A small, nonconducting ball of mass m=1.0mg and charge q=2.0*10^-8 C (distributed uniformly through it's volume) hangs from an insulated thread that makes an angle theta = 30 degrees with the vertical, uniformly charged nonconducting sheet. considering the gravitational force on the ball and assuming the sheet extends far vertically and into and out of the page, calculate the surface charge density of the sheet.
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To calculate the surface charge density of the sheet, we need to equate the electric force and gravitational force acting on the ball.
1. Electric Force (Fe): The electric force acting on the ball due to the charged sheet is given by:
Fe = k * (q1 * q2) / r^2
where k is the electrostatic constant (k = 9 * 10^9 N m^2/C^2), q1 and q2 are the charges on the ball and the sheet respectively, and r is the distance between the ball and the sheet.
In this case, q1 = 2.0 * 10^-8 C and q2 is the surface charge density of the sheet (σ).
2. Gravitational Force (Fg): The gravitational force acting on the ball is given by:
Fg = m * g
where m is the mass of the ball and g is the acceleration due to gravity (g ≈ 9.8 m/s^2).
3. Equating the forces:
Setting Fe = Fg, we have:
k * (q1 * q2) / r^2 = m * g
Substituting the given values:
k * (2.0 * 10^-8 C * σ) / r^2 = 1.0 * 10^-6 kg * 9.8 m/s^2
4. Solving for σ (surface charge density):
Rearranging the equation, we find:
σ = (m * g * r^2) / (2.0 * 10^-8 C * k)
Substituting the known values:
σ = (1.0 * 10^-6 kg * 9.8 m/s^2 * r^2) / (2.0 * 10^-8 C * 9 * 10^9 N m^2/C^2)
Simplifying further, we have:
σ = (4.9 * r^2) / (2.0 * 10^-8 * 9 * 10^9)
σ ≈ 2.7 * 10^-7 C/m^2
So, the surface charge density of the sheet is approximately 2.7 * 10^-7 C/m^2.
To calculate the surface charge density of the sheet, we need to consider the forces acting on the small ball hanging from the insulated thread. The forces include the gravitational force and the electrostatic force from the charged sheet.
Let's break down the problem and analyze the forces acting on the ball:
1. Gravitational force (weight):
The weight of the ball is given by the formula:
F_grav = m * g
Where:
- m = mass of the ball
- g = acceleration due to gravity (approximately 9.8 m/s^2)
2. Electrostatic force (Coulomb's Law):
The electrostatic force between the ball and the charged sheet is given by Coulomb's Law:
F_elec = k * (q_ball * q_sheet) / r^2
Where:
- k = Coulomb's constant (approximately 9 x 10^9 N m^2/C^2)
- q_ball = charge on the ball
- q_sheet = charge on the sheet (unknown variable we need to find)
- r = distance between the ball and the sheet
In this case, the electrostatic force acts in the horizontal direction, while the gravitational force acts vertically. Since the ball hangs at an angle θ = 30 degrees with respect to the vertical, we can consider the horizontal component of the electrostatic force to balance the weight component in the horizontal direction.
In other words, to maintain equilibrium:
F_elec_horizontal = F_grav_horizontal
Now, let's express these forces in terms of their components:
1. Gravitational force (weight):
F_grav_horizontal = F_grav * sin(θ)
2. Electrostatic force:
F_elec_horizontal = F_elec * cos(θ)
Since the forces are in equilibrium, their horizontal components are equal:
F_elec_horizontal = F_grav_horizontal
Now, let's substitute expressions for F_grav_horizontal and F_elec_horizontal:
F_elec * cos(θ) = F_grav * sin(θ)
Next, let's express F_elec in terms of q_ball and q_sheet:
F_elec = k * (q_ball * q_sheet) / r^2
And F_grav in terms of m and g:
F_grav = m * g
Substituting these expressions back:
k * (q_ball * q_sheet) / r^2 * cos(θ) = m * g * sin(θ)
Now, we can isolate the unknown charge on the sheet (q_sheet):
q_sheet = (m * g * sin(θ) * r^2) / (k * q_ball * cos(θ))
Substituting the known values:
- Mass of the ball, m = 1.0 mg = 1.0 x 10^-6 kg
- Charge on the ball, q_ball = 2.0 x 10^-8 C
- Acceleration due to gravity, g = 9.8 m/s^2
- Distance between the ball and sheet, r = unknown (given in the problem)
- Angle, θ = 30 degrees
Plug in these values and calculate q_sheet.
Calculate the E field due to the infinite sheet using Gauss' Law. Express it in terms of the charge density, sigma. The horizontal electrical force on the ball equals the horizontal component of the string tension, which is M g tan 30
E q = M g tan 30
sigma*q/(2*epsilono) = M g tan 30
sigma = 2 epsilono * M g tan 30/q