# Uniqueness of magnetic vector potential

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1. Jul 12, 2015

### fricke

I able to prove magnetic field is uniquely determined but I am confused how to prove that magnetic vector potential is also unique.

Can I say that magnetic vector potential is uniquely determined since magnetic field has unique solution?

Thanks.

2. Jul 12, 2015

### ShayanJ

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.

3. Jul 12, 2015

### fricke

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?

4. Jul 12, 2015

### ShayanJ

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 $\vec{A'}=\vec A+\vec\nabla \phi \Rightarrow \vec\nabla\times\vec {A'}=\vec\nabla\times\vec A$.

5. Jul 12, 2015

### fricke

thank you very much! I understand now! thank you.

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