91Ó°ÊÓ

The electric field is a vector quantity that represents the force experienced by a unit charge in a given space. It is measured in newtons per coulomb (N/C). In this exercise, the electric field has a uniform magnitude of \(1.44 \times 10^4\) N/C. An important aspect of an electric field is its direction, which influences how it interacts with surfaces. In calculations involving electric flux, the direction of the electric field relative to the surface normal is crucial for determining the angle \(\theta\). Understanding this relationship helps in comprehending the underlying physics of the problem.

Key points about electric fields:

• Electric fields exert force on charges within the field.
• They are described by both magnitude and direction, making them vector quantities.
• The angle between the electric field and a surface normal is essential when calculating flux.

These concepts are foundational when dealing with electromagnetic problems and are particularly useful in finding the angle between an electric field and a surface, as in this problem.

The circular surface area is critical in this exercise because it directly affects the electric flux calculation. The area of a circle is determined by the formula \( A = \pi r^2 \), where \( r \) is the radius. In this problem, the radius is given as 0.057 m. Substituting this into the area formula gives \( A \approx 0.0102 \; \mathrm{m}^2 \).

There are a few steps to calculate this:

• Square the radius: \( (0.057)^2 \).
• Multiply the squared radius by \(\pi\) (approximately 3.14159).

The calculated area provides the necessary input for the electric flux formula \( \Phi = E \cdot A \cdot \cos(\theta) \), which allows evaluating how the electric field distributes over the surface.

Angle calculation is a fundamental part of determining how the electric field interacts with the circular surface. In this context, the angle \( \theta \) is the angle between the electric field direction and the normal of the circular surface. The normal is a line perpendicular to the surface. To find \( \theta \), we rearrange the electric flux formula \( \Phi = E \cdot A \cdot \cos(\theta) \) to solve for \( \cos(\theta) \):

\[ \cos(\theta) = \frac{78}{1.44 \times 10^4 \times 0.0102} \]

By computing the value of \( \cos(\theta) \), we set the stage to use the inverse cosine function, which is essential for retrieving the actual angle \( \theta \). Calculating angles in physics often involves using trigonometric functions to relate different sides and angles of triangles created by vector representations.

The inverse cosine function, denoted as \( \cos^{-1}(x) \), helps find the angle when you know the cosine value. When you rearrange the electric flux equation to solve for \( \cos(\theta) \), obtaining \( \frac{78}{1.44 \times 10^4 \times 0.0102} \), the inverse cosine function is your tool to calculate \( \theta \).

Using the inverse cosine:

• Input the calculated \( \cos(\theta) \) value into \( \cos^{-1}() \) to get \( \theta \).
• Ensure the angle you find is in the correct range—here, less than 90 degrees.

This trigonometric function is fundamental for many physics problems, as it allows you to move from calculated values back to intuitive, angle-based insights about the system. In this exercise, it gives an angle of around 53.1 degrees, showing how the electric field aligns with the surface.