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## Homework Statement

In a model of an atomic nucleus, the electric field is given by:

__E__= α

__r__for r < a

where α is a constant and a is the radius of the nucleus.

Use the differential form of Gauss's Law to calculate the charge density ρ inside the nucleus.

**2. The attempt at a solution**

Using the simple version of Gauss's law :

[tex]\int_{S} \underline{E}.\underline{dS} = \int_{V} \frac{\rho}{\epsilon_{0}} dV [/tex]

Yields a result [tex]\rho = \frac{3E\epsilon_{0}}{r} [/tex] for 0<r<=a

## Homework Statement

However when using the differential form:

[tex]\nabla . \underline{E} = \frac{\rho}{\epsilon_{0}}[/tex]

[tex]\frac{1}{r} \frac{\partial (r E_{r})}{\partial r} = \frac{\rho}{\epsilon_{0}} [/tex]

and when integrating with respect to r from 0 to a,

[tex] \rho = {\epsilon_{0}} E (1 + ln a ) [/tex]

Any helpful advice would be appreciated.

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