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AKG

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Okay, not really, but suppose it were.

[tex]\mathbf{F} = \frac{1}{4\pi \epsilon _0}\frac{q_1q_2}{s^2}\left (1 + \frac{s}{\lambda}\right )e^{-s/\lambda}\mathbf {\hat {s}}[/tex]

Where [itex]\lambda[/itex] is a new physical constant which is very large, [itex]\mathbf{s}[/itex] is the separation vector between the two charges, and [itex]s = |\mathbf{s}|[/itex].

I have found that:

The electric field of a charge distribution [itex]\rho[/itex] is:

[tex]\mathbf{E}(\mathbf{r}) = \frac{1}{4\pi \epsilon _0}\int _{\mathcal{V}} \frac{\rho (\mathbf{r'})}{s^2}\mathbf{\hat{s}}\left (1 + \frac{s}{\lambda}\right )e^{-s/\lambda}\, d\tau '[/tex]

I have found that the potential of a point charge [itex]q[/itex] is:

[tex]V(\mathbf{r}) = \frac{q}{4\pi \epsilon _0 r e^{r/\lambda}}[/tex]

Where, of course, [itex]r = |\mathbf{r}|[/itex].

For a point charge [itex]q[/itex] at the origin, I found that:

[tex]\oint _{\mathcal{S}} \mathbf{E}\cdot d\mathbf{a} + \frac{1}{\lambda ^2}\int _{\mathcal{V}}V\, d\tau = \frac{q}{\epsilon _0}[/tex]

where [itex]\mathcal{S}[/itex] is the surface of a sphere [itex]\mathcal{V}[/itex] centered at the origin.

I now need to show that:

[tex]\oint _{\mathcal{S}} \mathbf{E}\cdot d\mathbf{a} + \frac{1}{\lambda ^2}\int _{\mathcal{V}}V\, d\tau = \frac{Q_{enclosed}}{\epsilon _0}[/tex]

Where [itex]Q_{enclosed}[/itex] is the charge enclosed within [itex]\mathcal{V}[/itex]. How do I do this?

I can rewrite the things as:

[tex]\int _{\mathcal{V}} (\mathbf{\nabla} \cdot \mathbf{E}) + \frac{V}{\lambda ^2}\, d\tau = \int _{\mathcal{V}}\frac{\rho}{\epsilon _0}\, d\tau[/tex]

Where [itex]\rho : \mathbb{R}^3 \to \mathbb{R}[/itex]. If I could show the integrands to be equal, that would be good. The second equation in this post gives an expression for E in terms of [itex]\rho[/itex], and I could, I believe, get an expression for V in terms of [itex]\rho[/itex], and then possibly use that to show that the integrands are equal, but that sounds like an enormous task. Is there a better way to do it?

For the real Gauss's Law, my book says that:

[tex]\oint \mathbf{E}\cdot d\mathbf{a} = \frac{q}{\epsilon _0}[/tex]

For a charge centered at the origin, and the surface of integration being the surface of a sphere centered at the origin. The book then says that any shape enclosing the charge will do by appealing to the diagram that shows the field lines that go through the surface, saying that "any closed surface, whatever its shape, would trap the same number of field lines." It then says that if we had a bunch of charges scattered about, rather than one at the origin, the total field is the sum of all individual fields, so a surface that encloses them all will have:

[tex]\oint \mathbf{E}\cdot d\mathbf{a} = \oint \left (\sum _{i = 1} ^n \mathbf{E}_i\cdot d\mathbf{a}\right ) = \sum _{i = 1} ^n \left ( \oint \mathbf{E}_i \cdot d\mathbf{a}\right ) = \sum _{i = 1} ^n \left (\frac{q_i}{\epsilon _0}\right ) = \frac{Q_{enclosed}}{\epsilon _0}[/tex]

The problem with that is that:

1) To me, the argument that says that it's okay for the shape to be something other than a sphere just sounds like hand-waving to me.

2) It assumes that [itex]\oint \mathbf{E}_i \cdot d\mathbf{a} = \frac{q_i}{\epsilon _0}[/itex], even if the charge is not found at the origin.

3) This only deals with discrete charge distributions, but what if [itex]Q_{enclosed}[/itex] is continuously distributed?

So on the one hand, I can't see how to prove the proposition myself, at least not without too much computation (some of which may not be possible), on the other hand, if I do it the way that the book appears to have done it, it doesn't seem like a proof at all. Suggestions please.

**This is what my assignment wants me to suppose**. Suppose that the actual force of interaction between two piont charges is found to be:[tex]\mathbf{F} = \frac{1}{4\pi \epsilon _0}\frac{q_1q_2}{s^2}\left (1 + \frac{s}{\lambda}\right )e^{-s/\lambda}\mathbf {\hat {s}}[/tex]

Where [itex]\lambda[/itex] is a new physical constant which is very large, [itex]\mathbf{s}[/itex] is the separation vector between the two charges, and [itex]s = |\mathbf{s}|[/itex].

I have found that:

The electric field of a charge distribution [itex]\rho[/itex] is:

[tex]\mathbf{E}(\mathbf{r}) = \frac{1}{4\pi \epsilon _0}\int _{\mathcal{V}} \frac{\rho (\mathbf{r'})}{s^2}\mathbf{\hat{s}}\left (1 + \frac{s}{\lambda}\right )e^{-s/\lambda}\, d\tau '[/tex]

I have found that the potential of a point charge [itex]q[/itex] is:

[tex]V(\mathbf{r}) = \frac{q}{4\pi \epsilon _0 r e^{r/\lambda}}[/tex]

Where, of course, [itex]r = |\mathbf{r}|[/itex].

For a point charge [itex]q[/itex] at the origin, I found that:

[tex]\oint _{\mathcal{S}} \mathbf{E}\cdot d\mathbf{a} + \frac{1}{\lambda ^2}\int _{\mathcal{V}}V\, d\tau = \frac{q}{\epsilon _0}[/tex]

where [itex]\mathcal{S}[/itex] is the surface of a sphere [itex]\mathcal{V}[/itex] centered at the origin.

I now need to show that:

[tex]\oint _{\mathcal{S}} \mathbf{E}\cdot d\mathbf{a} + \frac{1}{\lambda ^2}\int _{\mathcal{V}}V\, d\tau = \frac{Q_{enclosed}}{\epsilon _0}[/tex]

Where [itex]Q_{enclosed}[/itex] is the charge enclosed within [itex]\mathcal{V}[/itex]. How do I do this?

I can rewrite the things as:

[tex]\int _{\mathcal{V}} (\mathbf{\nabla} \cdot \mathbf{E}) + \frac{V}{\lambda ^2}\, d\tau = \int _{\mathcal{V}}\frac{\rho}{\epsilon _0}\, d\tau[/tex]

Where [itex]\rho : \mathbb{R}^3 \to \mathbb{R}[/itex]. If I could show the integrands to be equal, that would be good. The second equation in this post gives an expression for E in terms of [itex]\rho[/itex], and I could, I believe, get an expression for V in terms of [itex]\rho[/itex], and then possibly use that to show that the integrands are equal, but that sounds like an enormous task. Is there a better way to do it?

For the real Gauss's Law, my book says that:

[tex]\oint \mathbf{E}\cdot d\mathbf{a} = \frac{q}{\epsilon _0}[/tex]

For a charge centered at the origin, and the surface of integration being the surface of a sphere centered at the origin. The book then says that any shape enclosing the charge will do by appealing to the diagram that shows the field lines that go through the surface, saying that "any closed surface, whatever its shape, would trap the same number of field lines." It then says that if we had a bunch of charges scattered about, rather than one at the origin, the total field is the sum of all individual fields, so a surface that encloses them all will have:

[tex]\oint \mathbf{E}\cdot d\mathbf{a} = \oint \left (\sum _{i = 1} ^n \mathbf{E}_i\cdot d\mathbf{a}\right ) = \sum _{i = 1} ^n \left ( \oint \mathbf{E}_i \cdot d\mathbf{a}\right ) = \sum _{i = 1} ^n \left (\frac{q_i}{\epsilon _0}\right ) = \frac{Q_{enclosed}}{\epsilon _0}[/tex]

The problem with that is that:

1) To me, the argument that says that it's okay for the shape to be something other than a sphere just sounds like hand-waving to me.

2) It assumes that [itex]\oint \mathbf{E}_i \cdot d\mathbf{a} = \frac{q_i}{\epsilon _0}[/itex], even if the charge is not found at the origin.

3) This only deals with discrete charge distributions, but what if [itex]Q_{enclosed}[/itex] is continuously distributed?

So on the one hand, I can't see how to prove the proposition myself, at least not without too much computation (some of which may not be possible), on the other hand, if I do it the way that the book appears to have done it, it doesn't seem like a proof at all. Suggestions please.

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