F = del(p.E) and F = (p.del)E are equivalent

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In summary, the equations F = del(p.E) and F = (p.del)E are both forms of the force equation that describe the force acting on a charged particle in an electric field. They are considered equivalent and can be applied to other types of fields such as magnetic fields. The dot product in F = del(p.E) represents the direction of the force and is used in practical applications such as designing electrical devices and studying molecular interactions. These equations are fundamental in electromagnetism and are essential for understanding charged particle behavior.
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Question: In the electrostatic case, the expressions F = del(p.E) and F = (p.del)E are equivalent:

I am having trouble with how to show they are equivalent

In the second equation, I expanded it out to give F= px (dE/dx) + py (dE/dy) + pz(dE/dz)

Any help as to how to do this would be much appreciated

thanks
 
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You have to use the vector differential operator equation
[tex]\nabla({\bf p\cdot E)=p\times(\nabla\times E)+(p\cdot\nabla)E}[/tex],
and curl E=0.
 
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1. What is the equation F = del(p.E) and how is it related to F = (p.del)E?

The equation F = del(p.E) is known as the gradient form of the force equation, where F represents the force, del is the gradient operator, p is the electric dipole moment, and E is the electric field. This equation states that the force acting on a charged particle is equal to the gradient of the dot product of its dipole moment and the electric field. This equation is equivalent to F = (p.del)E, which is known as the divergence form of the force equation.

2. Why are F = del(p.E) and F = (p.del)E considered equivalent?

These equations are considered equivalent because they both describe the same physical phenomenon - the force acting on a charged particle in an electric field. The only difference is the mathematical representation of this force, with one being in gradient form and the other in divergence form. Both forms are valid and can be used interchangeably depending on the context.

3. What is the significance of the dot product in the equation F = del(p.E)?

The dot product in this equation represents the direction of the force acting on the charged particle. When the dipole moment and electric field are parallel, the force is at its maximum, and when they are perpendicular, the force is zero. This dot product also allows us to calculate the work done by the electric field on the charged particle, as it is equal to the dot product of the force and the displacement.

4. Can these equations be applied to other types of fields besides electric fields?

Yes, these equations can be applied to other types of fields, such as magnetic fields. The only difference is that the electric dipole moment would be replaced by the magnetic dipole moment, and the electric field would be replaced by the magnetic field. This highlights the versatility of these equations and their applicability to various physical systems.

5. How are these equations used in practical applications?

F = del(p.E) and F = (p.del)E are used in a variety of practical applications, such as in the design and analysis of electric motors, generators, and other electrical devices. These equations are also used in the study of molecular interactions and in understanding the behavior of charged particles in electric fields. Additionally, these equations are fundamental in the field of electromagnetism and are essential for understanding and predicting the behavior of electrically charged particles.

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