Problem 22 - University Physics with Modern Physics

Question

A point charge of -3.00 \(\mu\)C is located in the center of a spherical cavity of radius 6.50 cm that, in turn, is at the center of an insulating charged solid sphere. The charge density in the solid is \(\rho =\) 7.35 \(\times\) 10\(^{-4}\) C/m\(^3\). Calculate the electric field inside the solid at a distance of 9.50 cm from the center of the cavity.

Step-by-Step Solution

Answer

The electric field inside the sphere at 9.50 cm is determined by calculating the net charge enclosed and applying Gauss's Law.

Step 1: Understanding the Problem

We need to find the electric field inside a solid sphere with a point charge at its center. The sphere has a uniform volume charge density and we need to consider both the charge of the point and the charge due to the sphere.

Step 2: Calculate Enclosed Charge

Firstly, the charge inside the sphere up to a radius of 9.50 cm can be calculated using the charge density \( \rho \) and the volume of the sphere: \( Q = \rho \cdot \frac{4}{3}\pi (0.095)^3 \). Evaluate this to get the enclosed charge from the sphere.

Step 3: Consider the Point Charge

Since the point charge at the center is -3.00 \( \mu \)C, which equals \(-3.00 \times 10^{-6} \) C, we need to take this into account as part of the enclosed charge. The total enclosed charge \( Q_{enc} \) is the charge due to the sphere plus the point charge.

Step 4: Apply Gauss's Law

Using Gauss's Law, the electric field at any point inside the sphere is given by \( E = \frac{Q_{enc}}{4\pi\varepsilon_0 r^2} \), where \( r = 9.50 \) cm is the distance from the center, and \( \varepsilon_0 = 8.85 \times 10^{-12} \) C²/(N·m²) is the permittivity of free space. Substitute the values to find \( E \).

Step 5: Solve for Electric Field

Calculate the total enclosed charge and substitute it into the Gauss's Law equation. Evaluate the expression to find the electric field at the given distance.

Key Concepts

Electric Field

The electric field is a fundamental concept in electromagnetism, which describes how electric forces are exerted in the space surrounding electric charges. It's a vector field, meaning it has both magnitude and direction. The electric field

• is created by electric charges
• can exert force on other nearby charges
• is strongest near the charge that creates it

The strength of the electric field (\( E \)) created by a point charge (\( Q \)) at a distance (\( r \)) can be calculated using the formula: \[E = \frac{Q}{4\pi\varepsilon_0 r^2}\] where \( \varepsilon_0 \) is the permittivity of free space. This helps in understanding how influence spreads from the charge outward.

Charge Density

Charge density (\( \rho \)) is the amount of electric charge per unit volume in a region of space. It helps us comprehend how charge is distributed within a material. In our problem, the charge density of the solid sphere is given as 7.35 \times 10^{-4} C/myard³. This uniform distribution means:

• The charge is spread evenly throughout the sphere's volume.
• Calculations using this charge density are simplified, as simple geometric formulas can be used to find the enclosed charge.
• It affects how the electric field behaves inside insulators or conductors, impacting how charges influence one another in these materials.

Enclosed Charge

To find the electric field at a specified point, the concept of enclosed charge (\( Q_{enc} \)) is essential. It's the total charge enclosed within a given volume. For this exercise:

• Calculate the charge enclosed by the sphere at a radius of 9.50 cm using the equation: \( Q = \rho \cdot \frac{4}{3}\pi(0.095)^3 \)
• Include the contribution from the central point charge of -3.00 \( \mu \)C.This concept is crucial because Gauss's Law requires knowing all the charges affecting the region of interest to determine the electric field effectively.

Point Charge

A point charge is an idealization where a charge is assumed to be concentrated at a single point in space. In practice, it provides ease in calculations and theoretical models. In this problem:

• The given point charge is -3.00 \( \mu \)C, centrally located within the sphere.
• Its negative sign indicates that it will exert an attractive force on positive charges.
• This singular charge significantly influences the local electric field inside the surrounding material, and its effects are considered separately when calculating electric fields using Gauss's Law.

Permittivity of Free Space

The permittivity of free space (\( \varepsilon_0 \)) is a constant that characterizes the ability of the vacuum to permit electric field lines. Its value is approximately 8.85 \times10^{-12} C²/(N·m²). This constant appears frequently in equations involving electric fields and forces.

• It defines the strength of the electric field generated per unit charge in a vacuum.
• Essential for calculations involving electric fields as it bridges the conceptual understanding of fields with practical calculations.
• Frequently used in Gauss's Law to relate enclosed charge to the electric field.