A small spherical insulator of mass 8.00*10^-2kg and charge +6.00*10^-7C is hung by a thin wire of negligible mass. A charge of -9.00*10^-7C is held 0.450m away from the sphere and directly to the right of it, so the wire makes an angle with the vertical. Find (a) the angle and (b) the tension in the wire.
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You have to assume the small spherical insulator is a point charge.
The force horizontal is given by coulombs law, so looking at the diagram,
TanTheta=horizontalforce/vertical force.
Of course, the vertical force is mg.
Tension in the wire? SinTheta=horizonalforce/tension.
To find the angle and the tension in the wire, we can use the concept of electrostatic forces and equilibrium.
(a) Finding the angle:
The small spherical insulator is attracted to the charge held 0.450m away from it. The force of attraction 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 (9.0 x 10^9 Nm^2/C^2), q1 and q2 are the magnitudes of the charges, and r is the distance between the charges.
In this case, q1 = +6.00 * 10^-7C and q2 = -9.00 * 10^-7C, and r = 0.450m. Since the charges have different signs, the force between them is attractive.
Plugging in the values:
F = (9.0 x 10^9 Nm^2/C^2) * |(+6.00 * 10^-7C) * (-9.00 * 10^-7C)| / (0.450m)^2
F = (54 * 10^-9 Nm^2/C^2) / 0.2025m^2
F = 266.67N (approximately)
Now, let's analyze the forces acting on the small spherical insulator when it is in equilibrium. There are three forces: the force of gravity (mg), the tension in the wire (T), and the electrostatic force (F).
Since the small spherical insulator is hanging by the wire, the gravitational force is balanced by the tension in the wire:
mg = T
where m is the mass of the insulator (8.00 * 10^-2kg) and g is the acceleration due to gravity (approximately 9.8m/s^2).
Substituting the values:
T = (8.00 * 10^-2kg) * 9.8m/s^2
T ≈ 0.784N
Now, we can use trigonometry to find the angle θ between the wire and the vertical direction:
sin(θ) = F / T
θ = arcsin(F / T)
Substituting the values:
θ = arcsin(266.67N / 0.784N)
θ ≈ 77.5°
Therefore, the angle between the wire and the vertical is approximately 77.5 degrees.
(b) Finding the tension in the wire:
We have already found the tension in the wire using the equation mg = T. The tension is approximately 0.784N.