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When an electron enters an electric field, it experiences what we call an electric force. This force, denoted by the vector \(\overrightarrow{F}\), is key to determining how the electron will move within the field. The electric force exerted on a charged particle is calculated using the formula \(\overrightarrow{F} = q \overrightarrow{E}\), where \(q\) represents the charge of the electron. Typically, the charge of an electron is a known negative value of \(-1.6 \times 10^{-19}\) Coulombs. The vector \(\overrightarrow{E}\) is the electric field in which the electron is moving, measured in newtons per coulomb (N/C).
In the example given, the field vector is \((2.0 \hat{\mathbf{i}} + 8.0 \hat{\mathbf{j}}) \times 10^{4}\) N/C. This implies that the electric force experienced by the electron will be in both the \(\hat{\mathbf{i}}\) and \(\hat{\mathbf{j}}\) directions. This electric force will cause a change in the electron's motion, influencing its acceleration and direction of travel.

Vector acceleration is the change in velocity that an object undergoes due to the force acting upon it. In physics, Newton's Second Law \(\overrightarrow{F} = m \overrightarrow{a}\) helps describe this relationship. Here, \(\overrightarrow{a}\) is the acceleration vector, \(\overrightarrow{F}\) is the force applied to the object, and \(m\) is the object's mass.
For an electron, whose mass is notably small \(9.11 \times 10^{-31}\) kg, applying a force results in a substantial acceleration. Upon calculating the force as mentioned in the electric force section, we can substitute it into Newton's formula to find the acceleration.
Let's break it down further into components:

• The x-component of acceleration \(a_x\) is calculated using the x-component of the force and is found to be \(-3.51 \times 10^{15} \text{ m/s}^2\).
• Similarly, the y-component of acceleration \(a_y\) is calculated using the y-component of the force, resulting in \(-1.40 \times 10^{16} \text{ m/s}^2\).

This acceleration tells us how quickly and in what direction the electron's velocity is changing over time.

Once we know the acceleration, predicting the electron’s new velocity after a period is possible with the equation \(\overrightarrow{v}(t) = \overrightarrow{v}_0 + \overrightarrow{a} t\). This formula accounts for the initial velocity of the electron, \(\overrightarrow{v}_0 = 8.0 \times 10^{4} \hat{\mathbf{j}}\) m/s in this case, and its acceleration over a given time \(t\).
At \(t = 1\) ns, the velocity vector of the electron can be determined:
- The x-component \(v_x\) gets a boost from the acceleration and is \(-3.51 \times 10^{6} \text{ m/s}\).

The y-component \(v_y\) changes due to force in the field and becomes \(-13.92 \times 10^{6} \text{ m/s}\).

This results in a new moving direction for the electron. To figure out its exact direction, you can use the trigonometric relationship \(\tan \theta = \frac{v_y}{v_x}\). Calculating \(\theta\) tells us how the direction of the velocity vector differs from the original orientation along the \(\hat{\mathbf{j}}\) axis at any given time post-acceleration. Such adjustments are crucial in understanding the electron's dynamic behavior in an electric field.