * |q2*q3| / r23^2
F23 = (8.99 × 10^9) * (2.3 × 10^-9 * 2.0 × 10^-9) / (1.34)^2
F23 = 1.74 × 10^-8 N
The direction of this force is given by the angle theta between the line connecting q2 and q3 and the negative y-axis:
theta = arctan((0.60)/(1.20)) = arctan(0.5) = 26.57°
The x and y components of F23 are:
F23x = F23 * sin(theta) = 1.74 × 10^-8 * sin(26.57°) = 7.66 × 10^-9 N
F23y = -F23 * cos(theta) = -1.74 × 10^-8 * cos(26.57°) = -1.54 × 10^-8 N
So, F23 = (7.66 × 10^-9, -1.54 × 10^-8) N.
Finally, we'll find the net force acting on q3 by adding the individual forces vectorially:
Fx = F13x + F23x = 0 + 7.66 × 10^-9 = 7.66 × 10^-9 N
Fy = F13y + F23y = -5.12 × 10^-8 + (-1.54 × 10^-8) = -6.66 × 10^-8 N
So, the force acting on the negative charge is approximately (7.66 × 10^-9, -6.66 × 10^-8) N.
Well, well, well! It looks like we have ourselves some charged disco balls floating around. Let's get down to business and calculate the force acting on the negative charge.
The force between two charges can be calculated using Coulomb's Law:
F = (k * |q1 * q2|) / r^2
Where F is the force between the charges, k is the electrostatic constant (9 x 10^9 N·m^2/C^2), q1 and q2 are the charges, and r is the distance between the charges.
Now, let's calculate the forces between the charges and the negative charge one by one. We'll consider the forces in the x and y directions:
Force due to 1.7nC at the origin:
F1x = (k * |-2.0nC * 1.7nC|) / (1.2m)^2
F1y = (k * |-2.0nC * 1.7nC|) / (1.2m)^2
Force due to 2.3nC at (0.60m, 0):
F2x = (k * |-2.0nC * 2.3nC|) / (0.60m)^2
Force due to -2.0nC at (0, 1.20m):
F3y = (k * |-2.0nC * -2.0nC|) / (1.20m)^2
Now, let's add up the forces in the x and y directions:
Fx = F1x + F2x
Fy = F1y + F3y
And there you have it! The x-component and y-component of the force acting on the negative charge. Enter them separated by a comma.
But hey, remember to use two significant figures!
To find the force acting on the negative charge, we can use Coulomb's Law. Coulomb's Law states that the force between two charges is given by:
F = k * (|q1| * |q2|) / r^2
Where:
- F is the magnitude of the force between the charges
- k is the electrostatic constant with a value of 9.0 x 10^9 N m^2/C^2
- |q1| and |q2| are the magnitudes of the charges
- r is the distance between the charges
Let's calculate the force in the x and y components separately.
First, calculate the force in the x-component:
- The positive charge at the origin does not contribute to the x-component because it is along the y-axis.
- The force due to the positive charge at (0.60m, 0) in the x-component is given by:
F_x1 = (k * |q1| * |q3|) / r^2
Where:
- |q1| is the magnitude of the charge at the origin, 1.7nC = 1.7 x 10^-9 C
- |q3| is the magnitude of the negative charge at (0, 1.20m), 2.0nC = 2.0 x 10^-9 C
- r is the distance between the charges, which is the x-coordinate of the negative charge, 0.60m
Substituting the values, we have:
F_x1 = (9.0 x 10^9 N m^2/C^2 * 1.7 x 10^-9 C * 2.0 x 10^-9 C) / (0.60m)^2
Calculate F_x1.
Next, calculate the force in the y-component:
- The positive charge at the origin does not contribute to the y-component because it is along the y-axis.
- The force due to the positive charge at (0.60m, 0) in the y-component is given by:
F_y1 = (k * |q1| * |q3|) / r^2
Where:
- |q1| is the magnitude of the charge at the origin, 1.7nC = 1.7 x 10^-9 C
- |q3| is the magnitude of the negative charge at (0, 1.20m), 2.0nC = 2.0 x 10^-9 C
- r is the distance between the charges, which is the y-coordinate of the negative charge, 1.20m
Substituting the values, we have:
F_y1 = (9.0 x 10^9 N m^2/C^2 * 1.7 x 10^-9 C * 2.0 x 10^-9 C) / (1.20m)^2
Calculate F_y1.
Lastly, calculate the force in the x-component due to the negative charge at (0, 1.20m):
- The positive charge at the origin does not contribute to the x-component because it is along the y-axis.
- The force due to the negative charge at (0, 1.20m) in the x-component is given by:
F_x3 = -(k * |q1| * |q2|) / r^2
Where:
- |q1| is the magnitude of the charge at the origin, 1.7nC = 1.7 x 10^-9 C
- |q2| is the magnitude of the positive charge at (0.60m, 0), 2.3nC = 2.3 x 10^-9 C
- r is the distance between the charges, which is the y-coordinate of the negative charge, 1.20m
Substituting the values, we have:
F_x3 = -(9.0 x 10^9 N m^2/C^2 * 1.7 x 10^-9 C * 2.3 x 10^-9 C) / (1.20m)^2
Calculate F_x3.
Now we have the x-component force due to each charge. To find the total x-component force, we add the forces from each charge:
F_x_total = F_x1 + F_x3
Calculate F_x_total.
For the y-component force, it is only contributed by the charge at (0.60m, 0):
F_y_total = (k * |q1| * |q2|) / r^2
Where:
- |q1| is the magnitude of the charge at the origin, 1.7nC = 1.7 x 10^-9 C
- |q2| is the magnitude of the positive charge at (0.60m, 0), 2.3nC = 2.3 x 10^-9 C
- r is the distance between the charges, which is the x-coordinate of the positive charge, 0.60m
Calculate F_y_total.
Finally, write down the x and y components of the force separated by a comma.
To find the force acting on the negative charge, we can use Coulomb's law, which states that the force between two charges is given by:
F = k * (|q1| * |q2|) / r^2,
where F is the force, k is the Coulomb's 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 the charges.
Let's label the charges as follows:
q1 = 1.7 nC (charge at the origin)
q2 = -2.0 nC (negative charge at (0, 1.20m))
q3 = 2.3 nC (charge at (0.60m, 0))
We need to calculate the force due to q1 and q3 acting on q2. Since both q1 and q3 are positive charges, they will repel the negative charge q2.
To calculate the force components, we can break it down into x and y directions. The x-component and y-component of the force are given by:
F_x = F * cos(theta), F_y = F * sin(theta),
where theta is the angle that the force vector makes with the positive x-axis.
To find the x and y components, we'll follow these steps:
Step 1: Calculate the x and y components of the distances between the charges:
- For q1: The distance is (0 - 0) = 0, so both x and y components are 0.
- For q3: The distance is (0.60m - 0) = 0.60m, so the x component is 0.60m and the y component is 0.
Step 2: Calculate the x and y components of the forces due to q1 and q3:
- For q1: Since the distance is 0, there is no force in either the x or y direction.
- For q3: Since q3 is on the x-axis, its force only has an x-component, which is calculated using Coulomb's law.
Using Coulomb's law, we have:
F_q3 = k * (|q1| * |q3|) / r^2,
where r is the distance between q3 and q2.
The distance is given by:
r = sqrt((0.60m)^2 + (1.20m)^2) = sqrt(0.60^2 + 1.20^2) = sqrt(0.36 + 1.44) = sqrt(1.80) = 1.34m.
Substituting the values into the equation, we have:
F_q3 = (8.99 x 10^9 N m^2/C^2) * |1.7 nC| * |2.3 nC| / (1.34m)^2.
Calculating the x-component of the force, F_q3x:
F_q3x = F_q3 * cos(theta),
where cos(theta) = dx / r,
and dx is the x-component of the distance, which is 0.60m.
Finally, we have all the components needed to calculate the net force acting on the negative charge q2.
F_net_x = F_q3x, since there is no x-component from q1.
F_net_y = 0, since there is no y-component from either q1 or q3.
We can round the final answer to two significant figures and express it as "X, Y".