A point charge of charge 1 mC and mass 100 g is attached to a non-conducting massless rod of length 10 cm. The other end of the rod is attached to a two-dimensional sheet with uniform charge density σ and the rod is free to rotate. The sheet is parallel to the y-z plane (i.e. it's a vertical sheet). I lift the point charge so the rod is horizontal and release it. I observe that the point charge achieves its maximum speed when the rod makes an angle of 30∘ with respect to the vertical. What is σ in C/m2?

Details and assumptions
• The acceleration of gravity is −9.8 m/s2.
• 14πϵ0=9×109 Nm2/C2.

You need to use the idea the the change in potential energy is equal to the change in kinetic energy

To find the value of σ, the uniform charge density, we can use the concept of torque and equilibrium.

1. Let's start by determining the forces acting on the point charge when the rod is inclined at an angle of 30∘ with respect to the vertical. First, we have the gravitational force acting vertically downwards with a magnitude of mg, where m is the mass (100 g) and g is the acceleration due to gravity (-9.8 m/s^2).

2. The other force acting on the point charge is the electrostatic force due to the uniform charge density σ on the sheet. Since the sheet is parallel to the y-z plane, the electrostatic force will act along the x-axis. Let's call this force Fe.

3. At equilibrium, the sum of the torques acting on the system should be zero. The torque due to the gravitational force is zero because it acts along the line of the rod. The torque due to the electrostatic force can be calculated as follows:

τ = r * F * sin(θ)

Here, r represents the distance from the axis of rotation to the point where the force is applied (length of the rod = 10 cm = 0.1 m), F is the magnitude of the electrostatic force, and θ is the angle between the rod and vertical (30∘).

4. Now, we need to find the expression for the electrostatic force (Fe). The electrostatic force between two point charges can be calculated using Coulomb's law:

Fe = k |q1| |q2| / r^2

Here, k is the electrostatic constant (9×10^9 Nm^2/C^2), q1 is the charge on the point charge (1 mC = 1×10^-3 C), q2 is the charge density on the sheet (σ), and r is the distance between the charges (0.1 m).

5. Plugging the values into the formula for torque, we have:

τ = 0.1 * Fe * sin(30∘)
= 0.1 * (k |q1|σ / r^2) * sin(30∘)

6. At maximum speed, the net torque should be zero, so:

τ = 0

Solving the equation for σ, we get:

σ = -τ * r^2 / (k |q1| sin(30∘))
= 0 / (k |q1| sin(30∘))
= 0

Therefore, σ is equal to zero. This means the sheet has no charge density (no charge) and does not exert an electrostatic force on the point charge.

In conclusion, the value of σ in C/m^2 is zero.