How Does an Electric Field Affect the Equilibrium of a Torsion Pendulum?

[pianoman2700]

An electric dipole p is suspended as a torsion pendulum, which is allowed to pivot
about the nz-axis only. The dipole has moment of inertia I and the torsion spring
has Hooke constant K. In the absence of an electric Field the torsion pendulum's
equilibrium orientation theta-not is equal to zero. The dipole's orientation is allowed to change only in the direction permitted by the torsion pendulum (i.e., rotation 
about the nz axis). A uniform electric Field E is applied.

(a) Derive the equations of motion for the torsion pendulum for arbitrary dipole
moment p and uniform electric Field E.

(b) Find, from the equations of motion, the equilibrium orientation of the pendulum
Theta-not(E) when the uniform electric field is E

I'm not really even sure where to start with this problem. I know the equations:
torque = p x E (for a dipole)
torque = K*theta (for a torsion pendulum)
torque = I*alpha (for rotation)

Other than that, though, I don't know where to go. Am I supposed to solve for theta and derive the rest of the equations from there, or should I solve for alpha and integrate? Or am I on the wrong track completely?

Any help would be much appreciated.

 

[anathema]

To derive the equations of motion for the torsion pendulum with a dipole moment p and uniform electric field E, we can start by considering the torques acting on the dipole. As you mentioned, the torque due to the electric field is given by p x E, and the torque due to the torsion spring is K*theta. Since the dipole is only allowed to rotate about the nz-axis, these torques will cause the dipole to rotate in that direction. We can express this rotation using the equation torque = I*alpha, where I is the moment of inertia and alpha is the angular acceleration.

From this, we can write the following equation:

I*alpha = p x E + K*theta

Since the dipole's orientation is only allowed to change in the direction of rotation about the nz-axis, we can express theta as the angular displacement of the dipole. Therefore, we can rewrite the equation as:

I*alpha = p x E + K*theta

Next, we can use the definition of torque to rewrite the first term on the right-hand side as pEsin(theta). This gives us the following equation:

I*alpha = pEsin(theta) + K*theta

To find the equilibrium orientation of the pendulum, we need to set alpha equal to zero (since the pendulum is not rotating in equilibrium). This gives us the following equation:

0 = pEsin(theta) + K*theta

We can rearrange this equation to solve for theta:

pEsin(theta) = -K*theta

Using the small angle approximation sin(theta) ~ theta, we can simplify the equation to:

pE*theta = -K*theta

Solving for theta, we get:

theta = -(pE)/K

Therefore, the equilibrium orientation of the pendulum, Theta-not(E), is given by:

Theta-not(E) = -(pE)/K

I hope this helps you understand how to derive the equations of motion and find the equilibrium orientation for the torsion pendulum with a dipole moment and uniform electric field. Let me know if you have any further questions. Best of luck with your problem!

 

[Moetasim]

I would approach this problem by first understanding the physical principles involved. A dipole is a pair of equal and opposite charges separated by a distance, and it experiences a torque when placed in an electric field. A torsion pendulum is a system that oscillates due to the restoring torque of a spring. Keeping these in mind, I would proceed to derive the equations of motion for the torsion pendulum with an electric dipole.

(a) To derive the equations of motion, we need to consider the forces acting on the dipole in the presence of an electric field. The electric field exerts a torque on the dipole, given by the equation torque = p x E. This torque will cause the dipole to rotate about the nz-axis, which is the axis of rotation for the torsion pendulum. We also have the torque from the torsion spring, given by the equation torque = K*theta. Equating these two torques, we get:

p x E = K*theta

Using the definition of torque as I*alpha, where I is the moment of inertia and alpha is the angular acceleration, we can rewrite this equation as:

I*alpha = p x E

Now, we can use the equation of motion for a torsion pendulum, which is:

I*alpha = -K*theta

Combining these two equations, we get:

-p x E = K*theta

This is the equation of motion for the torsion pendulum with an electric dipole. We can solve this equation to find the angular acceleration alpha, and then integrate to find the angle theta as a function of time.

(b) To find the equilibrium orientation of the pendulum, we need to find the value of theta at which the net torque on the dipole is zero. This happens when the two torques, from the electric field and the torsion spring, are equal and opposite. So, we can set the two torques equal to each other and solve for theta:

p x E = K*theta

This gives us the equilibrium orientation of the pendulum, Theta-not(E), in the presence of a uniform electric field E. To find the specific value of Theta-not(E), we would need to know the values of p, K, and E.

In summary, as a scientist, I would approach this problem by first understanding the physical principles involved and then using the equations of motion to derive the necessary equations. I