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When examining the motion of electrons in the presence of electric fields, we encounter the concept of electric force. This force is the result of the electric field \(\overrightarrow{E}\) acting on a charged particle such as an electron. The formula for calculating the electric force \(\overrightarrow{F_E}\) is given by \(\overrightarrow{F_E} = e\overrightarrow{E}\), where \(e\) represents the elementary charge, approximately \(-1.60 \times 10^{-19}\) C, characteristic of electrons.

This force contributes to the electron's acceleration in the direction of the field if unopposed. However, in our scenario, the presence of a magnetic field exerts a simultaneous magnetic force, potentially countering the electric force to result in no net deflection of the electron's path. Understanding this balance is crucial in identifying conditions for a stable electron trajectory in combined electric and magnetic fields.

Magnetic force is another critical factor in the movement of electrons within a magnetic field. Unlike electric force, magnetic force depends on both the velocity of the electron and the magnetic field. The expression for magnetic force \(\overrightarrow{F_B}\) acting on a charged particle moving through a magnetic field \(\overrightarrow{B}\) is given by: \(\overrightarrow{F_B} = e\overrightarrow{v} \times \overrightarrow{B}\).

Notice the cross product, indicating that the magnetic force is perpendicular to both the velocity \(\overrightarrow{v}\) and the magnetic field vector \(\overrightarrow{B}\). It is this unique orientation that can lead to circular paths when no electric force counters it. In the exercise, the condition that there's no deflection means that the electron's perceived path remains straight in the combined fields, achievable when the magnitudes of the electric and magnetic forces are equal, thereby balancing each other out. This is expressed mathematically as \(eE = evB\), leading to a relation for the electron's velocity.

Once the electric field is removed, only the magnetic field remains to influence the electron. Under these circumstances, the electron begins to travel in a circular orbit. The radius of this orbit \(r\) in the magnetic field can be determined using the formula: \(r = \frac{mv}{eB}\), where \(m\) is the electron's mass (approximately \(9.11 \times 10^{-31}\) kg), \(v\) is the velocity previously calculated, and \(e\) is the elementary charge.

The magnetic force acts as the centripetal force, maintaining the electron's circular motion by pulling it inward. Calculating \(r\) provides insight into the physical path the electron will traverse under these conditions. This understanding is particularly useful in areas like designing cyclotrons and understanding particle tracks within magnetic fields.

Understanding the orientation of vectors is vital when visualizing the scenario described in the exercise. Here, the electric field vector \(\overrightarrow{E}\), magnetic field vector \(\overrightarrow{B}\), and electron velocity \(\vec{v}\) must all be perpendicular to one another. This mutual perpendicularity ensures that the forces can balance out, causing no net deflection of the electron's trajectory in the electric and magnetic fields.

To illustrate: if the electric field is aligned along the x-axis, the magnetic field might be along the y-axis, making the electron's velocity vector aligned along the z-axis. Each vector's perpendicular arrangement ensures the forces resulting from each field can counteract the others, allowing electrons to maintain a straight path without deflection. In physics, such vector orientations help solve complex problems, clarifying how these entities interact in space.