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MrMuscle
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How to find an expression for bound charge densities in griffiths
- Thread starter MrMuscle
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- Bound Charge Expression Griffiths
In summary, the differentiation being with respect to the source coordinates ##r'## causes the minus sign to be lost in the well known identity $$\nabla\frac{1}{|\vec{r}-\vec{r'}|}=-\nabla'\frac{1}{|\vec{r}-\vec{r'}|}$$ The correct identity involving divergence and dot product is $$\nabla\cdot (f\vec{A})=f\nabla\cdot\vec{A}+\vec{A}\cdot\nabla f$$ and it also holds for differentiating with source coordinates, represented by ##\nabla'##. Applying this identity for ##\vec{A
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- #2
Delta2
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The minus sign is lost because the differentiation is with respect to the source coordinates ##r'##. It is a well known identity that $$\nabla\frac{1}{|\vec{r}-\vec{r'}|}=-\nabla'\frac{1}{|\vec{r}-\vec{r'}|}$$
where in the above the second ##\nabla## (the one in the right) has a ##'## next to it which means just that the differentiation is with respect to ##r'## coordinates.
For the rest you got the wrong identity involving curl and cross product. You should have the identity involving divergence and dot product which is as follows
$$\nabla\cdot (f\vec{A})=f\nabla\cdot\vec{A}+\vec{A}\cdot\nabla f$$
this identity holds for differentiating with source coordinates as well that is with ##\nabla'## in the place of ##\nabla##. Apply this Identity for ##\vec{A}=\vec{P}## and ##f=\frac{1}{|\vec{r}-\vec{r'}|}##.
where in the above the second ##\nabla## (the one in the right) has a ##'## next to it which means just that the differentiation is with respect to ##r'## coordinates.
For the rest you got the wrong identity involving curl and cross product. You should have the identity involving divergence and dot product which is as follows
$$\nabla\cdot (f\vec{A})=f\nabla\cdot\vec{A}+\vec{A}\cdot\nabla f$$
this identity holds for differentiating with source coordinates as well that is with ##\nabla'## in the place of ##\nabla##. Apply this Identity for ##\vec{A}=\vec{P}## and ##f=\frac{1}{|\vec{r}-\vec{r'}|}##.
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FAQ: How to find an expression for bound charge densities in griffiths
1. What is bound charge density?
Bound charge density refers to the electric charge that is bound to the atoms or molecules within a material. It is not free to move and is therefore not affected by external electric fields.
2. How is bound charge density different from free charge density?
Free charge density refers to the electric charge that is free to move and is affected by external electric fields. Bound charge density, on the other hand, is not free to move and is only affected by internal electric fields within a material.
3. How can we find an expression for bound charge densities?
An expression for bound charge densities can be found by using the concept of polarization, which is the separation of positive and negative charges within a material. This can be represented mathematically using the polarization vector P, which is related to the bound charge density by the equation ρ_b = -∇ · P.
4. What is the role of Griffiths in finding an expression for bound charge densities?
David J. Griffiths is a physicist and author of the textbook "Introduction to Electrodynamics" which is commonly used in studying electromagnetism. In this textbook, he provides a thorough explanation of polarization and its relation to bound charge densities, making it a valuable resource for finding an expression for bound charge densities.
5. Are there any limitations to the expression for bound charge densities in Griffiths?
The expression for bound charge densities in Griffiths is based on the assumption that the material is linear and isotropic, meaning that its properties do not change with direction. In reality, most materials are not perfectly linear and isotropic, so this expression may not be accurate in all cases.