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Let's analyze and solve the problems in the image you provided.

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**Problem 6:**

The question asks for the electric field inside a region \( r < a \), where the charge is enclosed inside a Gaussian surface with radius \( r \). Since the region is within the sphere and we are likely dealing with a uniformly charged insulating sphere, the electric field inside this region would be zero.

This is because there is no enclosed charge within the Gaussian surface for \( r < a \) inside a conducting shell or hollow sphere. For a conducting sphere, all charge resides on the surface.

Therefore, **the correct answer is:**
\[ E = 0 \text{ for } r < a \]

Among the given options, it seems like this matches the answer with \( E = k \times 0 / r^2 \).

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**Problem 7:**

Now, this question asks about the electric field in the region \( a < r < b \), which is inside an insulating sphere between two radii.

In this region, the enclosed charge is not zero. The correct expression for the electric field inside a uniformly charged insulating sphere (non-conducting) can be found by integrating over the volume enclosed. The result depends on how much charge is enclosed as a function of \( r \).

For a region \( a < r < b \) inside a uniformly charged insulator, the electric field is proportional to \( r^3 \), as the enclosed charge depends on the volume of the sphere up to radius \( r \), minus the hollow portion.

Therefore, the electric field \( E \) in this region is typically given by:

\[
E = k \frac{Q_{\text{insulator}}(r^3 - a^3)}{r^2 (b^3 - a^3)}
\]

Thus, **the correct answer for Problem 7 is:**
\[
E = k \frac{Q_{\text{insulator}}(r^3 - a^3)}{r^2 (b^3 - a^3)}
\]

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If you would like more details or further clarification on these concepts, feel free to ask! Here are five related questions you can explore further:

1. How is Gauss's law applied to find the electric field inside and outside a conducting sphere?
2. What is the significance of charge distribution in a conducting versus non-conducting sphere?
3. How does the electric field behave at the boundary between a conductor and a dielectric?
4. Can you explain how the volume charge density affects the electric field inside an insulating sphere?
5. What is the relationship between electric potential and electric field for spherical charge distributions?

**Tip:** In problems involving Gauss's law, carefully consider whether the material is conducting or insulating, as this will significantly affect the charge distribution and electric field.

This problem explores the electric field inside spherical charge distributions using Gauss's Law. For r < a, the electric field is zero inside a conducting sphere. For a < r < b inside an insulating sphere, the electric field is determined by the enclosed charge, which is proportional to (r^3 - a^3). Learn more about these key concepts in electromagnetism.

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