Problem 32 - University Physics with Modern Physics

Question

Electric Field of the Rarth. 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) +3.92 C. (b) 1.39 × 10⁷ N repulsion; impractical for flight due to safety concerns and feasibility issues.

Step 1: Understanding the Electric Field Force

The earth's electric field exerts a force on a charge given by the formula \( F = qE \), where \( F \) is the force, \( q \) is the charge, and \( E \) is the electric field strength (150 N/C). We need the force exerted by the electric field to overcome the person's weight (\(mg\), where \(m = 60\, \text{kg}\) and \(g = 9.8\, \text{m/s}^2\)). Set \( F = mg \) to find \( q \).

Step 2: Deriving the Required Charge Sign and Magnitude

Using \( F = mg \) and \( F = qE \), set \( qE = mg \). Solve for \( q \): \( q = \frac{mg}{E} = \frac{60 \times 9.8}{150} \). This results in \( q \approx 3.92 \) C. To overcome gravity, the charge should be opposite to the field; in this case, a positive charge is needed (since the field is directed inward, the person must "push" upward).

Step 3: Calculating the Force of Repulsion Between Two Charges

Use the formula for the electric force between two point charges, \( F = \frac{k q_1 q_2}{r^2} \), where \( k \) is Coulomb's constant \( 8.99 \times 10^9 \; \text{Nm}^2/\text{C}^2 \), \( q_1 = q_2 = 3.92 \; \text{C} \), and \( r = 100 \; \text{m} \). So, \( F = \frac{8.99 \times 10^9 \times (3.92)^2}{100^2} \). This yields \( F \approx 1.39 \times 10^7 \text{N} \).

Step 4: Feasibility Analysis for Using Earth's Electric Field for Flight

Considering the massive amount of charge required and the resulting huge repulsive force between two charged persons, it is impractical to use the Earth's electric field for flight. Such a charge would pose safety risks, such as electric discharge and interference with electronic devices.

Key Concepts

Electric Force

Electric force is the interaction between charged objects. When you place a charged particle in an electric field, it experiences a force. This force can either pull the charges together or push them apart, depending on their charge types. Opposite charges attract each other, while like charges repel. This property is fundamental to understanding how objects behave in electric fields.

This force is described using the equation:

• \( F = qE \)

Here:

• \( F \) is the force
• \( q \) is the charge
• \( E \) is the electric field strength.

For instance, when a person with a charge experiences the Earth's electric field of 150 N/C, that field exerts a force pulling them inward. To counteract gravity, they would need an opposing force, which could be achieved with a sufficient charge directed against the field.

Coulomb's Law

Coulomb's Law helps us understand how strongly two charged objects interact. It states that the force between two point charges is proportional to the product of their charges and inversely proportional to the square of the distance between them.

The formula is given by:

• \( F = \frac{k \, q_1 \, q_2}{r^2} \)

Where:

• \( F \) is 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 magnitudes of the charges
• \( r \) is the distance between the two charges.

This principle was applied to determine the force between two charged people in the exercise. By assigning the same charge to both individuals, and placing them 100 meters apart, the immense repulsive force derived from the calculation indicates significant interaction despite the considerable distance.

Feasibility of Electrostatic Flight

The idea of using electric fields for flight is indeed captivating. It would require exploiting the force exerted by environmental electric fields upon objects with large charges. In the context of overcoming Earth's gravity, the challenge lies in creating a sufficient electric force to counteract weight force.

For a person weighing 60 kg, the charge required to produce an upward force equal to their weight was calculated to be approximately 3.92 C. However, consider the practicality:

• Such a charge is dangerously large and can result in harmful sparks or electric discharges.
• Electronic device interferences would become significant safety hazards.
• The obtained repulsive force between two similarly charged people was found to be extraordinarily high, about 1.39 × 10⁷ N, making any nearby interaction impractical.

Due to these concerns, while electrostatic flight sounds intriguing, current technological and safety challenges prevent its practical implementation. It remains an impractical dream until safer and more efficient methods are developed.