Electric dipole - derive torque from pot. energy

In summary, to derive the equation of torque, G = p X E, from the equation U = -p\cdotE, we can use the relationship between work and change of energy. This results in the equation G = - \frac{\partial U}{\partial\theta} = - p \times E.
  • #1
zimo
45
0

Homework Statement



Derive equation of torque: G = p X E from equation U = -p[itex]\cdot[/itex]E

Homework Equations



G = [itex]\int x \times \rho(x)E dV[/itex]

U = [itex]\int \rho(x)x \cdot \nabla\varphi dV [/itex]


The Attempt at a Solution


I tried to figure how to represent G in a way that better express the nature of the problem. so I figured maybe G = r x F but then got stuck again...
 
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  • #2
Use the relationship between the work and change of the energy.
 
  • #3
A shot:

[itex]G = - \frac{\partial U}{\partial\theta}
= - \frac{\partial(-p \cdot E)}{\partial\theta}
= \frac{\partial(pEcos\theta)}{\partial\theta}
= - pEsin\theta
= - p \times E
[/itex]


did I get it all-right?
 

1. What is an electric dipole?

An electric dipole is a pair of equal and opposite charges that are separated by a certain distance. It is a fundamental concept in electricity and can be represented by a positive charge and a negative charge that are connected by a straight line.

2. How is torque derived from potential energy in an electric dipole?

The torque on an electric dipole is derived from the potential energy between its two charges. The potential energy is a measure of the work required to bring the two charges from an infinite separation to their current distance. By taking the derivative of this potential energy with respect to the angle between the dipole and the electric field, we can derive the torque.

3. What is the formula for calculating torque in an electric dipole?

The formula for calculating torque in an electric dipole is given by τ = pEsinθ, where τ is the torque, p is the magnitude of the dipole moment, E is the magnitude of the electric field, and θ is the angle between the dipole moment and the electric field.

4. What is the significance of torque in an electric dipole?

Torque in an electric dipole is significant because it is the force that causes the dipole to rotate in an electric field. This rotation is important in many applications, such as in electric motors and generators.

5. Can torque be negative in an electric dipole?

Yes, torque can be negative in an electric dipole. This occurs when the angle between the dipole moment and the electric field is greater than 90 degrees, resulting in a negative sign in the torque formula. Negative torque causes the dipole to rotate in the opposite direction of a positive torque.

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