Axially symmetric B field vector potential?

• AxiomOfChoice
In summary, in a scenario where the azimuthal component B_phi is equal to zero in an axially symmetric magnetic field, there are possible vector potentials A that could produce this field. One such potential is A = (1/2)(B_r * z-hat - B_z * r-hat), where the components of B can be expressed in terms of the partial derivatives of A with respect to r and z.
AxiomOfChoice
Suppose you have an axially symmetric magnetic field for which the azimuthal component $$B_\phi = 0$$. This is all you know. What are some possible vector potentials $$\vec A$$ (such that $$\vec B = \nabla \times \vec A$$) that would produce this field? (So we're working in cylindrical coordinates.)

The obvious one I've thought of is just $$\vec A = A \hat \phi$$. But I'm not sure what form $$A$$ should take in terms of $$B_r$$ and $$B_z$$, where $$\vec B = B_r \hat r + B_z \hat z$$.

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One possible form of A is A = \frac{1}{2} (B_r \hat z - B_z \hat r). This gives B_\phi = 0, and B_r = \frac{\partial A_z}{\partial r} - \frac{\partial A_r}{\partial z}, and B_z = \frac{\partial A_r}{\partial r} - \frac{\partial A_z}{\partial z}.

One possible vector potential that would produce an axially symmetric magnetic field with no azimuthal component is \vec A = \frac{1}{2}B_r r^2 \hat \phi, where B_r is the radial component of the magnetic field. This potential satisfies the condition of \nabla \times \vec A = \vec B and also takes into account the fact that the magnetic field strength increases with radial distance from the axis of symmetry.

Another potential that could produce this magnetic field is \vec A = \frac{1}{2}B_z z^2 \hat \phi, where B_z is the axial component of the magnetic field. This potential takes into account the fact that the magnetic field strength also increases with axial distance from the axis of symmetry.

In general, there are many possible vector potentials that could produce an axially symmetric magnetic field with no azimuthal component. The choice of which potential to use would depend on the specific characteristics of the magnetic field and the system in question.

1. What is an axially symmetric B field vector potential?

An axially symmetric B field vector potential describes the magnetic field around a symmetrical object, such as a cylinder or sphere. It is a mathematical representation of the magnetic field that takes into account the symmetry of the object.

2. How is the axially symmetric B field vector potential calculated?

The axially symmetric B field vector potential is calculated using mathematical equations, such as the Biot-Savart law or Maxwell's equations, depending on the specific situation and geometry of the object. These equations take into account the properties of the object, such as its size, shape, and magnetic properties.

3. What are some applications of the axially symmetric B field vector potential?

The axially symmetric B field vector potential is commonly used in physics and engineering applications, such as in the design of electromagnets, magnetic sensors, and MRI machines. It is also used in theoretical studies of magnetism and in modeling the behavior of magnetic fields in different situations.

4. How does the axially symmetric B field vector potential differ from other types of magnetic fields?

The axially symmetric B field vector potential is unique in that it takes into account the symmetry of an object, whereas other types of magnetic fields may not. This allows for more accurate calculations and predictions of the magnetic field around symmetrical objects.

5. Are there any limitations or assumptions when using the axially symmetric B field vector potential?

Yes, there are some limitations and assumptions when using the axially symmetric B field vector potential. For example, it assumes that the object being studied is perfectly symmetrical, which may not always be the case in real-world scenarios. Additionally, it may not accurately represent the magnetic field in regions that are far from the object or in cases where the magnetic field is highly complex.

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