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One type of ink-jet printer, called an electrostatic ink-jet printer, forms the letters by using deflecting electrodes to steer charged ink drops up and down vertically as the ink jet sweeps horizontally across the page. The ink jet forms30μm diameter drops of ink, charges them by spraying 800,000 electrons on the surface, and shoots them toward the page at a speed of 20m/s. Along the way, the drops pass through two horizontal, parallel electrodes that are 6.0mmlong,4.0mm wide, and spaced 1.0mm apart. The distance from the center of the electrodes to the paper is 2.0cm. To form the tallest letters, which have a height of 6.0mm, the drops need to be deflected upward (or downward) by 3.0mm. What electric field strength is needed between the electrodes to achieve this deflection? Ink, which consists of dye particles suspended in alcohol, has a density of 800kg/m3 .

Short Answer

Expert verified

The electric field strength needed is7.06x105N/C

Step by step solution

01

Given Information and Formula used

Given :

The diameter of drops of ink : 30μm

Charge on the droplets : 800,000

Speed : 20 m/s.

Dimensions of electrodes : 6.0mmlong,4.0mmwide, 1.0mmapart.

The distance from the center of the electrodes to the paper : 2.0cm.

Height of the tallest letters :6.0mm

The drops are deflected upward (or downward) by :3.0mm

Ink has a density of : 800kg/m3

Theory used :

Mass : m=ρV

Maximum deflection : tanθ=vyvx

Newton's law : v1y=v0y+ay(t1-t0)

Field strength :E=mayq

02

Calculating the maximum deflection

First, we must locate mass :

m=ρV=ρ(4π3r3)=(800kg/m³)(4π3(30x10-6m)3)=9.04x10-11kg

Then there's charge q:

q=(8x105)(1.6×10-19)C=1.28x10-13C

We now have a value for maximum deflection v1y

role="math" localid="1649140142801" tanθ=vyvx=(2mm30mm)=0.15v1y=tanθ(Vx)=0.15(20m/s)=3.0m/s

03

Calculating the required electric field strength needed between the electrodes 

We can now calculate acceleration ay using the following relation:

v1y=v0y+ay(t1-t0)ay=v1y-v0y(t1-t0)=3.0m/s-0m/s6.0×10-320m/s=3.0m/s3.0×10-4say=1.0×104m/s2

Finally, field strength E:

E=mayq=(9.04x10-11kg)(1.0x104m/s2)1.28x10-13C=7.06x105N/C

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