The Conditions for Biot-Savart Law to Ampere's Law

In summary, the Biot-Savart and Ampere's Law are equivalent and require that all fields are time-independent and that the stationary current obeys the continuity equation. This can be applied to Faraday's Law, but it is more efficient to use Jefimenko's equation. These laws also have applications in aerodynamics for calculating velocity induced by vortex lines.
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Mathematically, what conditions must a B-field that obeys the Biot-Savart Law satisfy before it will obey Ampere's Law?

Additionally, what conditions must the B-field obey in order to satisfy Faraday's Law?
 
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  • #3
vanhees71 said:
The Biot-Savart and Ampere's Law are equivalent. The condition for their validity is that (a) all fields are time-independent and (b) that the stationary current obeys the (reduced) continuity equation ##\vec{\nabla} \cdot \vec{j}=0##. See

http://th.physik.uni-frankfurt.de/~hees/publ/ampere-law-discussion-ver2.pdf
http://dx.doi.org/10.1088/0143-0807/35/5/058001

thanks. I wanted to apply this to Faraday's Law since it also follows Stokes' theorem.
 
  • #4
Of course, you can do that, but that leads to very cumbersome and not very physically intuitive non-local equations. The local representation is given by Jefimenko's equation rather than by lumping field parts into the sources. It's also clear from relativistic covariance, that the electromagnetic field builds one "unit" in the sense that the electric and magnetic components together are the components of the electromagnetic field-strength (or Faraday) tensor ##F_{\mu \nu}## and the charge density and current density together are components of a four-vector field ##j^{\mu}=(c \rho,\vec{j})##.
 
  • #5
Both laws have applications in aerodynamics for calculating the velocity induced by vortex lines by using the magnetic induction current formula B=µH
 

1. What is the Biot-Savart Law?

The Biot-Savart Law is a mathematical equation that describes the magnetic field generated by a steady current, such as a current-carrying wire. It states that the magnetic field at a point is directly proportional to the current, the length of the current-carrying wire, and the sine of the angle between the wire and the point.

2. What are the conditions for the Biot-Savart Law to be applicable?

The Biot-Savart Law is applicable when the current is steady, the medium is a vacuum or air, and the distances involved are much larger than the dimensions of the current-carrying wire. Additionally, the wire must be thin and straight, and the magnetic field must be weak.

3. How is the Biot-Savart Law related to Ampere's Law?

Ampere's Law is a fundamental law in electromagnetism that relates the magnetic field around a closed loop to the electric current passing through the loop. The Biot-Savart Law is a special case of Ampere's Law when the loop is infinitesimal and the current is constant.

4. Can the Biot-Savart Law be used to calculate the magnetic field of a permanent magnet?

No, the Biot-Savart Law is only applicable to steady currents. Permanent magnets have a constantly changing magnetic field, so the Biot-Savart Law cannot be used to calculate their magnetic field.

5. How is the Biot-Savart Law important in practical applications?

The Biot-Savart Law is used in various practical applications, such as in the design of electromagnets, motors, and generators. It is also used in medical imaging, such as MRI machines, to create a magnetic field strong enough to produce images of the body's internal structures.

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