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Gravitational force is the natural phenomenon by which all things with mass attract each other. This force is one of the fundamental forces of nature and is responsible for the attraction objects experience from the Earth. It is mathematically expressed as:

• \( F_g = mg \)

where:

• \( F_g \) is the gravitational force (in Newtons),
• \( m \) is the mass of the object (in kilograms), and
• \( g \) is the acceleration due to gravity, approximately equal to \( 9.8 \, \text{m/s}^2 \) near the Earth's surface.

For the tiny ball in this exercise, the gravitational force can be calculated as \( 0.1176 \, \text{N} \). This indicates the force with which the Earth pulls on the ball. Understanding gravitational force is crucial for solving problems related to objects in the Earth’s gravitational field, such as determining the conditions needed to make an object float or levitate.

Coulomb's Law describes the force between two charged objects. It is a cornerstone of electrostatics and helps explain how electric forces operate. The force between two charges is directly proportional to the product of the magnitude of the charges and inversely proportional to the square of the distance between them. The formula for Coulomb’s law is:

• \( F = k \frac{|q_1 q_2|}{r^2} \)

where:

• \( F \) is the electric force,
• \( k \) is Coulomb's constant (\( 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \)),
• \( q_1 \) and \( q_2 \) are the point charges, and
• \( r \) is the distance between the charges.

In the context of this exercise, the electric field \( E \) causing the ball to float can be viewed as the influence of an electric force to counteract gravity. The electric force \( F_e \) is modeled as \( qE \), where \( q \) is the charge on the ball. Calculating \( E = 6533.33 \, \text{N/C} \) gives the necessary electric field to balance the gravitational force on the charged ball.

Equilibrium occurs when all forces acting on an object are balanced, resulting in no net force and thus no acceleration. This means the object will remain at rest or move at constant velocity. In the case of our floating charged ball, equilibrium is achieved when the upwards electric force due to the electric field precisely offsets the downwards gravitational force.

For equilibrium in our specific scenario, the key equations used were:

• Gravitational force: \( F_g = mg \)
• Electric force: \( F_e = qE \)

Setting these two equations equal, \( mg = qE \), allows us to solve for the electric field \( E \) needed to make the ball hover. It's crucial to note that the direction of the electric field must be such that it provides an upward force. Since the charge is negative, the field must be directed downward to achieve this effect. This concept ensures that all acting forces are in perfect balance, allowing for the ball to float.