What is the physical interpretaion of the vector potential.

[gursimran]

I was reading the text of electricity and magnetism by griffiths. Here I read a term called magnetic potential but I did not completely understood the physical essence of the term, neither it is explained in the book. It should have some physical interpretation as it is named a potential. In what sense it is a potential??

 

[Pengwuino]

As far as I know, it's simply a mathematical tool with no concrete physical meaning. I can't imagine it has physical meaning since for any configuration, the vector and scalar potentials are not uniquely defined.

 

[Studiot]

Hello girsimran,

There are many vector potential functions in Physics.

Basically if we can assign a scalar value to every point in some region of space, the vector potential is the gradient of this scalar as we pass from one point to another.

Mathematically if we have a scalar function of position, $\varphi$ at some point

then the vector a = $\nabla\varphi$ is the vector value and direction of the vector potential at this point.

All the scalars form a scalar field and all the vectors field over the region in question.

Examples are Gravitational potential, Electrostatic potential, Magnetostatic potential, Fluid velocity potential.

Very often (as with the fluid velocity field) we have the vector potential and infer the existence of a scalar from it.

go well

 

[gursimran]

Hello girsimran,

There are many vector potential functions in Physics.

Basically if we can assign a scalar value to every point in some region of space, the vector potential is the gradient of this scalar as we pass from one point to another.

Mathematically if we have a scalar function of position, $\varphi$ at some point

then the vector a = $\nabla\varphi$ is the vector value and direction of the vector potential at this point.

All the scalars form a scalar field and all the vectors field over the region in question.

Examples are Gravitational potential, Electrostatic potential, Magnetostatic potential, Fluid velocity potential.

Very often (as with the fluid velocity field) we have the vector potential and infer the existence of a scalar from it.

go well

Hey first of thanks for the answer but what you wrote about is the scalar potential. I'm asking about vector potential. This one http://en.wikipedia.org/wiki/Vector_potential

 

[Pengwuino]

Hey first of thanks for the answer but what you wrote about is the scalar potential. I'm asking about vector potential. This one http://en.wikipedia.org/wiki/Vector_potential

The idea is the same. Potentials have no significant meaning in classical e/m since they're just mathematical tools that are not unique to any given situation. Things get a little clouded when you start looking at quantum mechanical effects, however. You may want to look up the Aharonov-Bohm effect.

 

[Dickfore]

Because the magnetic field is solenoidal ($\mathrm{div} \vec{B} = 0$), it can be expressed as the curl of a vector field:
$$\vec{B} =\mathrm{curl} \, \vec{A}$$
Since the curl of a gradient of any scalar function is zero ($\mathrm{curl} \, \mathrm{grad} \, \phi = 0$), the vector potential is determined only up to a gradient of a scalar function:
$$\vec{A} = \vec{A'} + \mathrm{grad} \, \Lambda$$
The flux of the magnetic field through a closed contour is:
$$\Phi = \int_{S}{\vec{B} \cdot \hat{n} \, da} = \int_{S}{\mathrm{curl} \vec{A} \cdot \hat{n} \, da} = \oint_{C}{\vec{A} \cdot d\vec{l}}$$
is given by the circulation of the vector potential around its boundary. We see that the arbitrariness of the definition of the vector potential disappears since the circulation of a gradient around a closed contour is always zero:
$$\oint_{C}{\mathrm{grad} \, \Lambda \cdot d\vec{l}} = 0$$

 

[gursimran]

The idea is the same. Potentials have no significant meaning in classical e/m since they're just mathematical tools that are not unique to any given situation. Things get a little clouded when you start looking at quantum mechanical effects, however. You may want to look up the Aharonov-Bohm effect.

Thanks for answering. But I do not agree that potentials are just mathematical constructs. Gravitaional potentail or electric potential has significant well understood physical interpretation which is in accordance with its name, potential (ie potential to do work.. due to its location in space)

 

[Pengwuino]

Thanks for answering. But I do not agree that potentials are just mathematical constructs. Gravitaional potentail or electric potential has significant well understood physical interpretation which is in accordance with its name, potential (ie potential to do work.. due to its location in space)

That's why I specifically noted classical e/m potentials. I should have further noted this actually only applies to the vector potential as well.