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Hooke's Law is an essential principle in physics that explains how springs behave when force is applied to them. It signifies the relationship between the force exerted by a spring and the distance it is stretched or compressed. Formally, Hooke's Law is described by the equation, \( F_s = k \cdot \Delta x \), where \( F_s \) is the force exerted by the spring, \( k \) is the spring constant, and \( \Delta x \) is the change in the spring's length from its natural length. The spring constant \( k \) is crucial as it indicates how stiff or flexible the spring is. A large \( k \) means the spring is stiffer, requiring more force to stretch or compress. In the given exercise, the spring's constant is \( 2.4 \text{ N/m} \), showing the spring is relatively flexible since it alters its length under the sphere's weight and the external force.

The gravitational force is a fundamental force that attracts two bodies with mass. It's the reason objects fall towards the Earth when dropped. This force can be calculated with the formula \( F_g = m \cdot g \), where \( F_g \) is the gravitational force, \( m \) is the object's mass, and \( g \) is the acceleration due to gravity, approximately \( 9.81 \text{ m/s}^2 \) on Earth's surface. In the problem, the sphere has a mass of \( 5.1 \times 10^{-3} \, \text{kg} \), producing a downward force. Calculating \( F_g \) provides \( 0.049981 \text{ N} \), reflecting the gravitational pull on the sphere. This force is essential as it influences the equilibrium position of the spring-sphere system against the upward electric force.

Electric force is the force between charged objects. It can either attract or repel depending on the nature of the charges. The electric force acting on a charge in an electric field is given by \( F_e = q \cdot E \), where \( F_e \) is the electric force, \( q \) is the charge on the object, and \( E \) is the electric field strength. In our scenario, the sphere has a charge of \( +6.6 \mu \text{C} \) and interacts with an external electric field that counteracts the gravitational and spring forces. By rearranging the equilibrium equation to solve for the electric force, we obtain \( F_e = 0.013981 \text{ N} \). Using this, we can find \( E \), the desired magnitude of the electric field, equaling approximately \( 2118.33 \text{ N/C} \). Electric forces like this are integral in balancing other forces acting on the system, achieving equilibrium.

Equilibrium refers to the state of a system where all forces acting upon it balance out, resulting in no net force. This means the object remains stationary or moves at a constant speed. In the exercise, we observe a spring hanging with a sphere attached to its end, influenced by gravitational and electric forces. At equilibrium, the downward gravitational force equals the sum of the upward spring force and electric force. Mathematically, \( F_g = F_s + F_e \). This relationship allows us to understand where and why the sphere stops moving, showing that balance occurs when the net force is zero. Recognizing equilibrium is crucial in solving complex physics problems, as it guides you through understanding how forces interact and balance in various physical systems.