About the curl of B using Biot-Savart Law

In summary, The conversation is about understanding the step between two equations in Griffiths regarding the curl of B using the Biot-Savart Law. The question is about how to deduce a certain result using the product rule.
  • #1
yanyan_leung
2
0
I am reading the Griffiths about finding the curl of B using Biot-Savart Law. I do not understanding the step between equation (5.52) and (5.53) which finding the x components of the following:
[tex](\boldsymbol{J}\cdot\nabla^{\prime})\dfrac{\hat{\xi}}{\xi^{2}}[/tex]
where
[tex]\mathbf{\mathbf{\xi}}=\mathbf{r}-\mathbf{r}^{\prime}[/tex]
I don't know why it said using product rule 5 in his book, can get the following result:
[tex]\left(\boldsymbol{J}\cdot\nabla^{\prime}\right)\dfrac{x-x^{\prime}}{\xi^{3}}=\nabla^{\prime}\cdot\left[\dfrac{(x-x^{\prime})}{\xi^{3}}\mathbf{J}\right]-\left(\dfrac{x-x^{\prime}}{\xi^{3}}\right)\left(\nabla^{\prime}\cdot\mathbf{J}\right)[/tex]
Is there anyone know how to deduce this result?
 
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  • #2
Welcome to PF!

Hi yanyan_leung ! Welcome to PF! :smile:
yanyan_leung said:
[tex]\left(\boldsymbol{J}\cdot\nabla^{\prime}\right)\dfrac{x-x^{\prime}}{\xi^{3}}=\nabla^{\prime}\cdot\left[\dfrac{(x-x^{\prime})}{\xi^{3}}\mathbf{J}\right]-\left(\dfrac{x-x^{\prime}}{\xi^{3}}\right)\left(\nabla^{\prime}\cdot\mathbf{J}\right)[/tex]
Is there anyone know how to deduce this result?

That's ∑i (Ji∂/∂xi) (f) = ∑i ∂/∂xi (fJi) - f ∑i ∂/∂xi (Ji) …

now just use the product rule on the middle term. :wink:
 
  • #3

The Biot-Savart Law is a fundamental equation in electromagnetism that describes the magnetic field produced by a steady current. It is given by:

\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int \frac{\mathbf{J}(\mathbf{r}') \times (\mathbf{r}-\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|^3} d\tau'

Where \mathbf{B} is the magnetic field, \mathbf{r} is the position of the observation point, \mathbf{r}' is the position of the current element, \mu_0 is the permeability of free space, and \mathbf{J} is the current density.

To find the curl of \mathbf{B} using the Biot-Savart Law, we can use the vector identity:

(\mathbf{A}\cdot\nabla)\mathbf{B} = \nabla\cdot(\mathbf{A}\times\mathbf{B}) - \mathbf{A}\times(\nabla\times\mathbf{B})

Applying this identity to the Biot-Savart Law, we get:

\left(\mathbf{J}\cdot\nabla\right)\mathbf{B} = \nabla\cdot\left(\mathbf{J}\times\mathbf{B}\right) - \mathbf{J}\times(\nabla\times\mathbf{B})

Substituting \mathbf{B} from the Biot-Savart Law into this equation, we get:

\left(\mathbf{J}\cdot\nabla\right)\left(\frac{\mu_0}{4\pi} \int \frac{\mathbf{J}(\mathbf{r}') \times (\mathbf{r}-\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|^3} d\tau'\right) = \nabla\cdot\left(\mathbf{J}\times\left(\frac{\mu_0}{4\pi} \int \frac{\mathbf{J}(\mathbf{r}') \times (\mathbf{r}-\mathbf{r}')}{|\mathbf{
 

Related to About the curl of B using Biot-Savart Law

What is the Biot-Savart Law?

The Biot-Savart Law is a mathematical equation that describes the magnetic field generated by a steady current.

How is the Biot-Savart Law used to calculate the curl of B?

The Biot-Savart Law can be rearranged to solve for the curl of B, which is given by the cross product of the current and the distance vector between the current and the point of interest.

Why is the curl of B important?

The curl of B is important because it describes the circulation of the magnetic field around a point. This can be used to determine the direction and strength of a magnetic field.

What is the difference between the curl of B and the divergence of B?

The curl of B describes the circulation of the magnetic field, while the divergence of B describes the spreading or convergence of the magnetic field. In other words, the curl of B describes the rotational aspect of the field, while the divergence of B describes the expansion or contraction of the field.

Are there any limitations to using the Biot-Savart Law to calculate the curl of B?

Yes, the Biot-Savart Law is only valid for steady currents and does not account for any changing electric fields. Additionally, it assumes that the currents are confined to wires or other one-dimensional structures and does not account for three-dimensional current distributions.

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