A small spherical insulator of mass 1.90 10-2 kg and charge +0.600 µC is hung by a thin wire of negligible mass. A charge of −0.900 µC is held 0.150 m away from the sphere and directly to the right of it, so the wire makes an angle è with the vertical.
Find the angle è.
Find the tension in the wire
Again, as above, the horzontal electric force is equal to mg*tanTheta.
Tension= mg*cosTheta
Draw the force diagram to demonstrate this.
tension = mg/ cosTheta
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To find the angle è, we can use trigonometry and the electric force equation.
First, let's find the electric force acting on the small spherical insulator due to the charge of -0.900 µC. The electric force between two charges is given by Coulomb's law:
F = k * (q1 * q2) / r^2
Where F is the electric force, k is the electrostatic constant (8.99 * 10^9 N m^2/C^2), q1 and q2 are the charges, and r is the distance between the charges.
In this case, q1 is +0.600 µC and q2 is -0.900 µC. The distance r is 0.150 m. Substituting these values into Coulomb's law, we get:
F = (8.99 * 10^9 N m^2/C^2) * ((0.600 * 10^-6 C) * (-0.900 * 10^-6 C)) / (0.150 m)^2
Calculate F:
F = -39.48 N
Since the charge of the small spherical insulator is positive, it will be attracted towards the charge of -0.900 µC. Therefore, the tension in the wire will oppose this force and keep the insulator in equilibrium.
The tension force in the wire can be resolved into two components: one acting upwards to balance the weight of the insulator, and another component acting towards the charge to balance the electric force.
Let's call the angle between the wire and the vertical as θ. The vertical component of the tension force will be T * cos(θ), and the component towards the charge will be T * sin(θ).
Since the insulator is in equilibrium, the vertical component of the tension force must balance its weight. The weight force can be calculated using the formula:
Weight = mass * gravity
In this case, the mass of the insulator is 1.90 * 10^-2 kg. The acceleration due to gravity is approximately 9.8 m/s^2. Therefore, the weight force is:
Weight = (1.90 * 10^-2 kg) * (9.8 m/s^2)
Next, we need to set up an equation for the equilibrium of the forces in the horizontal direction. The electric force F and the horizontal component of the tension force (T * sin(θ)) must be equal in magnitude and opposite in direction (since they balance each other out).
So, we have:
F = T * sin(θ)
Substituting the value of F we calculated earlier (-39.48 N), we get:
-39.48 N = T * sin(θ)
Now, let's set up an equation for the equilibrium of forces in the vertical direction:
T * cos(θ) = Weight
Substituting the weight value we calculated earlier, we get:
T * cos(θ) = (1.90 * 10^-2 kg) * (9.8 m/s^2)
From here, we have two equations:
1) -39.48 N = T * sin(θ)
2) T * cos(θ) = (1.90 * 10^-2 kg) * (9.8 m/s^2)
We can solve these equations simultaneously to find both the angle è and the tension in the wire.