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A Sphere in a Sphere. A solid conducting sphere carrying charge \(q\) has radius \(a .\) It is inside a concentric hollow conducting sphere with inner radius \(b\) and outer radius \(c .\) The hollow sphere has no net charge. (a) Derive expressions for the electric-field magnitude in terms of the distance \(r\) from the center for the regions \(rc .\) (b) Graph the magnitude of the electric field as a function of \(r\) from \(r=0\) to \(r=2 c .\) (c) What is the charge on the inner surface of the hollow sphere? (d) On the outer surface? (e) Represent the charge of the small sphere by four plus signs. Sketch the field lines of the system within a spherical volume of radius \(2 c\).

Step by step solution

01

Electric Field for Region r < a

In this region, the electric field inside a solid conductor is zero because there are no free charges inside it. Thus:\[E = 0 \quad \text{for} \quad r < a\]

02

Electric Field for Region a < r < b

Using Gauss's Law, consider a Gaussian surface just outside the radius of the solid sphere. The electric field is only due to the charge on the smaller sphere.\[E = \frac{kq}{r^2} \quad \text{for} \quad a < r < b\]where \(k\) is the Coulomb's constant.

03

Electric Field for Region b < r < c

In this region, the hollow conducting sphere will have induced charges. According to Gauss's Law and the conducting property, the electric field inside the conductor is zero:\[E = 0 \quad \text{for} \quad b < r < c\]

04

Electric Field for Region r > c

For this region, apply Gauss's Law to a Gaussian surface outside the outer shell. The net charge enclosed is \(q\), so:\[E = \frac{kq}{r^2} \quad \text{for} \quad r > c\]

05

Graph of Electric Field Magnitude vs. Distance r

Plot the magnitude of the electric field with the following segments:- From \(r = 0\) to \(r = a\), \(E = 0\).- From \(r = a\) to \(r = b\), \(E = \frac{kq}{r^2}\), showing a decreasing curve.- From \(r = b\) to \(r = c\), \(E = 0\).- From \(r = c\) to \(r = 2c\), \(E = \frac{kq}{r^2}\), showing a further decreasing curve.

06

Charge on Inner Surface of Hollow Sphere

By Gauss's Law, the charge on the inner surface of the hollow sphere must be \(-q\) to cancel the field inside the conductor. Thus, the inner surface of the hollow sphere has charge \(-q\).

07

Charge on Outer Surface of Hollow Sphere

The hollow sphere has no net charge. As a result, if the inner surface has charge \(-q\), the outer surface must have charge \(+q\) to neutralize the total charge. Hence, the outer surface has charge \(+q\).

08

Field Lines and Charge Representation

Represent the charge on the small sphere with four plus signs and draw electric field lines emanating radially outward from the smaller sphere until the outside of the larger sphere. The lines should be spaced apart and symmetric around the sphere, extending until the boundary at \(r = 2c\). Field lines do not exist within the conductor region.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gauss's Law

Imagine Gauss's Law as a powerful tool for exploring electric fields, especially in symmetrical situations like spheres. It states that the total electric flux through a closed surface is proportional to the charge enclosed by that surface. Mathematically, this is expressed as:\[ \Phi = \frac{q_{enc}}{\varepsilon_0} \]where \( \Phi \) is the electric flux, \( q_{enc} \) is the enclosed charge, and \( \varepsilon_0 \) is the permittivity of free space.Gauss's Law simplifies calculations of electric fields around symmetric shapes. For our concentric spheres problem, we use it to find that the electric field depends on the charge within the Gaussian surface.

Conductors

When we talk about conductors, we're referring to materials where electric charges move freely. Inside a conductor, the electric field is always zero in electrostatic equilibrium because any excess charges redistribute themselves on the surface to neutralize the electric field within.In our sphere-in-a-sphere problem, this property explains why inside both the solid and hollow conducting spheres, the electric field is zero (\(r < a\) and \(b < r < c\)). Electrons adjust on the surfaces, ensuring the field inside remains null.

Electric Field in Spheres

The electric field around spheres varies depending on whether you are inside, on the surface, or outside the sphere. In electrostatics, the distribution of charge on a conductor means that there is no electric field inside a solid conducting sphere (\(r c\).

Charge Distribution

Charge distribution in a conductors problem dictates how charges are located around a system. In our example, the smaller sphere has a charge \(q\). The hollow sphere conducts no net charge, but charges redistribute due to influence.

• The inner surface of the hollow sphere has a charge of \(-q\), neutralizing the electric field inside the shell.
• The outer surface balances this with a charge of \(+q\) to ensure no net charge overall.

The configuration of charges creates an interesting field pattern, with field lines originating from the smaller charged sphere and extending outward beyond the larger sphere until reaching the environment.