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An electric field is a region where an electric charge experiences a force. When a charge enters an electric field, it feels a push or pull dependent upon the field's direction and the charge's polarity. Notably, electric fields are typically illustrated with field lines pointing in the direction a positive charge would move. However, since electrons have negative charges, they move opposite to these field lines.

In the given scenario, the electric field points along the negative y-axis. This means an electron, moving toward the positive x-axis, will experience a force pulling it upward, along the positive y-axis. The formula for calculating the force exerted by an electric field on a charge is given by:

• The electric force, \( \vec{F}_e = q \vec{E} \)
• Where \( q \) is the charge of the electron and \( \vec{E} \) is the electric field

This is crucial in understanding how the electron's movement and forces in multiple dimensions interact.

The right-hand rule is a convenient way to determine the direction of the force exerted by a magnetic field on a moving charge. This rule also applies to finding the direction of the magnetic field itself. Remember, electrons carry a negative charge, so the right-hand rule must be carefully adjusted.

When using the right-hand rule to determine the direction of the magnetic force on an electron, do the following:

• First, point your right thumb in the direction of the electron's velocity. Because the electron's charge is negative, reverse this direction. In this problem, while the electron moves positively along the x-axis, point your thumb along the negative x-axis to account for negativity.
• Next, suppose the lines curl toward the magnetic field's direction along the positive y-axis if we ignore reversal. This guides you to the correct thumb direction.
• Your thumb then indicates the force on a positive charge; reverse it for an electron. If the thumb initially was along what indicates the electric force direction, shift orientations to achieve cancellations.

Ultimately, according to this adjustment of the rule, the resultant cross-product for the correct field direction is directed along the negative z-axis to counterbalance the electric force.

The concept of net force is critical in physics, as it denotes the total effect of all forces acting upon a body. For equilibrium or non-movement, the net force on an object must be zero. This means that all involved forces counteract one another effectively.

When considering our problem involving the electron, the goal is to achieve a net force of zero. This requires balancing the electric and magnetic forces perfectly:

• The electric force acts in the positive y-direction, moving the electron upwards.
• The required magnetic force must act in the opposite direction to counterbalance this, thus, along the negative y-axis.
• The magnetic field direction that accomplishes this is along the negative z-axis.

Thus, by setting the forces against each other, equilibrium is established, resulting in a zero net force. This neat balancing ensures the electron's unimpeded travel and is central to resolving the magnetic forces interactively with electric forces.