91Ó°ÊÓ

When studying electron motion in an electric field, we often use kinematic equations. These equations help us describe the motion of an object when it is subject to acceleration. In our example, we used the kinematic equation:
\[ v_f^2 = v_i^2 + 2ad \]
This allows us to calculate the distance an electron travels before stopping. Here, \(v_f\) is the final velocity, \(v_i\) is the initial velocity, \(a\) is the acceleration, and \(d\) is the distance. We also use another kinematic equation to find the time it takes for the electron to stop: \[ v_f = v_i + at \]
By rearranging, we can solve for time: \[ t = \frac{v_f - v_i}{a} \]
In these equations:

• \(v_f\) is the final velocity, which is 0 when the electron stops momentarily.
• \(v_i\) is the initial velocity, given as \(5.00 \times 10^6 \mathrm{~m/s}\).
• \(a\) is the acceleration, calculated using the electric force acting on the electron.

Understanding these equations is crucial for analyzing how objects move under various forces, such as electric fields.

The electric force plays a significant role in the motion of an electron within an electric field. In our problem, the electric field (\(E\)) has a magnitude of \(1.00 \times 10^3 \mathrm{~N/C}\). The force (\(F\)) acting on the electron due to the electric field can be calculated using the formula:
\[ F = eE \]
Here:

• \(e\) is the charge of the electron, approximately \(1.60 \times 10^{-19} \mathrm{~C}\).
• \(E\) is the magnitude of the electric field, given as \(1.00 \times 10^3 \mathrm{~N/C}\).

Once we know the force, we can find the acceleration (\(a\)) using Newton's second law: \[ F = ma \]
Rearranging to solve for acceleration, we get: \[ a = \frac{F}{m} \]
Since the force (\(F\)) is equal to \(eE\), we have: \[ a = \frac{eE}{m} \]
By substituting the given values, we can determine the acceleration of the electron caused by the electric field. Understanding electric force is essential for analyzing the behavior of charged particles in fields and various applications, including semiconductors and accelerators.

Kinetic energy is the energy an object possesses due to its motion. In our problem, we need to calculate the initial and final kinetic energy of the electron as it moves through the electric field. The formula for kinetic energy (\(KE\)) is:
\[ KE = \frac{1}{2} mv^2 \]
The initial kinetic energy (\(KE_i\)) can be found using the initial speed (\(v_i = 5.00 \times 10^6 \mathrm{~m/s}\)).
The final kinetic energy (\(KE_f\)) is calculated using the final speed (\(v_f\)), which we find from the kinematic equation: \[ v_f^2 = v_i^2 + 2ad \]
By determining \(v_f\) and substituting it back into the kinetic energy formula, we can compare the initial and final kinetic energy. To find the fraction of kinetic energy lost, we use: \[ \frac{KE_i - KE_f}{KE_i} \]
Understanding the relationship between motion and energy helps us analyze how forces impact an object's speed and how energy is transferred or transformed within a system.