Problem 42 - University Physics with Modern Physics

Question

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 electricfield magnitude in terms of the distance \(r\) from the center for the regions \(r < a, a < r < b, b < r < c\), and \(r > c\). (b) Graph the magnitude of the electric field as a function of \(r\) from \(r =\) 0 to \(r =\) 2c. (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

Answer

(a) E=0 for rc. (c) -q, (d) +q. (b, e) Graph and sketch field lines.

Step 1: Identify Regions for Electric Field

We must identify four distinct regions to derive expressions for the electric field: (i) Inside the conducting sphere, where \( r < a \). (ii) In the space between the conducting sphere and the hollow shell, where \( a < r < b \). (iii) Inside the hollow part of the shell, where \( b < r < c \). (iv) Outside both spheres, where \( r > c \).

Step 2: Electric Field for r < a

For any point inside the solid conducting sphere, the electric field is zero due to electrostatic shielding in conductors. Thus, \( E = 0 \) for \( r < a \).

Step 3: Electric Field for a < r < b

In the region between \( a \) and \( b \), treat the solid sphere as a point charge. The electric field is given by: \[ E = \frac{kq}{r^2} \] where \( k \) is Coulomb's constant and \( r \) is the distance from the center.

Step 4: Electric Field for b < r < c

Within the shell, the electric field is zero due to the hollow conductor's electrostatic shielding. Hence, \( E = 0 \) for \( b < r < c \).

Step 5: Electric Field for r > c

Outside both spheres, treat the entire system as if the charge \( q \) is concentrated at the center. Thus, the electric field is: \[ E = \frac{kq}{r^2} \] for \( r > c \).

Step 6: Graphing the Electric Field

The graph for \( E \) vs \( r \) would be zero for \( r < a \), a curve representing inverse square law for \( a < r < b \), zero for \( b < r < c \), and another inverse square curve for \( r > c \).

Step 7: Charge on Inner Surface of Hollow Sphere

The inner surface of the hollow sphere must counteract the field from the solid sphere's charge, so it has a charge of \( -q \) to ensure the field within \( b < r < c \) remains zero.

Step 8: Charge on Outer Surface of Hollow Sphere

Since the hollow sphere has no net charge, the outer surface must have a charge of \(+q\) to balance the \(-q\) charge induced on the inner surface.

Step 9: Sketch Electric Field Lines

Inside the solid sphere, there are no field lines. Between \( a < r < b \), field lines radiate outward. These lines stop at \( r = b \) where the field inside is zero. Beyond \( r = c \), lines from the central sphere continue to radiate out, diminishing in density with distance.

Key Concepts

Gauss's Law

Gauss's Law is a fundamental principle used to determine the electric field distribution due to a given charge configuration. Imagine an imaginary closed surface, often referred to as a Gaussian surface. According to Gauss's Law, the net electric flux through this surface is proportional to the enclosed charge. Mathematically, it is expressed as:
\[ \Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0} \]where:

• \( \Phi_E \) is the electric flux through the Gaussian surface.
• \( \vec{E} \) is the electric field.
• \( d\vec{A} \) is a differential area vector on the Gaussian surface.
• \( Q_{\text{enc}} \) is the charge enclosed by the surface.
• \( \varepsilon_0 \) is the permittivity of free space.

Gauss's Law is especially useful where symmetry simplifies calculations. In the exercise, we used it to simplify understanding of the field around concentric spheres: identifying regions of zero electric field and determining net charges.

Electrostatic Shielding

Electrostatic shielding is a phenomenon where electric fields are counteracted or nullified within a conductor. When an external electric field impacts a conductor, free electrons in the conductor redistribute to cancel out the field within its material. This distribution ensures that the net field inside the conductor is zero.
In the described problem, electrostatic shielding occurs in two critical locations:

• Inside the solid conducting sphere (\(r
• In the hollow region of the shell (\(b

This principle allows conductors to protect enclosed regions from external static electric fields, making conductors excellent shields against electrostatic effects in designs like Faraday cages.

Conductors

Conductors are materials that permit the flow of electric charge with minimal resistance. They possess freely movable electrons, which respond efficiently to electric fields. This characteristic impacts how they interact with electric fields and charge distribution.
In the context of the exercise, the conducting nature of spheres plays a crucial role:

• The solid sphere holds an initial charge \(q\), set centrally to prevent any field within its own volume.
• The hollow sphere, although charge-neutral overall, adjusts its inner and outer surfaces through electrostatic induction. The inner surface gains a charge \(-q\) to counteract the internal field, ensuring \[E=0\] within \(b < r < c\).
• The outer surface conversely must acquire a \(+q\) charge, balancing the system since the hollow sphere itself is neutral.

Conductors react instantly to external fields, leading to quick adjustments in charge distribution, and these attributes make them a key focus in discussions of electrostatics.