A small sphere with a mass of 2.00×10−3 g and carrying a charge of 5.20×10−8 C hangs from a thread near a very large, charged insulating sheet, as shown in the figure . The charge density on the sheet is −2.40×10−9 C/m^2.
Find the angle of the thread.
Why did the charged sphere go to a party? Because it wanted to hang out with the insulating sheet! Now, to find the angle of the thread, let's unleash some physics humor.
To start, we can find the force acting on the charged sphere due to the electric field created by the insulating sheet. The electric force can be calculated using the equation:
F = q * E
where F is the force, q is the charge, and E is the electric field. Given that the charge on the sphere is 5.20×10−8 C and the charge density on the sheet is −2.40×10−9 C/m^2, we can calculate the electric field created by the sheet using the equation:
E = σ / ε₀
where σ is the charge density and ε₀ is the permittivity of free space. Plugging in the values, we get:
E = -2.40×10−9 C/m^2 / ε₀
Now, since we know the force and the mass of the sphere, we can use the equation:
F = m * g
where m is the mass and g is the acceleration due to gravity, to find the force acting on the sphere due to gravity. Plugging in the values, we get:
m * g = 2.00×10−3 g * 9.81 m/s²
Now, equating the force due to gravity to the force due to electric field, we get:
q * E = m * g
Simplifying, we have:
E / g = m / q
Now, we can use some trigonometry to find the angle of the thread. The angle can be calculated using the equation:
tan(θ) = (E / g) / (F_net / m)
where θ is the angle, E/g is the electric field by gravity ratio, and F_net/m is the net force due to gravity ratio. Plugging in the values, we can find the angle of the thread.
But you know what's really funny about this? No matter what angle the thread makes, the sphere will always hang out!
To find the angle of the thread, we can analyze the forces acting on the sphere. Since the sheet is negatively charged, it will induce a positive charge on the sphere, leading to an attractive force between the two.
1. Calculate the force due to the electric field:
The electric field due to the charged sheet can be calculated using:
E = σ/2ε₀
where E is the electric field, σ is the charge density, and ε₀ is the permittivity of free space.
Plugging in the given values:
E = (-2.40×10^-9 C/m^2) / (2ε₀)
The electric field exerts a force on the sphere given by:
F = qE
where F is the force, q is the charge on the sphere, and E is the electric field.
Plugging in the given values:
F = (5.20×10^-8 C) * (E)
2. Calculate the gravitational force:
The gravitational force is given by:
F_grav = mg
where F_grav is the gravitational force, m is the mass of the sphere, and g is the acceleration due to gravity.
Plugging in the given values:
F_grav = (2.00×10^-3 g) * g
3. Equate the forces and solve for the angle:
Since the sphere is in equilibrium, the two forces are equal. Therefore:
F_grav = F
(2.00×10^-3 g) * g = (5.20×10^-8 C) * (E)
Expressing g in terms of the acceleration due to gravity:
(2.00×10^-3 g) * (9.8 m/s^2) = (5.20×10^-8 C) * (E)
Solving for the electric field:
E = [(2.00×10^-3 g) * (9.8 m/s^2)] / (5.20×10^-8 C)
Substituting the value of E into the electric field equation:
E = (2.00×10^-3 g) * (9.8 m/s^2) / [(5.20×10^-8 C) / (2ε₀)]
Simplifying the expression and solving for the angle:
θ = arctan(E / g)
where θ is the angle of the thread.
Plugging in the values of E and g:
θ = arctan[(2.00×10^-3 g) * (9.8 m/s^2) / [(5.20×10^-8 C) / (2ε₀)]]
Finally, calculate the value of θ using the given values.
To find the angle of the thread, we need to analyze the forces acting on the charged sphere.
The sphere experiences two forces: the gravitational force pulling it downward and the electric force repelling it away from the charged sheet.
Let's break it down step by step:
1. Calculate the gravitational force:
The gravitational force acting on the sphere is given by the equation:
F_gravity = m * g
where m is the mass of the sphere and g is the acceleration due to gravity. In this case, m = 2.00×10^(-3) g and g = 9.8 m/s^2.
Converting the mass to kg:
m = 2.00×10^(-3) g = 2.00×10^(-6) kg
Calculating the gravitational force:
F_gravity = (2.00×10^(-6) kg) * (9.8 m/s^2)
2. Calculate the electric force:
The electric force acting on the sphere is given by the equation:
F_electric = q * E
where q is the charge of the sphere and E is the electric field. The electric field E is created by the charged insulating sheet. The electric field is constant and equal to the charge density (σ) divided by the permittivity of free space (ε0), which is a physical constant.
The equation for the electric field is:
E = σ / ε0
where σ is the charge density on the sheet and ε0 is approximately 8.85 x 10^(-12) C^2/N·m^2.
Converting the charge density to C/m^2:
σ = -2.40×10^(-9) C/m^2
Calculating the electric field:
E = (-2.40×10^(-9) C/m^2) / (8.85 x 10^(-12) C^2/N·m^2)
3. Determine the angle of the thread:
The total force experienced by the charged sphere is the vector sum of the gravitational force and the electric force. We can decompose this total force into two components: a horizontal component (Fx) and a vertical component (Fy).
The vertical component (Fy) represents the force that opposes the gravitational force and keeps the sphere in equilibrium. It is given by:
Fy = F_gravity - F_electric
To find the angle (θ) of the thread, we need to calculate the ratio of Fy to Fx and use the inverse tangent function:
θ = arctan(Fy / Fx)
4. Calculate Fy and Fx:
To calculate Fy and Fx, we use trigonometry. Let's assume the angle between the thread and the vertical as φ.
Fy = F_gravity * sin(φ)
Fx = F_gravity * cos(φ)
Now we have all the information we need to find the angle of the thread.
forces: mg downward, E*q=k*sigma*q
tanTheta (angle from vertical)= k*sigma*q/mg
where k= 1/2epislion