Problem 4 - Fundamentals of Physics

Question

In Fig. 23-32, a butterfly net is in a uniform electric field of magnitude \(E=3.0 \mathrm{mN} / \mathrm{C}\). The rim, a circle of radius \(a=11 \mathrm{~cm}\), is aligned perpendicular to the field. The net contains no net charge. Find the electric flux through the netting.

Step-by-Step Solution

Answer

Electric flux through the net is approximately \(1.14 \times 10^{-3} \text{ Nm}^2/\text{C}\).

Step 1: Understand the Concept of Electric Flux

Electric flux \( \Phi_E \) is a measure of the electric field passing through a given area. Mathematically, it can be calculated using the formula: \( \Phi_E = E \cdot A \cdot \cos(\theta) \), where \( E \) is the electric field, \( A \) is the area through which the field lines pass, and \( \theta \) is the angle between the field lines and the normal (perpendicular) to the surface.

Step 2: Calculate the Area of the Circular Rim

The net rim is a circle with radius \( a = 11 \) cm. The area \( A \) of the circle is given by the formula \( A = \pi a^2 \). First, convert the radius to meters: \( a = 0.11 \) m. Then, calculate \( A = \pi \times (0.11)^2 \) square meters.

Step 3: Substitute Values into the Electric Flux Formula

Since the rim is aligned perpendicular to the electric field, \( \theta = 0 \), and \( \cos(\theta) = 1 \). Substitute the values into the electric flux formula: \( \Phi_E = E \times A \times \cos(\theta) = 3.0 \times 10^{-3} \times \pi \times (0.11)^2 \times 1 \).

Step 4: Perform the Calculation

Calculate the expression: \( \Phi_E = 3.0 \times 10^{-3} \text{ mN/C} \times \pi \times 0.0121 \text{ m}^2 \). After performing the multiplication, \( \Phi_E \approx 1.14 \times 10^{-3} \text{ Nm}^2/\text{C} \).

Key Concepts

Electric Field

An electric field is a vector field that represents the force exerted per unit charge at a given point in space. It is denoted by the letter \( E \) and is measured in newtons per coulomb (N/C).
The direction of the electric field is defined as the direction a positive test charge would move if placed in the field. Thus, electric field lines point away from positively charged objects and toward negatively charged ones.
Consider a uniform electric field, like that in the case of the butterfly net exercise. Here, the field has the same magnitude and direction at every point in space.

• The given electric field magnitude is \( E = 3.0 \text{ mN/C} \), which can also be expressed as \( 3.0 \times 10^{-3} \text{ N/C} \).
• Understanding the uniformity and direction of the field is crucial when calculating the electric flux through any surface aligned with it.

Circular Area

To calculate the electric flux through a surface, knowing the area of that surface is essential. In this exercise, the surface is a circular rim of a butterfly net.
The formula for the area \( A \) of a circle is \( A = \pi a^2 \), where \( a \) is the radius of the circle. The radius given is \( a = 11 \text{ cm} \), which needs to be converted to meters for consistency with the SI unit system.

• Convert the radius: \( a = 11 \text{ cm} = 0.11 \text{ m} \).
• Calculate the area: \( A = \pi \times (0.11)^2 \).
• The result is approximately \( 0.038 \text{ m}^2 \).

Knowing the area allows you to proceed with finding the electric flux, as it directly factors into the electric flux formula with the electric field magnitude and the angle of incidence.

Angle of Incidence

The angle of incidence, in the context of electric flux, involves the angle \( \theta \) between the electric field lines and the normal (perpendicular) to the surface. This angle drastically affects the electric flux calculation because it influences how much field passes through the area.
In mathematical terms, it is denoted by \( \cos(\theta) \), where \( \theta \) impacts the formula: \( \Phi_E = E \cdot A \cdot \cos(\theta) \).

• For a surface perpendicular to the electric field, as in this exercise, \( \theta = 0 \).
• Thus, \( \cos(0) = 1 \), simplifying the formula to \( \Phi_E = E \cdot A \).

This simplification occurs because when the surface is fully perpendicular, all the field lines pass through, maximizing the flux. Therefore, understanding the angle's role helps predict how much of the electric field contributes to the flux, crucial for various applications in physics and engineering.