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gursimran

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- Thread starter gursimran
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In summary: In quantum mechanical systems, there are other potentials that also have physical interpretations. 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. In quantum mechanical systems, there are other potentials that also have physical interpretations.

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gursimran

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Pengwuino

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Studiot

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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, [itex]\varphi[/itex] at some point

then the vector

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

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gursimran

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Studiot said:

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, [itex]\varphi[/itex] at some point

then the vectora= [itex]\nabla\varphi[/itex] 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

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Pengwuino

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gursimran said: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.

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Dickfore

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Because the magnetic field is solenoidal ([itex]\mathrm{div} \vec{B} = 0[/itex]), it can be expressed as the curl of a vector field:

[tex]

\vec{B} =\mathrm{curl} \, \vec{A}

[/tex]

Since the curl of a gradient of any scalar function is zero ([itex]\mathrm{curl} \, \mathrm{grad} \, \phi = 0[/itex]), the vector potential is determined only up to a gradient of a scalar function:

[tex]

\vec{A} = \vec{A'} + \mathrm{grad} \, \Lambda

[/tex]

The flux of the magnetic field through a closed contour is:

[tex]

\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}}

[/tex]

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:

[tex]

\oint_{C}{\mathrm{grad} \, \Lambda \cdot d\vec{l}} = 0

[/tex]

[tex]

\vec{B} =\mathrm{curl} \, \vec{A}

[/tex]

Since the curl of a gradient of any scalar function is zero ([itex]\mathrm{curl} \, \mathrm{grad} \, \phi = 0[/itex]), the vector potential is determined only up to a gradient of a scalar function:

[tex]

\vec{A} = \vec{A'} + \mathrm{grad} \, \Lambda

[/tex]

The flux of the magnetic field through a closed contour is:

[tex]

\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}}

[/tex]

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:

[tex]

\oint_{C}{\mathrm{grad} \, \Lambda \cdot d\vec{l}} = 0

[/tex]

Last edited:

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gursimran

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Pengwuino said: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)

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Pengwuino

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gursimran said: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.

The vector potential is a mathematical concept used in electromagnetism to describe the direction and strength of the magnetic field at a given point in space. It is related to the magnetic field by the equation B = ∇ x A, where B is the magnetic field and A is the vector potential. Essentially, the vector potential tells us how the magnetic field is changing in space.

The vector potential plays a crucial role in Maxwell's equations, which are the fundamental equations that govern electromagnetism. It allows us to calculate the magnetic field in situations where the electric field is changing, such as when a current is flowing through a wire. Additionally, the vector potential is important in quantum mechanics, where it is used to describe the behavior of particles with spin.

The vector potential and the electric potential are related by the equation A = -∇φ, where φ is the electric potential. This means that the vector potential is the negative gradient of the electric potential. In other words, the vector potential is a measure of how quickly the electric potential changes in different directions.

Unlike the electric and magnetic fields, the vector potential cannot be directly measured. This is because it is a mathematical construct used to describe the behavior of the magnetic field. However, its effects can be observed and measured indirectly, such as through the deflection of a compass needle in the presence of a current-carrying wire.

The vector potential has many practical applications in electromagnetism, such as in the design of motors and generators, as well as in the fields of electronics and telecommunications. It is also used in theoretical physics to study the behavior of particles with spin, and in quantum mechanics to describe the behavior of particles in magnetic fields.

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