Name: solenoid vector potential explained
Uploaded: 2023-09-18T17:11:20.000Z
Duration: 1 min 15 s
Channel: JoVE

The divergence of a magnetic field is zero, and the divergence of the curl of any vector is always zero. So, a magnetic vector potential can be defined such that its curl equals the magnetic field.
The differential form of Ampere&#39;s Law states that the curl of the magnetic field equals the vacuum permeability times the current density.
Replacing the magnetic field with the magnetic vector potential, the equation gets modified.
Now, according to the vector product rule, the curl of the curl of a vector equals the gradient of its divergence minus its vector Laplacian.
Since the vector potential can be chosen to be non divergent, the Laplacian of the magnetic vector potential equals the permeability times the current density.
This differential equation bears an analogy to Poisson&#39;s equation in electrostatics.
Considering that the current density goes to zero at infinity, the solution of the Laplacian equation gives the vector potential.
The magnetic field due to any current source can be evaluated using the corresponding vector potential.

magnetic vector potential

Magnetic Vector Potential

In electrostatics, the electric field can be written as the negative gradient of the potential. In magnetostatics, the zero divergence of the magnetic field ensures that the magnetic field can be expressed as the curl of a vector potential. This potential is known as the magnetic vector potential.
Consider an ideal solenoid with n turns per unit length and radius R. If I is the current through the solenoid, the magnetic field inside the solenoid is expressed as the product of vacuum permeability, the number of turns per unit length, and the current. Conversely, the magnetic field outside the solenoid is zero. Considering this, what is the vector potential for an ideal solenoid?
The magnetic flux through the solenoid is given by
<img alt="Quantum mechanics principle diagram, illustrating Schrödinger's equation: iħ∂/∂t|Ψ⟩=H|Ψ⟩." src="/files/ftp_upload/14204/14204_Equation_1.png" style="text-align: center; height: 40px;" data-altupdatedat="1761830439000" data-alt="Equation1">
Since the magnetic field equals the curl of the vector potential, the magnetic flux can be rewritten in terms of the vector potential.
<img alt="Quantum harmonic oscillator potential, V(x)=1/2mω²x², wave function, ψ(x), physics diagram." src="/files/ftp_upload/14204/14204_Equation_2.png" style="text-align: center; height: 38px;" data-altupdatedat="1761830439000" data-alt="Equation2">
Thus, the line integral of the magnetic vector potential equals the surface integral of the magnetic field.
<img alt="Static equilibrium diagram with forces, ΣFx=0, ΣFy=0, showing force balance on a beam system." src="/files/ftp_upload/14204/14204_Equation_3.png" style="text-align: center; height: 40px;" data-altupdatedat="1761830439000" data-alt="Equation3">
Now consider a circular Amperian loop of radius r inside the solenoid. The magnetic flux through this loop is given by
<img alt="Static equilibrium concept; sum of forces ΣF=0, sum of moments Στ=0; balance beam diagram." src="/files/ftp_upload/14204/14204_Equation_4.png" style="text-align: center; height: 40px;" data-altupdatedat="1761830439000" data-alt="Equation4">
Equating the magnetic flux to the line integral of the magnetic vector potential, the expression for the vector potential can be obtained.
<img alt="Chemical equilibrium equations diagram demonstrating reaction balance and stoichiometry." src="/files/ftp_upload/14204/14204_Equation_5.png" style="text-align: center; height: 40px;" data-altupdatedat="1761830439000" data-alt="Equation5">
The vector potential mimics the magnetic field and acts along the circumference.

In electrostatics, the electric field can be written as the negative gradient of the potential. In magnetostatics, the zero divergence of the magnetic ...

Education

JoVE Core

Physics

Unit 1

Sources of Magnetic Fields

KEY TERMS AND CONCEPTS

solenoid vector potential explained

The magnetic vector potential can be used to describe the magnetic field inside an ideal solenoid. In magnetostatics, the magnetic field has zero divergence, so it can be written as the curl of a vector potential. This idea connects the magnetic field to a potential in the same way that electrostatics connects the electric field to an electric potential.

For an ideal solenoid, the magnetic field inside is given by the vacuum permeability, the number of turns per unit length, and the current. If the solenoid has n turns per unit length, radius R, and current I, the field inside follows that product. Outside the solenoid, the magnetic field is zero. That setup makes the solenoid a useful case for finding the vector potential.

The magnetic flux through the solenoid can be written in terms of the vector potential because the curl of the vector potential gives the magnetic field. In this way, the line integral of the magnetic vector potential equals the surface integral of the magnetic field. To find the potential, consider a circular Amperian loop of radius r inside the solenoid. Using the flux through that loop leads to an expression for the vector potential.

For this ideal case, the vector potential points along the circumference. It follows the same circular direction as the loop. In that sense, it mimics the magnetic field while remaining a vector potential rather than the field itself.

Solenoid Vector Potential Explained

biot-savart law

Magnetic Field from a Current Element

The Biot-Savart law links a current-carrying wire to the magnetic field it creates. It gives both the magnitude and the direction of that field. The law is named for Jean-Baptiste Biot and Félix Savart, who studied how a straight wire carrying current interacts with a permanent magnet. A current-carrying wire produces a magnetic field around it. To describe the field from a tiny section of wire, we use an infinitesimal current element, dl. The direction of dl follows the direction of the...

biot-savart law: problem-solving

Calculating Magnetic Fields with Biot-Savart

The Biot-Savart law is used to calculate the magnetic field made by a steady current. A mobile phone battery bank and the wire between the two can be used as an example of that current. At a field point P, the goal is to find the magnitude of the magnetic field created by the current in the wire. The first step is to look at a small current element of length dl. This element is at a distance r from the field point. The known values are the current, the length of the current element, the...

magnetic field due to moving charges

Magnetic Fields from Charges in Motion

A moving charge creates a magnetic field. For a point charge moving at constant velocity, the field at any point depends on the charge and the distance from the source point to the field point. It is directly proportional to the charge magnitude and inversely proportional to the square of that distance. The magnetic field also has a direction. Unlike the electric field, it is always perpendicular to the plane formed by the line from the source to the field point. Its strength depends on the...

magnetic field due to two straight wires

Net Magnetic Field at the Midpoint of Two Wires

Two parallel straight wires can create a net magnetic field at the point halfway between them. In this example, the wires carry currents of 10 A and 20 A and are separated by 20 cm. The midpoint, labeled P2, is 10 cm from each wire. The magnetic field lines around each wire form counterclockwise concentric circles. At the midpoint, the field from one wire can point in the opposite direction from the field from the other wire. The net magnetic field is found by magnetic field superposition,...

magnetic force between two parallel currents

magnetic-force-between-two-parallel-currents

Parallel Currents and the Pinch Effect

Parallel current-carrying wires can pull on each other with a magnetic force. The force has the same size on both conductors. Its direction depends on whether the currents flow in the same direction or in opposite directions. The magnetic field from one long, straight wire acts on a finite length of the other wire. The force on that wire comes from the product of its current and the cross product of its length vector with the field. Using the right-hand rule, the second conductor is pulled...

magnetic field of a current loop

Field Along the Axis of a Current Loop

A circular current loop creates a magnetic field along its axis. For a loop of radius a carrying current I, the field at a point P can be found with the Biot-Savart law. The point P lies a distance x from the center of the loop. The field from a tiny current element dl points at an angle θ. At every point on the loop, the current element and the unit vector toward P are perpendicular. Using that relationship and writing r in terms of x and a gives the size of the field from each element. The...

divergence and curl of magnetic field

Magnetic Field Loops and Ampere's Law

Magnetic field loops and Ampere's law describe how magnetic fields behave around current. For a volume current distribution, the magnetic field can be written using the Biot–Savart law. The divergence and curl of that field can then be found with vector analysis. To find the divergence, the divergence operator is applied to both sides of the Biot–Savart equation. After using the vector product rule, the integrand simplifies. The term with the curl of the current density becomes zero because...

ampere's law

Magnetic Circulation Around a Current

Magnetic circulation around a current is described by Ampère’s law. A static magnetic field is not conservative, unlike an electrostatic field. Instead, the magnetic field is linked to electric current, which is its source. Ampère’s law uses the line integral of the magnetic field around a closed path. This integral is proportional to the current enclosed by the loop. The law applies only when currents are steady and when no magnetic materials or time-varying electric fields are present. The...

ampere's law: problem-solving

Choosing an Amperian Loop for Magnetic Fields

Ampere's law connects a closed loop path with the current enclosed by that loop. The line integral of the magnetic field around the path equals the vacuum permeability times the enclosed current. A right-hand rule helps set the sign of the current. If the fingers curl with the integration path, the thumb gives positive current. Current opposite the thumb direction is negative. Solving magnetic field problems with Ampere's law starts by checking symmetry in the current distribution. If the...

solenoids

Solenoids as Controllable Electromagnets

A solenoid is a tightly wound coil of insulated conducting wire. It acts as an electromagnet, and its magnetic field comes from the combined field of each turn in the coil. In an ideal solenoid, the magnetic field inside is almost uniform and points along the solenoid axis. The magnetic field outside is nearly zero. Each turn can be treated as a circular current-carrying loop, and each loop creates a magnetic dipole moment. The dipole moment from one turn equals the current multiplied by the...

magnetic field of a solenoid

Solenoid Field Strength Example

A solenoid is a conducting wire coated with an insulating material and wound tightly into a helical coil. Its magnetic field comes from the vector sum of the fields from each turn. For an ideal solenoid, the field inside depends directly on the number of turns per unit length and the current. The magnetic field outside an ideal solenoid is zero. In this example, a solenoid has 100 turns wrapped around a cylinder that is 1 cm in diameter and 10 cm long. A current of 0.5 A is passed through the...

toroids

Toroid Magnetic Fields and Turn Count

A toroid is a donut-shaped coil made from a single conducting wire. It usually has many circular loops that sit perpendicular to its axis. When current flows through it, the magnetic field forms circular lines around the axis. The right-hand rule shows the field direction: curl the fingers of your right hand with the current, and your thumb points in the magnetic field direction. Inside a toroid, the magnetic field changes with distance from the axis. It varies inversely with that distance.

The magnetic vector potential can be used to describe the magnetic field inside an ideal solenoid. In magnetostatics, the magnetic field has zero divergence, so it can be written as the curl of a vector potential. This idea connects the magnetic field to a potential in the same way that electrostatics connects the electric field to an electric potential. For an ideal solenoid, the magnetic field inside is given by the vacuum permeability, the number of turns per unit length, and the current.

potential due to a magnetized object

Magnetization and Bound Currents

Magnetization describes how a magnetic material responds to an external magnetic field. In paramagnets and ferromagnets, magnetic dipoles align with the field. In diamagnets, the dipoles align opposite to the field. Magnetization is the dipole moment per unit volume, and it plays a role similar to polarization in electrostatics. The magnetic vector potential of a magnetized material can be written in terms of its aligned dipoles. This form is simplified with vector analysis. One term...

magnetostatic boundary conditions

Magnetic Fields at Material Interfaces

Magnetic fields change at the boundary between two magnetic materials, and the change depends on the surface conditions at the interface. A magnetic field can be discontinuous at a surface current, just as an electric field can change across a surface charge. The component of the magnetic field that points perpendicular to the boundary stays continuous. The component parallel to the boundary behaves differently. It becomes discontinuous by an amount equal to the vacuum permeability times the...

ampere's law in matter

Free Current and Magnetization in Ampere’s Law

Ampere’s law in matter links magnetic fields to the free current in a magnetized material. In such materials, the total current density has two parts. These are the free current density and the bound current density. The free current comes from the motion of free electrons inside the material. The bound current comes from the alignment of magnetic dipole moments. In vacuum, the differential form of Ampere’s law says that the curl of the magnetic field equals permeability times current density.

magnetic susceptibility and permeability

How Materials Respond to Magnetic Fields

Magnetic susceptibility and permeability describe how materials respond to a magnetic field. In linear magnetic materials, magnetization is proportional to the magnetic field intensity. The proportionality constant is magnetic susceptibility, which is a dimensionless number. The value of susceptibility depends on the material. Diamagnetic materials develop moments opposite to the applied field. Their susceptibility is slightly negative, usually about 10^-5 to 10^-6. This effect comes from the...

magnetic moment of an electron

Electron Spin and Bohr Magneton

Electrons create magnetic effects through both orbital motion and spin. When an electron moves around a nucleus, it acts like a tiny current loop. That current produces a magnetic dipole moment that is proportional to the electron's orbital angular momentum. The orbital angular momentum is quantized in units of the reduced Planck's constant. Because of that, the magnetic dipole moment is also quantized in the Bohr magneton. The Bohr magneton has a value of 9.27 x 10 -24 Am 2. Electrons also...

diamagnetism

Diamagnetism and Magnetic Repulsion

Diamagnetism is the magnetic response of materials that are repelled by an external magnetic field. These materials contain paired electrons, so they have no net magnetic moment at first. When a magnetic field is applied, moments opposite to the field are induced. Anton Brugmans discovered this effect in 1778 after observing that bismuth is repelled by magnetic fields. He used this observation to suggest that diamagnets are pushed away from magnets. In a non-uniform magnetic field, diamagnets...

paramagnetism

Paramagnetism and Magnetic Alignment

Paramagnetism and magnetic alignment explain how some materials respond to an outside magnetic field. Paramagnets contain unpaired electrons, so they have a finite magnetic moment. When no magnetic field is present, these moments point in random directions, and the net moment is zero. When an external field is applied, a torque acts on the magnetic moments and pulls them toward the field direction. At the same time, thermal motion of the electrons creates a competing torque that tries to...

ferromagnetism

Magnetic Domains and Hysteresis

Ferromagnetic materials like iron, nickel, and cobalt contain magnetic domains. Inside each domain, magnetic dipoles line up in the same direction. Quantum mechanical coupling keeps this alignment strong, so room-temperature thermal motion usually cannot break it. As a result, each domain has a net dipole moment. Some materials have weaker coupling and become ferromagnetic only at lower temperatures. When an external magnetic field is applied, the dipoles rotate to match the field. At the same...

key terms and concepts

Learn how the magnetic vector potential is found for an ideal solenoid, using magnetic flux, a circular Amperian loop, and the field inside the solenoid.

Solenoid Vector Potential Explained | JoVE K12

physics

JoVE Core Physics introduces the theory and application of fundamental physics topics, including kinematics, dynamics, fluid mechanics, statics, waves, and acoustics, with concise and easy-to-understand animated videos. Additionally, the scientist-in-action videos demonstrate the application of related concepts in classical and original research experiments performed in today's laboratories worldwide.

Solenoid Vector Potential Explained

Description

Transcript

Tags