The Law of Biot and Savart again

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In summary, the Biot and Savart integral can be used to calculate the magnetic field of a steady current in a closed loop. The corresponding formula for the vector potential is given by \vec A(\vec r) = \frac{\mu_0 I}{4\pi} \oint \limits_{\mathcal{C}} \mathrm d\vec r^{\, \prime} \, \frac{1}{|\vec r - \vec r^{\, \prime}|}, where the curve \mathcal{C} is parameterized through \vec r^{\, \, \prime}(t) and must be closed for the integral to converge. It is also possible to calculate the vector potential for an arbitrary divergence
  • #1
wofsy
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The magnetic field of a steady current in a loop is given by the Biot and savart integral which is

1/4pi Integral[((x-y)/|x-y|^3) x dy] = B(x)

What is the corresponding formula for the vector potential?
 
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  • #2
Although i can't decrypt the formula you have stated, i give it a try


[tex]
\vec A(\vec r) = \frac{\mu_0 I}{4\pi} \oint \limits_{\mathcal{C}} \mathrm d\vec r^{\, \prime} \, \frac{1}{|\vec r - \vec r^{\, \prime}|}
[/tex]

The curve [tex]\mathcal{C}[/tex] is parameterized through [tex]\vec r^{ \, \, \prime}(t)[/tex] !

It is necessary to mention, that the curve [tex]\mathcal{C}[/tex] must be closed, otherwise the integral diverges!


Best regards...
 
  • #3
saunderson said:
Although i can't decrypt the formula you have stated, i give it a try


[tex]
\vec A(\vec r) = \frac{\mu_0 I}{4\pi} \oint \limits_{\mathcal{C}} \mathrm d\vec r^{\, \prime} \, \frac{1}{|\vec r - \vec r^{\, \prime}|}
[/tex]

The curve [tex]\mathcal{C}[/tex] is parameterized through [tex]\vec r^{ \, \, \prime}(t)[/tex] !

It is necessary to mention, that the curve [tex]\mathcal{C}[/tex] must be closed, otherwise the integral diverges!


Best regards...

thanks I will try to prove it works.

BTW: how do you do the math notation?
 
  • #4
saunderson said:
Although i can't decrypt the formula you have stated, i give it a try


[tex]
\vec A(\vec r) = \frac{\mu_0 I}{4\pi} \oint \limits_{\mathcal{C}} \mathrm d\vec r^{\, \prime} \, \frac{1}{|\vec r - \vec r^{\, \prime}|}
[/tex]

The curve [tex]\mathcal{C}[/tex] is parameterized through [tex]\vec r^{ \, \, \prime}(t)[/tex] !

It is necessary to mention, that the curve [tex]\mathcal{C}[/tex] must be closed, otherwise the integral diverges!Best regards...

Thanks that works.
What about if you have an arbitrary divergence free field defined in space minus possibly a finite number of loops?

For instance if I have two magnetic fields generated by two non-linking current loops their cross product is divergence free. If there an integral formula for the vector potential of the cross product?

Or - suppose the magnetic field is confined to the interior of a closed tube as in a magnetic filament.
 
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  • #5


The corresponding formula for the vector potential is given by the Biot-Savart law as:

A(x) = 1/4pi Integral[(I/|x-y|) x dy]

where I is the current flowing through the loop and x is the position vector of the point where the vector potential is being calculated. This formula is derived from the Biot-Savart integral and represents the magnetic vector potential at a point due to a steady current in a loop. It is a fundamental equation in electromagnetism and is used to calculate the magnetic field and vector potential for various applications, such as inductance and magnetostatics.
 

1. What is the Law of Biot and Savart again?

The Law of Biot and Savart is a fundamental law in electromagnetism that describes the relationship between a small section of a current-carrying wire and the magnetic field it produces. It states that the magnetic field at a point is directly proportional to the current in the wire, the length of the wire, and the sine of the angle between the wire and the line connecting the point to the wire.

2. How is the Law of Biot and Savart used in scientific research?

The Law of Biot and Savart is used in many areas of scientific research, particularly in the study of electromagnetism and magnetism. It is used to calculate the magnetic field produced by a current-carrying wire, which is important in designing devices such as motors and generators. It is also used in the study of Earth's magnetic field and in medical imaging techniques like MRI.

3. What are the key principles of the Law of Biot and Savart?

The key principles of the Law of Biot and Savart are the relationship between current and magnetic field, the importance of the length of the wire in determining the strength of the field, and the impact of the angle between the wire and the point on the field. The law also follows the right-hand rule, where the direction of the magnetic field can be determined by pointing the thumb of the right hand in the direction of the current and curling the fingers towards the wire.

4. Can the Law of Biot and Savart be applied to non-current carrying wires?

While the Law of Biot and Savart is specifically formulated for current-carrying wires, it can be extended to non-current carrying wires by considering the current as a distribution of charges moving along the wire. In this case, the current is considered to be a series of infinitesimal current elements, and the law can still be applied.

5. How does the Law of Biot and Savart relate to other laws in electromagnetism?

The Law of Biot and Savart is closely related to the laws of Coulomb's Law and Ampere's Law. Coulomb's Law describes the relationship between electric charge and electric force, while Ampere's Law relates the magnetic field to the current enclosed by a loop. The Law of Biot and Savart is a combination of these two laws and can be derived from them.

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