About the curl of B using Biot-Savart Law

[yanyan_leung]

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?

 

[tiny-tim]

Welcome to PF!

Hi yanyan_leung ! Welcome to PF!

[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 (J_i∂/∂x_i) (f) = ∑_i ∂/∂x_i (fJ_i) - f ∑_i ∂/∂x_i (J_i) …

now just use the product rule on the middle term. ​

 

[charng]

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{