Problem 28 - University Physics with Modern Physics

Question

The earth has a net electric charge that causes a field at points near its surface equal to 150 N\(/\)C and directed in toward the center of the earth. (a) What magnitude and sign of charge would a 60-kg human have to acquire to overcome his or her weight by the force exerted by the earth's electric field? (b) What would be the force of repulsion between two people each with the charge calculated in part (a) and separated by a distance of 100 m? Is use of the earth's electric field a feasible means of flight? Why or why not?

Step-by-Step Solution

Answer

The person needs a charge of +3.92 C. Force of repulsion is 13.88 N. Flying with the Earth's electric field is not feasible.

Step 1: Calculate Force due to Weight

The weight of a person is given by the equation \( F_{gravity} = m \times g \) where \( m = 60 \ \text{kg} \) and \( g = 9.8 \ \text{m/s}^2 \). So, \( F_{gravity} = 60 \times 9.8 = 588 \ \text{N} \).

Step 2: Calculate Required Electric Force

The electric force needed must equal the gravitational force for the person to be levitated. Therefore, \( F_{electric} = 588 \ \text{N} \).

Step 3: Calculate Charge to Overcome Weight

The electric force is also given by \( F_{electric} = q \times E \), where \( E = 150 \ \text{N/C} \). To find the charge \( q \), rearrange as \( q = \frac{F_{electric}}{E} = \frac{588}{150} = 3.92 \ \text{C} \).

Step 4: Apply Newton's Third Law for Same Charges

The sign of the charge needed should be positive. This is because electric field lines point towards negative charges, and to have an outward force against the field direction (inwards), the charge must be positive.

Step 5: Calculate Force Between Two Charged Individuals

Use Coulomb's Law: \( F = \frac{k \times q_1 \times q_2}{r^2} \) where \( k = 8.99 \times 10^9 \ \text{Nm}^2/\text{C}^2 \), \( q_1 = q_2 = 3.92 \ \text{C} \), and \( r = 100 \ \text{m} \). Calculate: \[ F = \frac{8.99 \times 10^9 \times (3.92)^2}{(100)^2} \approx 13.88 \ \text{N} \].

Step 6: Evaluation of Feasibility

The required charge is extremely large compared to typical static charges, and producing such a charge poses significant practical and safety challenges. The relatively small repulsive force between two such charged individuals further highlights the difficulty in achieving flight using this method.

Key Concepts

Electrostatic Force

Electrostatic force is a fundamental force acting between electrically charged objects. It's a non-contact force, meaning charged objects can exert forces on each other even when they're not physically touching.
This force can be either attractive or repulsive, depending on the types of charges involved. Like charges repel each other, while opposite charges attract.

The magnitude of the electrostatic force between two point charges is determined by combining the magnitudes of the charges and the distance between them. The force is much more significant over shorter distances, but can still pack a punch even at a distance, depending on the charge magnitude. This is why understanding electrostatic forces is crucial, especially in applications like levitation or electronics.

Coulomb's Law

Coulomb's Law is the mathematical formulation that describes the electrostatic force between two charged particles. It was named after Charles-Augustin de Coulomb, who first expressed the force's inverse-square law.
The law is expressed by the formula: \[ F = \frac{k \times |q_1 \times q_2|}{r^2} \] where:

• \( F \) is the magnitude of the force between the charges,
• \( k \) is Coulomb's constant \( (8.99 \times 10^9 \, \text{Nm}^2/\text{C}^2) \),
• \( q_1 \) and \( q_2 \) are the amounts of the charges,
• \( r \) is the distance between the centers of the two charges.

This equation shows that the force increases with larger charges or closer distances and decreases with smaller charges or larger gaps. Coulomb's Law is crucial for calculating and understanding the behavior of charged particles in various physical situations.

Electric Charge

Electric charge is a property of matter that causes it to experience a force when placed in an electric field. Charges are either positive or negative and are measured in coulombs (C). Positive charges are associated with protons, while negative charges correlate with electrons.
Charge conservation is an important principle, indicating that the total electric charge in an isolated system remains constant regardless of changes within the system.

Objects become charged by gaining or losing electrons. When an object has more electrons than protons, it is negatively charged. When it has fewer electrons, it is positively charged. This transfer of charge is the essence of electrostatic phenomena, influencing how objects interact through electric forces.

Gravitational Force

Gravitational force is the attractive force that acts between any two masses. It’s what keeps people on the surface of the Earth and governs relationships between celestial bodies.
This force is calculated using Newton's law of universal gravitation, given by:\[ F_{gravity} = \frac{G \times m_1 \times m_2}{r^2} \]where:

• \( F_{gravity} \) is the gravitational force between two masses,
• \( G \) is the gravitational constant \((6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2)\),
• \( m_1 \) and \( m_2 \) are the masses,
• \( r \) is the distance between the centers of the two masses.

While the gravitational force is weaker compared to the electrostatic force on a particle scale, it is dominant at larger scales like planets and stars. It's a crucial force for understanding the dynamics of objects on Earth and in the cosmos.

Feasibility of Electromagnetic Levitation

Electromagnetic levitation involves using magnetic and electric fields to lift an object, counteracting the force of gravity. While it's an intriguing concept, there are various challenges in its practical application.
In the outlined exercise, the charge needed for levitation is unreasonably high. Achieving a charge sufficient to counteract the gravitational pull on a human is not practical with our current technology and safety considerations.

• The required charge is difficult to obtain or sustain due to the massive electric capacitance involved.
• Generous expenses and technological limitations make everyday use of electromagnetic levitation infeasible for large objects like humans.
• Despite this, the concept has found success in specialized areas like magnetic trains and industrial applications, where controlled environments and setups are possible.

Electromagnetic levitation remains a field of active research, with ongoing studies to improve its accessibility for broader applications.