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1

You just made some math mistakes. You made a mistake when you did $Q = h\int_A kr$. You got $Q = h\pi k r^2$, but you should have gotten $Q = \frac{2}{3} h\pi k r^3$. Notice how this second expression has units of charge while the first one doesn't. Another mistake you make is that you say \$\frac{1}{r} \frac{\partial rE(r)}{\partial r} = \frac{1}{r} ...

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A non-zero electric field on your Gaussian surface does not mean a non-zero flux. There are positive and negative contributions to the flux due to the electric field pointing in & out in different places. Your cosine term confirms this. Apparently the contributions must cancel in this case since the net enclosed charge is zero.

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Every field line from the dipole must begin on one charge and end on the other. That means that if a field line passes out of your surface it must pass back in through it again. The surface as a whole will have the same number of field lines going in as out, so the net flux through the surface will be zero.

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What you have done is calculating the densities for which the net charge density is zero. That does not mean the field is zero. You may be confused with the the principle that there is no field inside a conductor (and hence no net charge). This is not the case here. What the hint implies is that you need to calculate the field each object produces inside ...

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Great question. Almost all authors don't show that further justification is is needed to get Gaus's law for induced(time dependent) electric fields The third Hertz’s equation for electrostatic field is a generalization of the Gauss law for electrostatic fields arrived at as follows: $$\nabla \cdot E_{static} = \frac{Q}{e}$$ - Gaus's law for ...

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