91Ó°ÊÓ

Imagine a field where tiny invisible arrows surround a charged object. This is the electric field, an essential concept in physics that describes how electric forces act in space. An electric field is represented by the symbol \( E \) and it's a vector quantity, meaning it has both magnitude and direction. In simple terms, it tells us how strong the electric force is and in which direction it would push a positive test charge.

• The strength of the electric field is measured in newtons per coulomb \( (\mathrm{N/C}) \).
• In this exercise, the strength of the electric field is \( 1.44 \times 10^{4} \mathrm{N/C} \).

Electric fields are caused by charges; the field points away from positive charges and towards negative charges. Understanding this helps in analyzing forces between charged bodies and predicting how objects will interact in an electric field. Having a grasp of electric fields opens doors to understanding more complex electromagnetic phenomena around us.

What happens when an electric field passes through a surface? To understand this, visualize a flat, pancake-shaped surface lying in the electric field. This surface is described as a circular surface in the given problem.Key points include:

• The surface is circular, with a radius \( r \).
• For a circle, area \( A \) is calculated using the formula \( A = \pi r^2 \).
• In our problem, the radius is given as \( 0.057 \mathrm{m} \).
• This gives an area of \( 0.0102 \mathrm{m}^2 \), after applying the formula.

The circular surface acts like a net catching the electric field lines. The concept of area is crucial here because electric flux, the measure of how much electric field passes through a surface, depends on it. The larger the surface, the more field lines it can "catch," affecting the flux calculation.

When evaluating the interaction between the electric field and the circular surface, one crucial aspect is the angle between them. This angle, \( \theta \), is defined between the direction of the electric field and the normal (a perpendicular line) to the surface.This matters because the angle dictates how much of the electric field penetrates the surface fully. To find the angle:

• Use the electric flux formula: \( \Phi = E \cdot A \cdot \cos(\theta) \).
• Given values include \( \Phi = 78 \mathrm{N \cdot m^2/C} \), \( E = 1.44 \times 10^4 \mathrm{N/C} \), and \( A = 0.0102 \mathrm{m}^2 \).
• Rearrange to find \( \cos(\theta) \): \( \cos(\theta) = \frac{\Phi}{E \cdot A} \).
• Substitute known values: \( \cos(\theta) \approx 0.5294 \).
• Determine \( \theta \) using the inverse cosine function: \( \theta = \arccos(0.5294) \approx 58.1^{\circ} \).

Understanding this calculation helps determine how completely the electric field lines pass through the surface, influencing the electric flux measurement.