Consider four masses arranged as shown. Each is 1 cm from the center of the cross. Mass A has a charge +1 uC, mass B has a charge of -1uC, and mass C has a charge of +2uC. At the center the electric field points 30 degrees from the vertical axis as shown.
What is the charge of mass D in uC?
In order to calculate the charge of mass D, we need to find the net electric field at the center of the cross.
Since the electric field at the center points 30 degrees from the vertical axis, we can resolve the electric field into vertical and horizontal components. Let's assume the vertical component is EV and the horizontal component is EH.
Considering the symmetry of the system, we can see that the horizontal components of the electric fields due to masses A and C will cancel each other out because they are equidistant from the center along the horizontal axis. So, EH = 0.
Now let's consider the vertical components of the electric fields. The electric field due to mass A points directly downward, so the vertical component is -EA.
The electric field due to mass C forms a 30-60-90 triangle with the vertical and horizontal axis. The vertical component of its electric field is EC = EC * cos(30 degrees) = 2uC * cos(30 degrees) = sqrt(3)uC.
Now, to find the net electric field at the center, we can add up the vertical components:
Net vertical component = -EA + EC = -1uC + sqrt(3)uC = (sqrt(3) - 1)uC.
Since the net electric field at the center is given in the question as pointing 30 degrees from the vertical, we can write:
Net vertical component = Net electric field * sin(30 degrees).
Therefore, we have:
(sqrt(3) - 1)uC = Net electric field * sin(30 degrees).
Simplifying this equation, we find:
Net electric field = (sqrt(3) - 1) * sin(30 degrees) = (sqrt(3) - 1) * 0.5 = (sqrt(3) - 1)/2.
Now, the electric field due to a charge is given by the equation:
Electric field = (k * charge) / (distance^2),
where k is the electrostatic constant.
Since each mass is equidistant from the center at 1 cm, we can write:
Net electric field = (k * (1uC - 2uC + D))/ (1 cm)^2.
Plugging in the values we know, we can solve for D:
(sqrt(3) - 1)/2 = k * (1uC - 2uC + D).
Now, we need to use the formula for k, which is given by:
k = 8.99 * 10^9 N * m^2 / C^2.
Substituting in this value for k and solving for D, we get:
(sqrt(3) - 1)/2 = (8.99 * 10^9 N * m^2 / C^2) * (-1uC + D).
Simplifying this equation, we find:
D - 1 = [(sqrt(3) - 1) / 2] * [(-1)(8.99 * 10^9 N * m^2 / C^2)].
Solving for D, we get:
D = 1 - [(sqrt(3) - 1) / 2] * [(-1)(8.99 * 10^9 N * m^2 / C^2)].
Evaluating this expression, we find the charge of mass D:
D = 1 - [(sqrt(3) - 1) / 2] * [8.99 * 10^9] ≈ 0.824 uC.