Two point charges, q1 and q2, of 4.00 uc each, are placed -6.0 cm and 6.0 cm away from the origin on the x-axis. A charge q3 of -1.00 uC is placed 8.0 cm away from the origin on the y-axis.
a. Find the distance from q3 to q1 and from q3 to q2.
b, Find the magnitude and the direction of the force F13 exerted by q1 on q3.
c. Find the magnitude and the direction of the force F23 exerted by q2 on q3.
d. Find the magnitude and the direction of the force F12 exerted by q1 on q2.
e. In the space below, sketch the vectors representing forcesF13 and F23. f Find he angle between the q1-q3 line and the x-axis.
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a. The distance from q3 to q1 can be found using the Pythagorean theorem:
Distance^2 = (distance on x-axis)^2 + (distance on y-axis)^2
Distance to q1 = √((-6.0 cm)^2 + (8.0 cm)^2) = √(36.0 cm^2 + 64.0 cm^2) = √(100.0 cm^2) = 10.0 cm
The distance from q3 to q2 is the same as the distance from q3 to q1 since they are equidistant from the origin.
Distance to q2 = 10.0 cm
b. To find the magnitude and direction of the force F13 exerted by q1 on q3, we can use Coulomb's law:
F13 = (k * q1 * q3) / r^2
Where k is the Coulomb's constant (k = 9.0 x 10^9 N m^2/C^2), q1 and q3 are the magnitudes of the charges, and r is the distance between them.
F13 = (9.0 x 10^9 N m^2/C^2) * (4.00 x 10^-6 C) * (-1.00 x 10^-6 C) / (10.0 cm)^2
F13 = -3.24 x 10^-5 N
The negative sign indicates that the force is attractive. Since q1 is positive and q3 is negative, they attract each other.
c. To find the magnitude and direction of the force F23 exerted by q2 on q3, we can use Coulomb's law again:
F23 = (k * q2 * q3) / r^2
F23 = (9.0 x 10^9 N m^2/C^2) * (4.00 x 10^-6 C) * (-1.00 x 10^-6 C) / (10.0 cm)^2
F23 = -3.24 x 10^-5 N
The force F23 is also attractive, as q2 and q3 have opposite charges.
d. To find the magnitude and direction of the force F12 exerted by q1 on q2, we can again use Coulomb's law:
F12 = (k * q1 * q2) / r^2
F12 = (9.0 x 10^9 N m^2/C^2) * (4.00 x 10^-6 C) * (4.00 x 10^-6 C) / (12.0 cm)^2
F12 = 3.00 x 10^-4 N
The force F12 is repulsive because both q1 and q2 carry positive charges.
e. Vectors representing the forces F13 and F23 would originate from q1 and q2 respectively, and point towards q3. The length of the vectors would represent the magnitude of the forces.
f. To find the angle between the q1-q3 line and the x-axis, we can use trigonometry. The tangent of the angle is given by:
Tan(θ) = (distance on y-axis) / (distance on x-axis)
θ = Tan^(-1)((8.0 cm) / (-6.0 cm))
θ ≈ -50.19 degrees (approximate)
To solve this problem, we can use the principles of Coulomb's law to calculate the forces and distances involved. Coulomb's law states that the 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. The equation for the force is given by:
F = k * |q1 * q2| / r^2,
where F is the force between the charges, k is the electrostatic constant, q1 and q2 are the charges, and r is the distance between the charges.
a. The distance from q3 to q1 is the distance between the origin (0,0) and the point (0, -8), which is 8.0 cm.
The distance from q3 to q2 is the distance between the origin (0,0) and the point (6, 0), which is 6.0 cm.
b. To find the magnitude and direction of the force F13 exerted by q1 on q3, we can use Coulomb's law. The equation for F13 is:
F13 = k * |q1 * q3| / r13^2,
where F13 is the force, q1 is the charge of q1, q3 is the charge of q3, and r13 is the distance between q1 and q3. Substituting the given values:
F13 = (9.0 x 10^9 N*m^2/C^2) * |(4.00 x 10^-6 C) * (-1.00 x 10^-6 C)| / (0.08 m)^2.
Solving this equation will give you the magnitude of the force F13.
To find the direction, you can consider that the force will act along a line connecting q1 and q3, so the direction will be towards q1.
c. Similarly, to find the magnitude and direction of the force F23 exerted by q2 on q3, you can use Coulomb's law. The equation for F23 is:
F23 = k * |q2 * q3| / r23^2,
where F23 is the force, q2 is the charge of q2, q3 is the charge of q3, and r23 is the distance between q2 and q3. Substituting the given values:
F23 = (9.0 x 10^9 N*m^2/C^2) * |(4.00 x 10^-6 C) * (-1.00 x 10^-6 C)| / (0.06 m)^2.
Solving this equation will give you the magnitude of the force F23.
The direction of the force will be along the line connecting q2 and q3, so it will be towards q3.
d. To find the magnitude and direction of the force F12 exerted by q1 on q2, you can again use Coulomb's law. The equation for F12 is:
F12 = k * |q1 * q2| / r12^2,
where F12 is the force, q1 is the charge of q1, q2 is the charge of q2, and r12 is the distance between q1 and q2. Substituting the given values:
F12 = (9.0 x 10^9 N*m^2/C^2) * |(4.00 x 10^-6 C) * (4.00 x 10^-6 C)| / (0.12 m)^2.
Solving this equation will give you the magnitude of the force F12.
The direction of the force will be along the line connecting q1 and q2, so it will be towards q2.
e. To sketch the vectors representing forces F13 and F23, draw arrows starting at the position of q3 and pointing in the direction of the forces (towards q1 for F13, and towards q3 for F23). The length of the arrows can be representative of the magnitude of the forces.
f. To find the angle between the q1-q3 line and the x-axis, you can consider the right triangle formed by the x-axis, the line connecting q1 and q3, and the line connecting (6,0) to (0,-8). You can calculate the angle using the tangent function:
tanθ = opposite/adjacent
tanθ = 8/6
Solving for θ will give you the angle between the q1-q3 line and the x-axis.