Problem 16 - University Physics with Modern Physics

Question

22.16. A solid metal sphere with radius 0.450 \(\mathrm{m}\) carries a net charge of 0.250 \(\mathrm{nC}\) . Find the magnitude of the electric field (a) at a point 0.100 \(\mathrm{m}\) outside the surface of the sphere and \((\mathrm{b})\) at a point inside the sphere, 0.100 \(\mathrm{m}\) below the surface.

Step-by-Step Solution

Answer

Outside: 7.42 N/C, Inside: 0 N/C.

Step 1: Understand the Problem

We are given a solid metal sphere with a radius of 0.450 m carrying a net charge of 0.250 nC. We need to find the electric field at certain points relative to this sphere.

Step 2: Determine the Field Outside the Sphere

The electric field at a point outside a charged metal sphere can be treated as if all the sphere's charge is concentrated at its center. Use the formula for the electric field due to a point charge: \[ E = \frac{kQ}{r^2} \]where \( k = 8.988 \times 10^9 \, \text{N m}^2/\text{C}^2 \) (Coulomb's constant), \( Q = 0.250 \times 10^{-9} \, \text{C} \), and \( r = 0.450 + 0.100 = 0.550 \, \text{m} \).

Step 3: Calculate the Electric Field Outside

Substitute the values into the formula:\[ E = \frac{8.988 \times 10^9 \, \text{N m}^2/\text{C}^2 \times 0.250 \times 10^{-9} \, \text{C}}{0.550^2 \, \text{m}^2} \]Calculate to find the magnitude of the electric field.

Step 4: Result for the Electric Field Outside

After calculation, the electric field at 0.100 m outside the surface is approximately 7.42 N/C.

Step 5: Determine the Field Inside the Sphere

Inside a conducting sphere, the electric field is zero. The charges on the surface cause any point within to have no net electric field due to the symmetrical distribution of charge.

Step 6: Result for the Electric Field Inside

The electric field at 0.100 m inside the sphere is 0 N/C.

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental principle of electromagnetism that describes the force between two charged objects. It states that the electric force between two charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This can be expressed with the formula: \[F = \frac{k \cdot |q_1 \cdot q_2|}{r^2}\]where:

• \( F \) is the force between the charges,
• \( k \) is Coulomb's constant \( (8.988 \times 10^9 \, ext{N m}^2/ ext{C}^2) \),
• \( q_1 \) and \( q_2 \) are the magnitudes of the charges,
• \( r \) is the distance between the charges.

The force is attractive if the charges have opposite signs and repulsive if they have the same sign. This principle is crucial when determining electric forces and fields for both point charges and continuous charge distributions.

Conducting Sphere

A conducting sphere is an important subject in electrostatics. It is made of a material that allows electric charges to move freely across its surface. When a charge is placed on a conducting sphere, the charge uniformly distributes itself over the surface due to repulsion among like charges. The result is:

• The electric charge is only present on the surface of the sphere.
• Within the sphere, the electric field is zero because the charges distribute themselves symmetrically, canceling out any internal fields.
• The sphere can be treated as a point charge when calculating the electric field outside it, with the entire charge effectively acting as if it is located at the center.

These properties make conducting spheres ideal for understanding fundamental electrostatic principles.

Point Charge

A point charge is an idealized model in physics used to simplify and understand electric fields and forces. It is a hypothetical charge with an infinitely small size, so its shape doesn't affect calculations. Point charges are useful when:

• Analyzing electric fields generated by simple charge distributions.
• Using Coulomb's law to calculate forces and fields around charged objects, treating the charge as located at a single point.
• Imagining the charge as concentrated in a very small region, particularly useful in symmetry arguments and under conditions where the exact location doesn't change the outcome, as seen in the electric field calculations for spherical distributions.

Point charges simplify complex electrostatic problems, especially when dealing with spherical objects, as all external electric effects can be computed as if the charge were collected at one central point.

Electric Field Inside a Conductor

Understanding the electric field inside a conductor is vital for grasping electrostatic equilibrium concepts. In the electrostatic state (where charges are stationary), the electric field inside a conductor must be zero. Any non-zero electric field would cause charges within the conductor to move, disrupting this equilibrium. Considerations include:

• Electrostatic shielding: A conductor can shield its interior from external electric fields.
• Charge distribution occurs on the surface, ensuring no internal electric field.
• Because of this, if you were to look for an electric field within the body of a conducting sphere, you'd find none.

This principle is applied in practice, such as in Faraday cages, which protect sensitive equipment from external electric fields.