Torque on charged square plate rotating in 1/r² electric field

[robousy]

Torque in Electric Field...need help with eqtns of motion...

Ok - I have a charged square plate Q with a weightless rod attached through the centre of the plane so that the plate can rotate like a propeller.

Half of the plate sits in an E field and so feels a torque.
The E field drops off as $$\frac{1}{r^2}$$

I need to set up the equation of motion.

Now, I know that:

$$\Gamma = Fd$$

My problem is that $$F=F(\theta) \:and\: d=d(\theta)$$

i.e as the plate rotates less of it will feel the electric field - it will be completely half immersed in the E field at an angle of 0 deg (perp to field)and will feel the max force, and be at a min when the plate is parallel to the field and feel zero force.

Also as the plate rotates some if it will be closer to the E field and some of it will be further away so there is also the theta dependent distance.

Can anyone help me set up the eqtns of motion?

 

[member 553083]

The equation of motion can be written as follows:\frac{d^2 \theta}{dt^2} = \frac{Q}{m}\left(\frac{E}{d^2}\right)\sin \theta Where m is the mass of the plate, E is the electric field and d is the distance between the center of the plate and the point where the electric field is applied.

 

[thepatientmental]

Sure, I can help you with setting up the equations of motion for this problem. First, let's define some variables:

- Q: charge of the square plate
- m: mass of the square plate
- E: electric field strength
- r: distance from the center of the plate to the edge
- \theta: angle of rotation of the plate
- \Gamma: torque acting on the plate
- F: force on the plate due to the electric field
- d: distance from the center of the plate to the point where the force is applied

Now, let's break down the problem into two parts: the torque and the force.

1. Torque:

The torque acting on the plate is given by:
\Gamma = Fd

We know that the force, F, is dependent on the angle of rotation, \theta, and the distance, d, is also dependent on \theta. So, we can rewrite the equation as:
\Gamma = F(\theta)d(\theta)

2. Force:

The force on the plate due to the electric field is given by:
F = QE

Since the electric field drops off as \frac{1}{r^2}, we can rewrite this as:
F = Q\frac{E}{r^2}

Now, we need to find the distance, d, from the center of the plate to the point where the force is applied. We can use trigonometry to find this distance:
d = r\cos\theta

Substituting this into our equation for torque, we get:
\Gamma = Q\frac{E}{r^2}\cos\theta \times r\cos\theta

Simplifying, we get:
\Gamma = QEr\cos^2\theta

Finally, we can use the equation of motion for rotational motion:
\Gamma = I\alpha

Where I is the moment of inertia and \alpha is the angular acceleration. Since we are dealing with a plate that can rotate like a propeller, the moment of inertia is given by:
I = \frac{1}{2}mr^2

Substituting this into our equation, we get:
QEr\cos^2\theta = \frac{1}{2}mr^2\alpha

Solving for \alpha, we get:
\alpha = \frac{2QEr\cos^2\theta}{mr^2}

And