A small 8.00g metal ball is suspended by a 14.0cm long string in a uniform electric field of 1200 N/C [right] (like in question #20). If the ball has a charge of 3.5x10^-5 C, what angle will the string make in the electric field?
To determine the angle that the string makes in the electric field, we can use trigonometry. The gravitational force pulling the ball downward is balanced by the electric force acting on the ball.
1. Find the gravitational force acting on the ball:
Use the equation: F_gravity = m * g
where m is the mass of the ball and g is the acceleration due to gravity.
Given m = 8.00g = 8.00 * 10^(-3) kg (convert grams to kilograms)
g = 9.8 m/s^2
F_gravity = 8.00 * 10^(-3) kg * 9.8 m/s^2
2. Find the electric force acting on the ball:
Use the equation: F_electric = q * E
where q is the charge on the ball and E is the electric field strength.
Given q = 3.5 * 10^(-5) C
E = 1200 N/C
F_electric = 3.5 * 10^(-5) C * 1200 N/C
3. Equate the gravitational force and the electric force:
F_gravity = F_electric
8.00 * 10^(-3) kg * 9.8 m/s^2 = 3.5 * 10^(-5) C * 1200 N/C
4. Solve for the angle θ:
Divide both sides of the equation by the product of the mass and the electric field strength.
This will eliminate the units and leave us with the sine of the angle.
(8.00 * 10^(-3) kg * 9.8 m/s^2) / (3.5 * 10^(-5) C * 1200 N/C) = sin(θ)
5. Take the inverse sine of both sides to find θ:
θ = sin^(-1) [(8.00 * 10^(-3) kg * 9.8 m/s^2) / (3.5 * 10^(-5) C * 1200 N/C)]
6. Calculate the value of θ using a calculator to find the angle in radians.
θ ≈ sin^(-1) [0.11753]
7. Convert the angle from radians to degrees by multiplying by 180/π (radians to degrees conversion factor):
θ ≈ (0.11753) * (180/π) degrees
To find the angle that the string makes in the electric field, we need to consider the forces acting on the ball. In this case, the two forces acting on the ball are the gravitational force and the electrostatic force.
Let's start by finding the gravitational force on the ball. The gravitational force is given by the formula:
F_gravity = m * g,
where m is the mass of the ball and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Given that the mass of the ball is 8.00 g, we need to convert it to kilograms:
m = 8.00 g = 0.008 kg.
So, the gravitational force F_gravity is:
F_gravity = (0.008 kg) * (9.8 m/s^2).
Next, let's find the electrostatic force on the ball. The electrostatic force is given by the formula:
F_electric = q * E * sin(θ),
where q is the charge on the ball, E is the electric field strength, and θ is the angle between the electric field and the string.
Given that the charge on the ball is 3.5x10^-5 C and the electric field strength is 1200 N/C, we can find the electrostatic force F_electric.
Now, the ball is in equilibrium, which means there is no net force acting on it. Therefore, the gravitational force and the electrostatic force must cancel each other out. Since the gravitational force acts downward, and the electrostatic force acts to the right, we can set up the following equation:
F_gravity = F_electric.
By substituting the values we found, we can now solve for the angle θ:
(m * g) = (q * E * sin(θ)).
Substituting the known values:
(0.008 kg * 9.8 m/s^2) = (3.5x10^-5 C * 1200 N/C * sin(θ)).
Now, we can solve for sin(θ):
sin(θ) = (0.008 kg * 9.8 m/s^2) / (3.5x10^-5 C * 1200 N/C).
Finally, to find the angle θ, we can take the inverse sine (sin^-1) of the value we just calculated:
θ = sin^-1[(0.008 kg * 9.8 m/s^2) / (3.5x10^-5 C * 1200 N/C)].
Using a calculator or software that can perform trigonometric functions, you can find the exact value for θ.