91Ó°ÊÓ

An electric field is a region of space around a charged object where forces are exerted on other charged objects. The electric field generated by a large insulating sheet with uniform surface charge density, denoted as \( \sigma \), is constant and can be calculated using the formula:
\[E = \frac{\sigma}{2\varepsilon_0}\]where \( \varepsilon_0 \) is the permittivity of free space, valued at \( 8.85 \times 10^{-12} \, \text{C}^2/(\text{N} \, \text{m}^2) \). This equation indicates that the electric field strength is directly proportional to the surface charge density. This consistency means that any charged particle, such as a proton, will experience the same amount of force at any given distance from the sheet.
By evaluating the electric field, we understand the influence it has, which in turn affects the motion of the proton as it moves parallel to the sheet.

Protons, like other charged particles, move in response to electric fields. Understanding proton motion in this scenario requires considering the direction of both the proton's velocity and the electric field. Since the proton is initially moving parallel to the charged sheet, it's essential to focus on how the electric field's force impacts its velocity in a perpendicular direction.
The proton has an initial velocity of \( 9.70 \times 10^2 \, \text{m/s} \). The electric field doesn't affect this parallel component of velocity, but it does exert a force that changes the speed by accelerating the proton in the direction of the field's force. To understand the proton's exact motion, it is crucial to calculate the change in velocity induced by the electric field over the given time.

Surface charge density \( \sigma \) quantifies the amount of charge per unit area on a surface. It is measured in coulombs per square meter (C/m\(^2\)). In this exercise, understanding surface charge density helps determine the electric field's strength, as they are directly related through the formula:
\[E = \frac{\sigma}{2\varepsilon_0}\]The given uniform surface charge density of the sheet is \( 2.34 \times 10^{-9} \, \text{C/m}^2 \), which directly influences the electric field's magnitude.
This uniform distribution means that the electric field is constant around the sheet, providing a predictable environment for the proton's motion. This stability allows for straightforward calculations of the forces acting on the proton as it navigates the area adjacent to the charged sheet.

Newton's second law of motion describes how the velocity of an object changes when it is subjected to an external force. This law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration, given as:
\[F = ma\]In this exercise, we applied Newton's second law to find the acceleration of a proton under the influence of the electric field. Using the force calculated from the electric field formula, and knowing the proton's mass \((1.67 \times 10^{-27} \, \text{kg})\), we derived its acceleration:
\[a = \frac{F}{m}\]By knowing the acceleration and the time interval of proton motion, we could compute the change in velocity and thus determine the final speed of the proton after \(5.00 \times 10^{-8} \, \text{s}\). This showcases how Newton's second law facilitates understanding the dynamic behavior of charged particles in electromagnetic fields.