You are still fascinated by the process of inkjet printing,...

You are still fascinated by the process of inkjet printing, as described in the opening storyline for this chapter. You convince your father to take you to his manufacturing facility to see the machines that print expiration dates on eggs. You strike up a conversation with the technician operating the machine. He tells you that the ink drops are created using a piezoelectric crystal, acoustic waves, and the Plateau-Rayleigh instability, which creates uniform drops of mass $m=1.25 \times 10^{-8}$ g. While you don't understand the fancy words, you do recognize mass! The technician also tells you that the drops are charged to a controllable value of $q$ and then projected vertically downward between parallel deflecting plates at a constant terminal speed of $18.5 \mathrm{~m} / \mathrm{s} .$ The plates are $\ell=2.25 \mathrm{~cm}$ long and have a uniform electric field of magnitude $E=6.35 \times 10^{4} \mathrm{~N} / \mathrm{C}$ between them. Noting your interest in the process, the technician asks you, "If the position on the egg at which the drop is to be deposited requires that its deflection at the bottom end of the plates be $0.17 \mathrm{~mm},$ what is the required charge on the drop?" You quickly get to work to find the answer.

Key Concepts

Piezoelectricity

Piezoelectricity is a property of certain materials in which an applied mechanical stress produces an electric charge and, conversely, an electric field can produce mechanical deformation. This effect is exploited in various devices, including precision actuators used in applications like inkjet printing to create controlled vibrations that generate fluid motion.

Plateau-Rayleigh Instability

The Plateau-Rayleigh instability is a fluid dynamics phenomenon explaining how a continuous stream of liquid breaks up into droplets due to the fluid's surface tension. This principle ensures the formation of uniformly sized droplets, a critical aspect in technologies such as inkjet printing and spray formation.

Uniform Electric Field and Force on Charge

A uniform electric field exerts a constant force on any charged particle within it, given by the equation F = qE. This concept is essential in understanding how charged droplets can be deflected or steered by electric fields in various applications, including the precise targeting seen in inkjet printing.

Terminal Velocity

Terminal velocity refers to the constant speed that a falling object reaches when the force of gravity is balanced by the resistance of the medium through which it falls. In the context of charged droplets, achieving terminal velocity means that the droplet's motion due to gravity is steady, allowing for predictable deflection calculations.

Kinematics of Uniform Acceleration

Kinematics of uniform acceleration involves studying the motion of objects under constant acceleration. This framework is used to calculate displacement, velocity, and time, which is crucial when analyzing the deflection of particles or droplets subjected to a constant force, such as that from an electric field.

Transcript

00:01 So problem 27 is about ink jet printing.

00:07 Okay, without getting into the technicalities here, we can understand this ink jet printing as little drops of ink dropped from above.

00:29 Okay um there's a little little drops have mass of uh 1 .25 times 10 to the minus 8 grams which translates to 1 .25 times 10 to the minus 11 11 kilograms this is the si unit okay and which drops have a controllable charge meaning that it changes depending on on the purpose or depending on where we wanted to deposit on the egg specifically for this exercise is about printing on eggs anyway so these drops are projected vertically downwards they go downward with a terminal speed meaning a constant speed of 18 .5 meters per second okay and they they are there and they and they and they they they fall between these two charged parallel plates of size 2 .25 centimeters meaning 0 .0 225 meters okay when it says size this is my this is i can think of them as a square plates anyway it's both the height and in both sides anyway and because they are charged, probably they act like capaceros, they generate an electric field.

02:12 The direction of the electric field is not specifically relevant, but we'll be given their magnetic, their magnitude.

02:19 The magnitude, excuse me, the magnetic field, the 6 .35 times 10 to the 4 neutrons per globe.

02:27 Okay.

02:28 Okay, and it says that if what we want is, if what we want is at the very end, at the very bottom of this capacitor, what we want is a deflection of exactly, let's call it d, deflection of 0 .17 millimeters.

02:50 Yeah, which translates to, let's write, 0 .17 times 10 to minus 3.

03:01 Meters or even better 1 .7 times 10 to the minus 4 meters if this is what we want what is the charge that needs to be deposited on the ink drop okay let's let's move on the solution of this exercise let's draw again really quickly the sketch so okay so the ink drops from here up to the bottom necessarily it goes all the way from the top to the bottom okay this is the one thing that we have to have in mind and it falls with a constant speed therefore the time that needs to reach the bottom from the top because it's a constant speed it's straightforward just the size the size the site of the capacitors of the plates divided by the terminal speed okay let's let's calculate it really quickly so if if the size is 25 meters divided by 18 .5 meters squared.

04:56 This gives us a time of 0 .00121622 seconds.

05:13 Okay, this is the time.

05:15 Even though it will be deflected, the force accepted on it, would be perpendicular to that direction to the electrostatic force on it will be perpendicular to the vertical therefore it won't have any effect on the terminal speed the terminal speed remains the same the component of the speed the component of the speed of the of the ink drop while falling down always remains v -term terminal terminal and now we know the time that needs to which the bottom from top okay and now we also know that it needs to deflect by a certain d needs to deflect and you see we don't care where it's initially the the drop we don't care we know we just want to we know the deflection with the vertical and this enough okay, so we know there's a uniform because of the plates there's a uniform electric field between the plates.

06:34 Therefore using this formula we know now that the force on the drop is the same everywhere...

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