Understanding Ampere's Law: A Step-by-Step Derivation

In summary, the author is discussing how two different laws, Ampere's law and Biot Savart law, are related. The author is having difficulty understanding how the second term in Ampere's law vanishes to leave us with Biot Savart law.
  • #1
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This must be a pretty standard proof but I'm having difficulty with part of it.

So we have from Biot Savart law that [itex]\vec{B}(\vec{r})=\frac{\mu_0}{4 \pi} \int_V dV' \vec{J}(\vec{r'}) \times \nabla(\frac{1}{r})[/itex]

we take the curl of this and show the second term vanishes to leave us with [itex]\nabla \times \vec{B}(\vec{r})=\mu_0 \vec{J}(\vec{r})[/itex] which is Ampere's law in differential form.

however we take the curl of this using the identity a x (b x c) = b(a.c)-c(a.b). here this gives

[itex]\nabla \times \left(\vec{J}(\vec{r'}) \times \nabla(\frac{1}{r})\right) = \nabla^2(\frac{1}{r}) \vec{J}(\vec{r'}) - (\nabla \cdot \vec{J}(\vec{r'})) \nabla(\frac{1}{r})[/itex]

but in the book they have the scond term as [itex](\vec{J}(\vec{r'}) \cdot \nabla) \nabla(\frac{1}{r})[/itex]. why is this allowed?

the dot product doesn't commute when a grad is involved does it?
 
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  • #2
Try this identity: [tex] \nabla \times (\mathbf{A} \times \mathbf{B}) = \mathbf{A} (\nabla \cdot \mathbf{B}) - \mathbf{B} (\nabla \cdot \mathbf{A}) + (\mathbf{B} \cdot \nabla) \mathbf{A} - (\mathbf{A} \cdot \nabla) \mathbf{B} [/tex].
 
  • #3
ok so when i do that i get everything to work except i have an extra term [itex]-(\mathbf{J} \cdot \nabla) \nabla(\frac{1}{r})[/itex]. why does this vanish?
 
  • #4
surely here though, the components of [itex]\nabla (\frac{1}{r})[/itex] are changing and so their gradient is non-zero?
 
  • #5
Isn't Ampere's law more fundamental than Biot-Savart law? That's how it was presented in my E&M course...
 
  • #6
i think so. you have the fundamental equations of magnetostatics [itex]\nabla \cdot \mathbf{B}=0 , \nabla \times \mathbf{B}=\mu_0 \mathbf{J}[/itex] from which you can establish that the magnetic flux through a closed surface is zero and also ampere's law in integral form.

however in my notes, Biot Savart law is constructed from the example of force between two wires and from that we establish div and curl of B.

isn't it pretty copmlicated to establish Biot savart formula for B using ampere's law though?
 
  • #7
Yeah, the prof derived it in class over the course of two lectures!
 

1. What is 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.

2. How is Ampere's Law derived?

Ampere's Law is derived from Maxwell's equations, specifically the equation for the curl of the magnetic field. This equation is then integrated over a closed loop to obtain Ampere's Law.

3. What is the significance of Ampere's Law?

Ampere's Law is significant because it allows us to calculate the magnetic field around a closed loop due to a given electric current passing through the loop. It is also a key component in understanding the relationship between electricity and magnetism.

4. What are the limitations of Ampere's Law?

Ampere's Law only applies to steady currents and does not take into account the effects of changing electric fields or non-steady currents. It also assumes that the magnetic field is constant and does not take into account any variations in the field.

5. Can Ampere's Law be used to analyze complex systems?

Yes, Ampere's Law can be used to analyze complex systems as long as the current passing through the loop can be broken down into simpler components. In cases where the current is not constant or the magnetic field is not constant, modifications to the original form of Ampere's Law may be necessary.

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