Question

The earth has a downward-directed electric field near its surface of about 150 N\(/\)C. If a raindrop with a diameter of 0.020 mm is suspended, motionless, in this field, how many excess electrons must it have on its surface?

Short answer

The raindrop must have approximately 1,706,250 excess electrons.

Step-by-step solution

Convert Units

Convert the raindrop's diameter from millimeters to meters. Since 1 mm = 0.001 m, a diameter of 0.020 mm is equivalent to 0.020 mm × 0.001 m/mm = 0.000020 m.

Calculate the Volume of the Raindrop

Assume the raindrop is spherical. The formula for the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \). First, find the radius: \( r = \frac{0.000020}{2} = 0.000010 \) m. Substitute into the volume formula: \( V = \frac{4}{3} \pi (0.000010)^3 \approx 4.19 \times 10^{-15} m^3 \).

Calculate the Mass of the Raindrop

Use the density of water, which is \( 1000 \text{ kg/m}^3 \). The mass \( m \) is given by \( m = \text{density} \times \text{volume} \). Therefore, \( m = 1000 \times 4.19 \times 10^{-15} = 4.19 \times 10^{-12} \text{ kg} \).

Calculate the Weight of the Raindrop

The weight \( W \) is the mass times the acceleration due to gravity \( g = 9.8 \text{ m/s}^2 \). Thus, \( W = 4.19 \times 10^{-12} \times 9.8 \approx 4.11 \times 10^{-11} \text{ N} \).

Calculate the Force Due to the Electric Field

The electric force \( F \) required to suspend the raindrop is equal to the weight of the raindrop. Therefore, \( F = W = 4.11 \times 10^{-11} \text{ N} \).

Find Charge Using Electric Field Equation

Use the formula for electric force: \( F = qE \), where \( q \) is the charge and \( E \) is the electric field (\(150 \text{ N/C}\)). Therefore, \( q = \frac{F}{E} = \frac{4.11 \times 10^{-11}}{150} \approx 2.74 \times 10^{-13} \text{ C} \).

Calculate the Number of Electrons

The charge of a single electron is \( e = 1.6 \times 10^{-19} \text{ C} \). The number of excess electrons \( N \) is given by \( N = \frac{q}{e} = \frac{2.74 \times 10^{-13}}{1.6 \times 10^{-19}} \approx 1.71 \times 10^6 \). Round to the nearest whole number, \( N \approx 1,706,250 \).

Key concepts

Raindrop Suspension

A raindrop is deemed 'suspended' when it remains motionless in the air. This situation occurs due to the balance of forces acting upon it. For a raindrop suspended near the Earth's surface, two main forces are at play. One is gravitational force, which pulls the drop downward, while the other is the electric force exerted by the surrounding electric field, which acts upward if the drop is negatively charged.

In the scenario described, the Earth's downward electric field (150 N/C) plays a crucial role. If a raindrop is negatively charged, this field provides an upward force that can counteract the downward gravitational pull. When these forces balance out perfectly, the drop achieves suspension and remains still. Understanding the conditions for suspension involves grasping how electric forces can counteract gravity, an essential concept in electrostatics.

This balance doesn't just apply to raindrops. It is a principle used in various applications, such as in designing devices that suspend objects using electromagnetic fields.

Excess Electrons Calculation

Calculating the excess electrons on the surface of a raindrop is a fascinating intersection of concepts like electric fields, forces, and basic charge principles. The formula \( q = \frac{F}{E} \) helps determine the charge required for suspension, where \( q \) is the electric charge, \( F \) is the force, and \( E \) is the electric field.

In this problem, we'd calculated the needed charge \( q \) to balance the gravitational force with the electric force. This results in a charge of \( 2.74 \times 10^{-13} \) Coulombs. However, charges come in discrete units, specifically electrons in this case. To find the number of excess electrons, divide the total charge by the charge of a single electron (\( 1.6 \times 10^{-19} \) C). This division gives the number of electrons needed to create the required charge for suspension.

This process provides a clear illustration of converting continuous electric charge into discrete units of electrons. As contexts become more complex or change scales, handling these concepts becomes essential for fields like electronics and chemistry.

Electric Force

Electric force is a central concept in understanding how charged objects interact. It's defined as the force between charged entities, described by Coulomb's Law. The strength and direction of the electric force depend on the charge and the electric field around the object.

In this exercise, the electric force is used to balance the gravitational force on a raindrop. When charges are suspended within an electric field, the electric force acting on those charges can be described by the equation \( F = qE \), where \( F \) is the force, \( q \) the charge, and \( E \) the electric field.

The electric field direction and the sign of the charge determine whether the force will oppose or reinforce gravity. Here, for suspension to occur, the field exerts an upward force equal to the gravitational force acting downward. This demonstrates an intriguing use of physics principles, showcasing the manipulation of electric fields to achieve specific conditions, like suspending a charged object.