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hokhani
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The work done by magnetic force, [itex]F=qv\times B[/itex], is zero. Can we infer that the magnetic force is conservative? If so, how can we show that [itex]\nabla \times F=0 [/itex]?
No. This in not homework. The magnetic force is perpendicular to the velocity of the particle so the work down by the force is zero always. Therefore the work doesn't depend on the path between the two points, say A and B. I want to know whether the magnetic force is conservative or not.Vanadium 50 said:Is this homework?
Where have you started?
hokhani said:I want to know whether the magnetic force is conservative or not.
Thanks, how to show [itex]\nabla \times F=0[/itex] with [itex]F=qv\times B[/itex]?DrStupid said:Conservative forces conserve energy. That's where the name comes from. You already know that this is the case for Lorentz force.
hokhani said:Thanks, how to show [itex]\nabla \times F=0[/itex] with [itex]F=qv\times B[/itex]?
I did that using the Formula [itex]\nabla \times (v\times B) =(\nabla . B)v-(\nabla . v)B+(B. \nabla)v-(v. \nabla)B[/itex] and I can only eliminate the first term ([itex] \nabla .B=0[/itex]). How to show that the other three terms are zero or cancel each other?vela said:Calculate ##\nabla \times (q\vec{v}\times\vec{B})##. It's your job to figure out how to do that — or to at least make an attempt.
As pointed out earlier (DrStupid), the magnetic force is not a force field, since it depends on the velocity. There is good discussion about this in wikipedia: https://en.wikipedia.org/wiki/Conservative_force, and also in the book on Classical Mechanics by John R Taylor.hokhani said:I did that using the Formula [itex]\nabla \times (v\times B) =(\nabla . B)v-(\nabla . v)B+(B. \nabla)v-(v. \nabla)B[/itex] and I can only eliminate the first term ([itex] \nabla .B=0[/itex]). How to show that the other three terms are zero or cancel each other?
One hears this frequently, but it is a misconception. The point is that the fundamental quantity in analytical mechanics isn't the potential energy function but the work function. As long as the work function ##U(q_1,q_2,\cdots, q_n,\dot{q}_1, \dot{q}_2, \dot{q}_n)## is independent of time, the ##\textit{generalised}## forces are conservative and may be calculated from this single scalar function ##U##, taking the form ##F_i = \frac{\partial U}{\partial q_i} - \frac{d}{dt}\frac{\partial U}{\partial \dot{q}_i}##. For time independent systems (##U## independent of t as well as the kinetic energy function, i.e. the constraint equations do not contain t explicitly) the total energy is conserved and can still be expressed as the sum of the kinetic energy and the ##\textit{potential}## energy ##V##, provided however that ##V## is defined as a Legendre transformation of the work function: $$V=\sum_i \frac{\partial U}{\partial \dot{q}_i}\dot{q}_i - U.$$Chandra Prayaga said:Neither of these forces is conservative
They are conservative as long as one generalises the common ##F_i=\frac{\partial U}{\partial q_i}## (which is appropriate for velocity independent work functions) to the more general case of velocity dependent functions ##F_i = \frac{\partial U}{\partial q_i} - \frac{d}{dt}\frac{\partial U}{\partial \dot{q}_i}##. If the work function is time-independent, we obtain a class of forces called "conservative" because they satisfy the conservation of energy, as the name suggests and was pointed out by Dr.Stupid.vanhees71 said:No, these forces are not conservative but energy is conserved, if ##U## is not explicitly time dependent. Again, a force is called conservative iff it's the gradient of a scalar potential, independent of the (generalized) velocities (or canonical momenta in the Hamiltonian formulation).
vanhees71 said:By definition a conservative force is a force that's expressible (at least locally) as the gradient of a scalar field.
vanhees71 said:
Not that this is an official definition or anything, but the end of the first chapter of Lanczos book discusses this: https://books.google.fr/books?id=cmPDAgAAQBAJ&pg=PT21&lpg=PT21&dq=lanczos+variational+principles+chapter+1&source=bl&ots=QBhzOVPOtd&sig=lKVm9oEL3BfDhK94jMBf8xCrzQc&hl=en&sa=X&ved=0CCMQ6AEwAGoVChMI9orY4N6CyQIVgT0UCh2L4gBY#v=onepage&q=lanczos variational principles chapter 1&f=falseDrStupid said:I do not find a clear definition in this reference. The second paragraph says that "A conservative force is dependent only on the position of the object." but it is not clear if this is part of the definition or if the author just had force fields in mind. I also checked the German version of this article and there is no such limitation. Both articles have in common that the work done by a conservative force is independent of the taken path. Thats the definition as I know it and it also applies to the Lorentz force.
It seems Wikipedia is no proper reference for this particular question. Where can we find the official definition of conservative forces? Does such a definition exist at all or is it just a matter of personal preferences?
That's always right ;-).DrStupid said:It seems Wikipedia is no proper reference for this particular question. Where can we find the official definition of conservative forces? Does such a definition exist at all or is it just a matter of personal preferences?
vanhees71 said:Take any textbook on classical mechanics, where you'll find the definition of conservative forces quite in the beginning.
Magnetic force is the force exerted by a magnet on another magnet or on a moving electric charge. It is one of the fundamental forces of nature.
A conservative force is a force that does not depend on the path taken by an object and only depends on the initial and final positions of the object. The work done by a conservative force is independent of the path taken and only depends on the initial and final positions of the object.
Yes, magnetic force is a conservative force. This means that the work done by magnetic force on an object is independent of the path taken by the object and only depends on the initial and final positions of the object.
Magnetic force is a conservative force because it follows the principle of energy conservation. The work done by magnetic force is always equal to the change in potential energy of the object, which is independent of the path taken by the object.
Aside from magnetic force, other examples of conservative forces include gravitational force, electric force, and elastic force. These forces all follow the principle of energy conservation and are independent of the path taken by the object.