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When discussing electric fields, particularly in the context of charged surfaces, it's important to understand the concept of **uniform surface charge density**. This term refers to the distribution of electric charge across a surface. A uniform surface charge density indicates that the charge is spread evenly over the surface, which simplifies the calculations for electric fields.

The electric field created by a uniformly charged infinite sheet is particularly notable. For an infinitely large sheet, the electric field is calculated as:

• Independent of distance from the sheet
• Uniform and constant across space

These characteristics result because the field lines are parallel and uniformly distributed. The formula to calculate this constant electric field **E** for an infinite sheet with surface charge density \(\sigma\) is:\[ E = \frac{\sigma}{2 \varepsilon_0} \]where \(\varepsilon_0\) represents the vacuum permittivity, a constant value.

This consistent electric field plays a crucial role in determining how a proton or any charged particle will behave when introduced into this field.

Acceleration occurs when a force acts on a mass. In the context of charged particles, like a proton, moving in an electric field, this force can lead to changes in motion. **Proton acceleration** is determined by the electric force derived from the field.

The force **F** experienced by a proton in an electric field **E** is computed using the relation:\[ F = qE \]where **q** is the charge of the proton, equal to \(1.6 \times 10^{-19} \text{ C}\). Once we have the force, we apply Newton's second law, \( F = ma \), where **m** is the proton's mass \(1.67 \times 10^{-27} \text{ kg}\), to find the acceleration **a**:\[ a = \frac{F}{m} \]This allows us to calculate how fast the proton speeds up in response to the constant push it receives from the electric field.

Understanding this acceleration is critical because it tells us how the proton's velocity changes over time while it moves in the field.

In physics, kinematics deals with the motion of objects without discussing the forces that cause such motion. The **kinematic equation** used here relates the initial velocity, acceleration, and time to determine the final velocity of an object.

Given an initial velocity **u**, an object's velocity **v** after time **t** can be calculated if it undergoes constant acceleration **a**. The kinematic equation is expressed as:\[ v = u + at \]This equation allows us to predict the final velocity of the proton as it moves through the electric field.

• **u**: Initial speed of the proton
• **a**: Acceleration calculated from the electric force
• **t**: Time duration in motion

Using this equation, we can conclude that the proton's speed will increase linearly over the stated time interval if the acceleration remains constant. This is crucial for calculating how the proton's speed changes as a result of the uniform electric field, enabling us to find its new speed after **5.00 \(\times\) 10\(^{-8}\)** seconds.