Problem 32 - University Physics with Modern Physics

Question

Electric Field of the Earth. The earth has a net electric charge that causes a field at points near its surface equal to 150 \(\mathrm{N} / \mathrm{C}\) and directed in toward the center of the earth. (a) What magnitude and sign of charge would a \(60-\mathrm{kg}\) 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 \(\mathrm{m} ?\) Is use of the earth's electric field a feasible means of flight? Why or why not?

Step-by-Step Solution

Answer

A charge of approximately +3.92 C is needed. The force of repulsion is about 14,000 N, making flight impractical due to safety risks.

Step 1: Determine the Weight of the Person

The weight of the person is given by the formula: \( W = mg \), where \( m = 60 \, \mathrm{kg} \) is the mass of the person, and \( g = 9.8 \ \mathrm{m/s^2} \) is the acceleration due to gravity. Thus, \( W = 60 \times 9.8 = 588 \, \mathrm{N} \).

Step 2: Balance the Electric Force with the Weight

The electric field exerts a force \( F = Eq \) on a charge \( q \), where \( E = 150 \, \mathrm{N/C} \) is the electric field strength. To balance the weight with the electric force, set \( F = W \): \( 150q = 588 \, \mathrm{N} \). Solving for \( q \), \( q = \frac{588}{150} \approx 3.92 \, \mathrm{C} \).

Step 3: Determine the Sign of the Charge

Since the electric field is directed toward the center of the Earth, and the person needs to overcome the gravitational pull pointing downward, the charge \( q \) must be positive (since the electric field force on a positive charge is directed against the electric field).

Step 4: Calculate the Force of Repulsion Between Two People

Using Coulomb's law, the force between two charges is given by: \( F = \frac{kq_1q_2}{r^2} \), where \( k = 8.988 \times 10^9 \, \mathrm{N\,m^2/C^2} \). Both people have a charge of \( 3.92 \, \mathrm{C} \) and are \( 100 \, \mathrm{m} \) apart, so substituting these values: \( F = \frac{8.988 \times 10^9 \times (3.92)^2}{(100)^2} \approx 14,000 \, \mathrm{N} \).

Step 5: Evaluate Feasibility of Flight

For practical flight, the force needed to overcome gravitational pull should be manageable and relatively less hazardous. Given the enormous charge required and the large force of repulsion, this method is not feasible for flight, as it would be dangerous and unwieldy.

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental principle that describes the force between two point charges. It is a cornerstone of electrostatics, helping us understand how charged objects interact.
The law states that the magnitude of the electrostatic force between two charges is directly proportional to the product of the absolute magnitudes of the charges, and inversely proportional to the square of the distance between them.
The formula is given by

• \[ F = \frac{kq_1q_2}{r^2} \]

Where:

• \( F \) is the force between the charges,
• \( k \approx 8.988 \times 10^9 \, \mathrm{N\,m^2/C^2} \) is Coulomb's constant,
• \( q_1 \) and \( q_2 \) are the amounts of the charges, and
• \( r \) is the distance between the centers of the two charges.

Coulomb's Law shows us that like charges repel and unlike charges attract.
In the context of the exercise, it helps calculate the repulsive force between two humans given a theoretical charge that overcomes gravitational weight using Earth's electric field.

Electrostatics

Electrostatics is the study of electric charges at rest. It's the branch of physics that deals with phenomena and forces resulting from stationary or slow-moving electric charges.
One of its key interests is understanding how charges generate electric fields and how they interact with each other. Key concepts include:

• Electric field (\( E \)): a region around a charged object where another charge experiences a force.
• Electric force (\( F \)): the push or pull experienced by a charge in an electric field.

In the exercise you looked at, electrostatics is used to balance the force due to the Earth's electric field against the gravitational force on a person.
This illustrates the principle that static electric fields can exert forces powerful enough to counteract gravity, albeit requiring impractically large charges in this scenario.

Gravitational Force

Gravitational Force is a natural phenomenon by which all things with mass or energy are brought toward one another. On Earth, it gives weight to physical objects and is the force that draws you towards the center of our planet.
The gravitational force can be calculated with the formula:

• \[ W = mg \]

Where:

• \( W \) is the weight of the object,
• \( m \) is the mass of the object, and
• \( g \approx 9.8 \, \mathrm{m/s^2} \) is the acceleration due to gravity.

In the context of the exercise, a human's gravitational force is balanced against the electric force from Earth's electric field. The analysis asks whether it's possible to use this electric field to "fly" by counteracting gravity, concluding that it's impractical given the vast charge needed.

Electric Charge

Electric charge is a property of matter that causes it to experience a force when placed in an electric field. It can be positive or negative, and like charges repel while opposite charges attract.
Charges come in discrete natural units, with the elementary charge denoted as \( e \), roughly equal to \( 1.602 \times 10^{-19} \, \mathrm{C} \).

• Positive charge denotes a deficiency of electrons, and negative charge indicates an excess of electrons.

In the exercise, the electric charge needed for a human to overcome gravity involves understanding how charges interact within an electric field.
The computed charge in part 'a' exemplifies how an electric force could theoretically counterbalance weight, probing the feasibility of manipulating electric charge for practical engineering solutions like flight.