The combustion of fossil fuels produces micron-sized particles of soot, one of the major components of air pollution. The terminal speeds of these particles are extremely small, so they remain suspended in air for very long periods of time. Furthermore, very small particles almost always acquire small amounts of charge from cosmic rays and various atmospheric effects, so their motion is influenced not only by gravity but also by the earth's weak electric field. Consider a small spherical particle of radius , density , and charge . A small sphere moving with speed v experiences a drag force , where is the viscosity of the air. (This differs from the drag force you learned in Chapter 6 because there we considered macroscopic rather than microscopic objects.)
a. A particle falling at its terminal speed is in equilibrium with no net force. Write Newton's first law for this particle falling in the presence of a downward electric field of strength , then solve to find an expression for .
b. Soot is primarily carbon, and carbon in the form of graphite has a density of . In the absence of an electric field, what is the terminal speed in of a -diameter graphite particle? The viscosity of air at is .
c. The earth's electric field is typically (150 N/C , downward). In this field, what is the terminal speed in of a -diameter graphite particle that has acquired 250 extra electrons?
The terminal speed of a particle falling from the sky is when it hits the earth at a constant speed.
The particle's acceleration and net force are both zero if it moves at a constant speed.
The particle descending from the sky is being acted upon by three forces. That's gravitational force, dragging force, and the force exerted by the earth's surface.
Since the force acting in the upward direction is positive and the force acting in the descending direction is negative, the force acting in the upward direction is considered positive.
The force's equation is as follows:
The weight of the particle is , the dragging force operates in the upward direction is , and the electric field strength is E.
Insert for in the force equation.
Rewrite the above equation for .
The following is the formula for mass in terms of density and volume:
Substitute for m in the equation .
Thus, the terminal speed is .
In the absence of an electric field, calculate the terminal speed as follows:
The value of denotes the absence of an electric field.
In the given equation, replace E with 0.
In terms of density and volume, the formula for mass is as follows:
Insert for $m$ in the equation
Substitute for for r and for for g in the above equation.
Thus, the terminal speed in the absence of electric field is
In the presence of an electric field, calculate the terminal speed as follows:
calculate the term as follows:
Insert for for for r and for
Insert for and for in the equation
Therefore, the terminal speed in the presence of electric field is
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