Relationship between Bit-Savart and Ampere laws

In summary: However, if the surface is in motion, then you must take the time derivative of both sides of the equation to obtain the correct law for the electric field:$$\oint_{\partial A} \mathrm{d} \vec{r} \cdot \vec{B}=\mu_0 \vec{J}+\frac{1}{c^2} \oint_A \mathrm{d}^2 \vec{a} \cdot \partial_t \vec{E}.$$
  • #1
carllacan
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I've tried to relate Biot-Savart's Law to Ampere's and I've found a contradiction, which I guess is due to a naive use of Ampere's.

If ## \int \vec B · d\vec l = \mu_0 I_{enc} ## is applied to a circle of radius R around a current element ##Id\vec l## we have ## B·2\pi R = \mu_{0} I_{enc} ##, which gives ## B = \frac{\mu_0 I_{enc}}{2\pi R} ##, different from Biot-Savart. I'm guessins we can't use AMpere in this situation, but I can't put my finger on the exact reason.
 
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  • #2
An isolated current element is impossible, it does not conserve charge. Instead of a current element you need to use a loop. Try an infinitely long straight wire with a return path at infinity.
 
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  • #3
carllacan said:
If B⃗ ·dl⃗ =μ0Ienc \int \vec B · d\vec l = \mu_0 I_{enc} is applied to a circle of radius R around a current element IdlId\vec l

One problem here is with the concept of Ienc which is often stated in first-year textbooks as "the current through the loop" (circle in this case). What this means precisely is: "the current that passes through any surface whose boundary is the loop." The problem is that with a finite current element, you can construct some surfaces that the current "pierces", and some that that the current does not "pierce" (i.e. the surface is curved in such a way as to avoid the current element entirely). So "current through the loop" is ambiguous for a finite current element. It depends on which surface you use. If you use a complete current loop instead of a finite element, you avoid the ambiguity.

As DaleSpam also noted, an isolated current element (or any finite-length current-carrying wire) does not conserve charge all by itself. You can make it conserve charge by attaching a charged object to each end of the wire. As the current flows, one charge becomes more negative and the other becomes more positive. As these charges change, the electric field that they produce also changes. This changing electric field is associated with a magnetic field (in addition to the magnetic field associated with the current), according to the term that Maxwell added to Ampere's Law in order to make his equations "complete." This term compensates for the different possible values of "current through the loop".
$$\oint {\vec B \cdot d \vec l} = \mu_0 \int {\vec J \cdot d \vec a} + \mu_0 \epsilon_0 \frac {d}{dt} \int {\vec E \cdot d \vec a} \\
\oint {\vec B \cdot d \vec l} = \mu_0 i_\textrm{enc} + \mu_0 \epsilon_0 \frac {d}{dt} \int {\vec E \cdot d \vec a}$$
 
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  • #4
Wow, that was enlightening. Many thanks you both!
 
  • #5
Caveat concerning #3: This analysis is only valid for a surface on the right-hand side at rest. Otherwise there's an additional line integral missing. You find the correct laws always starting from the local Maxwell equations which are the fundamental equations anyway. Here you start from the Ampere-Maxwell law (written in SI units, sigh):
$$\vec{\nabla} \times \vec{B}=\mu_0 \vec{J} + \frac{1}{c^2} \partial_t \vec{E}.$$
Now using Stokes's integral theorem this gives the correct general form of the corresponding integral equation,
$$\int_{\partial A} \mathrm{d} \vec{r} \cdot \vec{B}=\mu_0 \vec{J} + \frac{1}{c^2} \int_A \mathrm{d}^2 \vec{a} \cdot \partial_t \vec{E}.$$
Now to take the time derivative outside of the integral, one must note that
$$\frac{\mathrm{d}}{\mathrm{d} t} \int_A \mathrm{d}^2 \vec{a} \cdot \vec{E}=\int_{A} \mathrm{d}^2 \vec{a} \cdot \left [\partial_t \vec{E}+\vec{v} (\vec{\nabla} \cdot \vec{E}) \right ] - \int_{\partial A} \mathrm{d} \vec{r} \cdot (\vec{v} \times \vec{E}).$$
Here ##\vec{v}## is the velocity field of the moving surface. If the surface is at rest, there's of course no problem, and you can simply put the time derivative out of the integral!
 

1. What is the Bit-Savart law?

The Bit-Savart law is a fundamental law in electromagnetism that describes the magnetic field produced by a steady current in a wire. It states that the magnetic field at a specific point is directly proportional to the current, the length of the wire, and the sine of the angle between the wire and the point. This law is named after French mathematician and physicist, Jean-Baptiste Biot, and French physicist, Félix Savart.

2. What is the Ampere law?

The Ampere law is another fundamental law in electromagnetism that relates the magnetic field around a closed loop to the electric current passing through that loop. It states that the magnetic field is directly proportional to the current passing through the loop and the length of the loop. This law is named after French mathematician and physicist, André-Marie Ampère.

3. What is the relationship between Bit-Savart and Ampere laws?

The Bit-Savart and Ampere laws are closely related and are used to describe different aspects of the same phenomenon. The Bit-Savart law is used to calculate the magnetic field produced by a steady current in a wire, while the Ampere law is used to calculate the magnetic field around a closed loop. Both laws involve the current and the length of the wire or loop, but the angle between the current and the point or the shape of the loop can affect the calculations.

4. How are the Bit-Savart and Ampere laws derived?

The Bit-Savart law can be derived from the Biot-Savart law, which is a more general form of the law that describes the magnetic field produced by a current-carrying element. The Ampere law can be derived from the Biot-Savart law and the principle of superposition, which states that the total magnetic field at a point is the vector sum of the magnetic fields produced by each current-carrying element in the system.

5. What are the practical applications of the Bit-Savart and Ampere laws?

The Bit-Savart and Ampere laws have many practical applications in engineering and physics. They are used in the design of electromagnets, motors, and generators, as well as in the study of electromagnetic fields and their effects on objects. These laws also play a crucial role in the development of technologies such as magnetic resonance imaging (MRI) and particle accelerators.

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