Problem 57 - Fundamentals of Physics

Question

A thin-walled metal spherical shell has radius \(25.0 \mathrm{~cm}\) and charge \(2.00 \times 10^{-7} \mathrm{C}\). Find \(E\) for a point (a) inside the shell, (b) just outside it, and (c) \(3.00 \mathrm{~m}\) from the center.

Step-by-Step Solution

Answer

(a) 0 N/C; (b) 2.88 × 10⁴ N/C; (c) 199.78 N/C

Step 1: Understanding the Electric Field Inside a Conductive Shell

According to Gauss's Law, the electric field inside a conductor in electrostatic equilibrium is zero. This is because any excess charge resides on the surface of the conductor, not inside. For a spherical shell, this implies that for any point inside the shell, the electric field \(E\) is \(0\).

Step 2: Calculate the Electric Field Just Outside the Shell

According to Gauss's Law, the electric field just outside a charged spherical shell (or any conductor) can be calculated using the formula: \[ E = \frac{kQ}{r^2} \] where \(k\) is the Coulomb's constant \((8.99 \times 10^9 \, \text{N m}^2/\text{C}^2)\), \(Q\) is the total charge on the shell, and \(r\) is the radius of the shell. Substituting the given values: \[ E = \frac{8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \times 2.00 \times 10^{-7} \, \text{C}}{(0.25 \, \text{m})^2} \approx 2.88 \times 10^4 \, \text{N/C} \]

Step 3: Calculate the Electric Field 3.00 m from the Center

For points outside a charged spherical shell, the shell behaves like a point charge located at its center. Again, apply Gauss's Law using the formula: \[ E = \frac{kQ}{r^2} \] where \(r\) this time is the distance from the center of the sphere to the point (3.00 m in this case). Substituting the given values: \[ E = \frac{8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \times 2.00 \times 10^{-7} \, \text{C}}{(3.00 \, \text{m})^2} \approx 199.78 \, \text{N/C} \]

Key Concepts

Gauss's Law

Gauss's Law is a fundamental principle used to relate the electric flux passing through a closed surface to the charge enclosed by that surface. It states that the total electric flux across a closed surface is proportional to the enclosed charge, mathematically expressed as:\[ \Phi_E = \frac{Q_{enc}}{\varepsilon_0} \]where \(\Phi_E\) is the electric flux, \(Q_{enc}\) is the enclosed charge, and \(\varepsilon_0\) is the permittivity of free space.
In practical terms, this law can simplify calculations of electric fields, especially in symmetrical situations. When applied to a spherical shell, Gauss's Law highlights that any charge placed on a conductor will reside on its surface, resulting in zero electric field inside the shell. This is because the electric field lines do not penetrate through the conductor, leaving the interior fieldless. This simplifies the understanding of charge distribution and electric field behavior in and around conductors.

Conductive Shell

A conductive shell is a hollow, conducting shape, often spherical in problems related to electrostatics. In electrostatics, a conductive shell has some unique properties:

• Charge Redistribution: When charge is placed on a conductive shell, it spreads uniformly across the surface due to electron repulsion.
• Zero Internal Field: Electric field inside the conductive shell is zero when in electrostatic equilibrium because the internal charges cancel each other's effects.

These properties mean that any point inside the hollow portion of a conductive shell has no net electric field. It is as though the inner space is shielded from electric forces. This characteristic is essential for understanding situations where sensitive electronic devices are placed inside conductive enclosures for protection from external electric fields.

Coulomb's Law

Coulomb's Law gives the fundamental principle of the electric force between two point charges. It is described by the equation:\[ F = \frac{k |Q_1 Q_2|}{r^2} \]where \(F\) is the magnitude of the force between the charges, \(Q_1\) and \(Q_2\) are the magnitudes of the charges, \(r\) is the distance between the centers of the two charges, and \(k\) is Coulomb's constant \((8.99 \times 10^9 \, \text{N m}^2/\text{C}^2)\).
In the context of a spherical shell, this law explains how the shell's overall charge interacts with external charges. Importantly, the field outside a spherical shell acts like it's coming from a point charge, located at the center of the sphere with the same magnitude as the total charge on the shell. This simplification is pivotal when determining the electric field at any given distance outside the shell.

Spherical Shell

A spherical shell is a three-dimensional hollow object, often used in physics problems related to electric fields and potentials. Some crucial characteristics include:

• Homogeneous Charge Distribution: The surface charge is evenly spread over the shell's outer surface.
• Electric Field Behavior:
  - Inside the Shell: Due to the symmetrical distribution of charge, the electric field inside is zero.
  - Just Outside: The electric field behaves like that from a point charge and can be calculated using \( E = \frac{kQ}{r^2} \).
• External Appearance as Point Charge: Beyond the outer surface, the shell can be treated as a point charge located at its center for calculating electric fields. This makes it simpler to analyze forces and fields interacting with it from a distance.

Understanding the properties of spherical shells is essential in electrostatics and helps to simplify complex interactions in problems involving charged objects.