Shyan said:For any magnetic field ## \vec B ##, there are an infinite number of equivalent magnetic vector potentials##\vec A##(##\vec B=\vec \nabla\times\vec A ##), related by ## \vec {A'}=\vec A+\vec \nabla \phi ## for some scalar field ## \phi ##. So the magnetic vector potential of a magnetic field is not uniquely determined.
As I said, all those infinite number of vector potentials are equivalent, which means they all give the same magnetic field. That's because ## \vec \nabla \times \vec \nabla \phi=0 ## for any scalar field ## \phi ##. So ## \vec{A'}=\vec A+\vec\nabla \phi \Rightarrow \vec\nabla\times\vec {A'}=\vec\nabla\times\vec A ##.fricke said:thank you very much for the reply!
As what you have explained, magnetic vector potential is not uniquely determined since there are an infinite number of equivalent vector potential A, for magnetic field B. But does it mean B also not uniquely determined since we could choose any scalar field of A?
Shyan said:As I said, all those infinite number of vector potentials are equivalent, which means they all give the same magnetic field. That's because ## \vec \nabla \times \vec \nabla \phi=0 ## for any scalar field ## \phi ##. So ## \vec{A'}=\vec A+\vec\nabla \phi \Rightarrow \vec\nabla\times\vec {A'}=\vec\nabla\times\vec A ##.
The magnetic vector potential is a mathematical quantity used to describe the magnetic field in a given region of space. It is a vector quantity and is denoted by the symbol A.
The magnetic vector potential is a mathematical construct used to describe the magnetic field, while the magnetic field is a physical quantity that describes the force exerted on a moving charge in the presence of a magnetic field.
The magnetic vector potential is considered unique because it is not uniquely determined by the magnetic field. Different choices of the vector potential can lead to the same magnetic field, making it a non-unique quantity.
The uniqueness of the magnetic vector potential is important in physics because it allows for the application of gauge transformations, which can simplify mathematical calculations and provide a better understanding of the underlying physical principles at play.
No, the magnetic vector potential cannot be measured directly. It is a mathematical construct used to describe the magnetic field and does not have a physical existence on its own.