
An air ionizer fillers particles of dust, pollen, and other allergens from the air using electric forces. In one type of ionizer (see diagram), a stream of air is drawn in with a speed of 3.0 m/s. The air passes through a fine, highly charged wire mesh that transfers electric charge to the particles. Then the air passes through parallel “collector” plates that admet the charged particles and trap them in a filter. Consider a dust particle of radius 6.0 μm, mass 2.0 × 10–13 kg, and charge l000e. The plates are 10 cm long and are separated by a distance of 10 cm. (a) Ignoring drag forces, what would be the minimum potential difference between the plates to ensure that the particle gets trapped by the filter? (b) At what speed would the particle be moving relative to the stream of air just before hitting the filter? (c) Calculate the viscous drag force on the particle when moving at the speed found in (b). (d) Is it realistic to ignore drag? Taking drag into consideration, is the minimum potential difference larger or smaller than the answer to (a)?
(a)

The minimum potential difference between the plates to ensure that the particle gets trapped by the filter.
Answer to Problem 125P
The minimum potential difference between the plates to ensure that the particle gets trapped by the filter is
Explanation of Solution
Write the equation for the time taken by the dust particle to pass through the length of the collector plates.
Here,
Write the equation of motion in vertical direction.
Here,
The initial speed of the particle in the vertical direction is zero.
Substitute
Put equation (I) in equation (II).
Write the equation for the net force on the particle in the vertical direction.
Here,
Rewrite the above equation for
The net force in the vertical direction is equal to the electric force.
Write the relationship between the net force and the electric force.
Here,
Put the above equation in equation (IV).
Write the equation for the electric force.
Here,
Write the equation for the electric field.
Here,
Put the above equation in equation (VI).
Put equation (VII) in equation (V).
Equate equations (III) and (VIII) and rewrite it for
Substitute
Here,
Conclusion:
The value of
Substitute
Therefore, the minimum potential difference between the plates to ensure that the particle gets trapped by the filter is
(b)

The speed at which the particle would be moving relative to the stream of air just before hitting the filter.
Answer to Problem 125P
The speed at which the particle would be moving relative to the stream of air just before hitting the filter is
Explanation of Solution
Write the equation of motion in vertical direction.
Here,
Substitute
Put equation (II) in the above equation.
Put equation (I) in the above equation.
Conclusion:
Substitute
Therefore, the speed at which the particle would be moving relative to the stream of air just before hitting the filter is
(c)

The viscous drag force on the particle when moving at the speed found in part (b).
Answer to Problem 125P
The viscous drag force on the particle when moving at the speed found in part (b) is
Explanation of Solution
Write the equation for the drag force.
Here,
Conclusion:
The viscosity of air is
Substitute
Therefore, the viscous drag force on the particle when moving at the speed found in part (b) is
(d)

Whether it is realistic to ignore the drag and whether the minimum potential difference larger or smaller than the answer to part (a) when drag is considered.
Answer to Problem 125P
It is not realistic to ignore the drag and the minimum potential difference should be larger when drag is considered.
Explanation of Solution
It is realistic to ignore the drag if the drag force is much smaller than the electric force.
Substitute
Conclusion:
Substitute
The value of
Therefore, it is not realistic to ignore the drag and the minimum potential difference should be larger when drag is considered.
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