A charge of -3.66C is fixed at the center of a compass. Two additional charges are fixed on the circle of the compass (radius = 0.0978 m). The charges on the circle are -4.04C at the position due north and +6.20C at the position due east. What is (a) the magnitude and (b) direction of the net electrostatic force acting on the charge at the center? Specify the direction as an angle relative to due east.
I get (a) to be 25.5, but (b) I cannot get, I keep getting -56.9 degrees. But it is wrong. why?
Let me check your angle: If the reference is East, then
tanTheta=-4.04/6.20
Theta=-.578 radians or -33.1 deg
and the angle is S of E
Do a little thinking on how I simplified the angle to just those two numbers..
Somehow, I got the right magnitude, but my x and y are different. my x=-21.35 and y=13.9 how?
I am not certain what your x and y are, what was given was forces along N-S, and E-W.
the N-S force is not as strong as the E-
w because of the charge magnitude.
Not knowing what x, y is, my guess is that x is the force along E, and y is the force S.
theta=arctan(13.9/21.35)=exactly the answer I got above.
To solve this problem, we can use Coulomb's Law, which states that the magnitude of the electrostatic force between two charges is given by:
F = (k * |q1 * q2|) / r^2
Where:
- F is the magnitude of the electrostatic force
- k is the electrostatic constant, approximately equal to 9 × 10^9 N m^2/C^2
- q1 and q2 are the charges involved
- r is the distance between the charges
Now, let's calculate the magnitudes of the electrostatic forces between the charges at the center and each of the charges on the circle.
Force due to -4.04C charge:
F_north = (k * |q1 * q3|) / r^2
Plugging in the values:
F_north = (9 * 10^9 * |-3.66C * -4.04C|) / (0.0978m)^2
F_north = (9 * 10^9 * 14.7944C^2) / 0.00956564m^2
F_north = 13,017.6784 N
Force due to +6.20C charge:
F_east = (k * |q1 * q3|) / r^2
Plugging in the values:
F_east = (9 * 10^9 * |-3.66C * 6.20C|) / (0.0978m)^2
F_east = (9 * 10^9 * 22.692C^2) / 0.00956564m^2
F_east = 19,979.1660 N
Now, to find the net force acting on the charge at the center, we need to find the vector sum of the forces. Since the forces are at right angles to each other (due north and due east), we can use the Pythagorean theorem to find the magnitude of the net force:
|F_net| = sqrt((F_north)^2 + (F_east)^2)
Plugging in the values:
|F_net| = sqrt((13,017.6784 N)^2 + (19,979.1660 N)^2)
|F_net| = sqrt(169,652,000.0446176 N^2 + 399,166,252.93076 N^2)
|F_net| = sqrt(568,818,252.9753776 N^2)
|F_net| ≈ 23823.645 N ≈ 2.38x10^4 N (rounded to 3 significant figures)
So, the magnitude of the net electrostatic force acting on the charge at the center is approximately 2.38x10^4 N.
To find the direction of the net force, we can use trigonometry. Since the force due north and force due east form a right triangle, we can use the tangent function to calculate the angle:
tan(theta) = (F_north / F_east)
Plugging in the values:
tan(theta) = (13,017.6784 N / 19,979.1660 N)
tan(theta) ≈ 0.6514
To find the angle theta, we need to take the inverse tangent (arctan) of 0.6514:
theta ≈ arctan(0.6514)
theta ≈ 33.5 degrees
The direction of the net electrostatic force acting on the charge at the center is approximately 33.5 degrees relative to due east. Please note that the angle should be positive since it is measured clockwise from due east.
So, the correct answer for (a) the magnitude of the net electrostatic force is 2.38x10^4 N, and (b) the direction of the net electrostatic force is approximately 33.5 degrees relative to due east.