one charge of +10 uC is placed at the origin, a second charge of +5uC is placed at (2cm, 4cm) and a third charge (-15 uC) is placed at (4cm, 1 cm). What is the magnitude and direction of the electric force exerted on the first charge by the other two?
I drew out the triangle, determined the distance between q1 and q2, then q1 and q3; after that I found the force of q3 on q1 and q2 on q1... not sure if that is right. The final answer is supposed to be F = 670.16 N @ -0.8 degrees.
Any help would be greatly appreciated.
did you add the forces as vectors?
Am I supposed to take the x and y component for each, then add them together to determine the overall force on q1? that would suggest I need to determine all the angles as well?
To find the magnitude and direction of the electric force exerted on the first charge by the other two charges, you can use the equation for the electric force:
F = k * (|q1| * |q2|) / r^2 * e,
where F is the magnitude of the electric force, k is the electrostatic constant (9 x 10^9 N m^2/C^2), q1 and q2 are the magnitudes of the charges, r is the distance between the charges, and e is the unit vector pointing from one charge to the other.
First, let's find the magnitude and direction of the electric force exerted on q1 by q2.
The distance between q1 and q2 can be calculated using the distance formula:
r12 = √((x2 - x1)^2 + (y2 - y1)^2).
Given:
q1 = +10 μC at the origin (0, 0).
q2 = +5 μC at (2 cm, 4 cm) = (0.02 m, 0.04 m).
r12 = √((0.02 - 0)^2 + (0.04 - 0)^2)
= √(0.0004 + 0.0016)
= √(0.002)
≈ 0.045 m.
Now, let's calculate the magnitude of the electric force exerted on q1 by q2:
F12 = (k * |q1| * |q2|) / r12^2
= (9 x 10^9 N m^2/C^2) * (10 x 10^-6 C) * (5 x 10^-6 C) / (0.045 m)^2
= 5000 N.
To find the direction of the electric force, we need to calculate the unit vector e12 pointing from q2 to q1:
e12 = (x1 - x2, y1 - y2) / r12
= (0 - 0.02, 0 - 0.04) / 0.045
= (-0.02, -0.04) / 0.045
≈ (-0.444, -0.889).
Next, let's find the magnitude and direction of the electric force exerted on q1 by q3.
The distance between q1 and q3 can be calculated using the distance formula:
r13 = √((x3 - x1)^2 + (y3 - y1)^2).
Given:
q1 = +10 μC at the origin (0, 0).
q3 = -15 μC at (4 cm, 1 cm) = (0.04 m, 0.01 m).
r13 = √((0.04 - 0)^2 + (0.01 - 0)^2)
= √(0.0016 + 0.0001)
= √(0.0017)
≈ 0.0412 m.
Now, let's calculate the magnitude of the electric force exerted on q1 by q3:
F13 = (k * |q1| * |q3|) / r13^2
= (9 x 10^9 N m^2/C^2) * (10 x 10^-6 C) * (15 x 10^-6 C) / (0.0412 m)^2
≈ 8227 N.
To find the direction of the electric force, we need to calculate the unit vector e13 pointing from q3 to q1:
e13 = (x1 - x3, y1 - y3) / r13
= (0 - 0.04, 0 - 0.01) / 0.0412
= (-0.04, -0.01) / 0.0412
≈ (-0.970, -0.243).
To find the net force exerted on q1, we need to add the forces F12 and F13 together. This can be done by vector addition:
F = F12 + F13.
F = (5000 N * e12) + (8227 N * e13).
Now, let's calculate the net force F exerted on q1:
F = (5000 N * (-0.444, -0.889)) + (8227 N * (-0.970, -0.243))
≈ (-4821 N - 5633.29 N, -9648 N - 2000.06 N)
≈ (-10454.29 N, -11648.06 N).
The magnitude of the net force F is given by:
|F| = √((-10454.29 N)^2 + (-11648.06 N)^2)
≈ √(109242912.8 N^2 + 135578609.1 N^2)
≈ 1865.48 N.
The direction of the net force F can be found using trigonometry. We calculate the angle θ as:
θ = atan2(Fy, Fx).
θ = atan2(-11648.06 N, -10454.29 N)
≈ -0.83 rad.
Finally, let's convert the angle θ from radians to degrees:
θ_deg = θ * (180/π)
≈ -0.83 * (180/3.14159)
≈ -47.49 degrees.
Therefore, the magnitude and direction of the electric force exerted on the first charge by the other two charges is approximately 1865.48 N at an angle of -47.49 degrees.
To find the magnitude and direction of the electric force exerted on the first charge by the other two charges, you can use Coulomb's Law. Coulomb's Law states that the magnitude of 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.
Here are the steps to solve the problem:
1. Calculate the distance between the first charge (+10 uC) at the origin and the second charge (+5 uC) at (2 cm, 4 cm). To find the distance, you can use the distance formula:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Plugging in the values:
distance = sqrt((2 cm - 0 cm)^2 + (4 cm - 0 cm)^2)
distance = sqrt(4 cm^2 + 16 cm^2)
distance = sqrt(20 cm^2)
distance = 2sqrt(5) cm
2. Similarly, calculate the distance between the first charge at the origin and the third charge (-15 uC) at (4 cm, 1 cm):
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Plugging in the values:
distance = sqrt((4 cm - 0 cm)^2 + (1 cm - 0 cm)^2)
distance = sqrt(16 cm^2 + 1 cm^2)
distance = sqrt(17 cm^2)
3. Calculate the magnitude of the electric force exerted by the second charge on the first charge using Coulomb's Law:
force = (k * |q1| * |q2|) / distance^2
Here, k is the electrostatic constant (k = 9 x 10^9 Nm^2/C^2).
|q1| represents the magnitude of the first charge (+10 uC).
|q2| represents the magnitude of the second charge (+5 uC).
distance is the distance calculated in step 1.
Plugging in the values:
force = (9 x 10^9 Nm^2/C^2 * 10^-6 C * 5 x 10^-6 C) / (2sqrt(5) cm)^2
force = (9 x 10^9 N * 5 x 10^-12 C^2) / (20 cm^2)
force = 45 x 10^-3 N / (20 cm^2)
force = 2.25 x 10^-3 N/cm^2
4. Repeat step 3 to calculate the magnitude of the electric force exerted by the third charge on the first charge, using the distance calculated in step 2.
force = (k * |q1| * |q3|) / distance^2
Here, |q3| represents the magnitude of the third charge (-15 uC).
5. Add the two electric forces calculated in steps 3 and 4 using vector addition. Since the forces are acting in different directions, you need to consider their vector components. To find the magnitude and direction of the total force, you can use trigonometry.
magnitude of total force = sqrt(F1^2 + F2^2)
direction of total force = arctan(F2/F1)
Plugging in the values:
magnitude of total force = sqrt((2.25 x 10^-3 N/cm^2)^2 + (F2)^2)
direction of total force = arctan(F2 / (2.25 x 10^-3 N/cm^2))
Solve for F2 using the information given in the question and the fact that the net force is zero.
6. Finally, convert the magnitude of the total force to newtons and the direction to degrees.
magnitude in newtons = (magnitude in cm^2) * (1 N/cm^2)
direction in degrees = direction in radians * (180 degrees / pi radians)
By following these steps, you should be able to obtain the desired answer of magnitude 670.16 N at -0.8 degrees.