# Electric field strength in an ink-jet printer

• pbj_sweg
In summary: Then use the equilibrium relationship between the forces on the drop and solve for the value of Δy.The Δy should be greater than 0.5 mm.
pbj_sweg

## Homework Statement

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 forms 29.0 μm diameter drops of ink, charges them by spraying 800,000 electrons on the surface, and shoots them toward the page with a horizontal velocity of 16.0 m/s. Along the way, the drops pass through the long axis of two horizontal, parallel electrodes that are 6.0 mm long, 4.0 mm wide, and spaced 1.0 mm apart. The distance from the center of the electrodes to the paper is 2.80 cm. To form the letters, which have a maximum height of 6.0 mm, the drops need to be deflected up or down a maximum of 3.0 mm. Ink, which consists of dye particles suspended in alcohol, has a density of 800 kg/m3.

What electric field strength is needed between the electrodes to achieve this deflection?

## Homework Equations

$$y_f = y_i+v_{0y}t+\frac{1}{2}a_yt^2$$
$$\vec{F}_{net}=m\vec{a}$$
$$F_E=qE$$
$$d = v_0t$$
$$\rho = \frac{m}{V}$$
##\Delta{x}## = 6.0 mm, ##\Delta{y}## = 3.0 mm

## The Attempt at a Solution

(1) Found time that a drop of ink is inside the electrode by ##d=v_0t##.
##t = \frac{\Delta{x}}{v_0}## = 0.000375 sec

(2) Found the acceleration in the y-direction with ##y_f = y_i+v_{0y}t+\frac{1}{2}a_yt^2##
##a_y = \frac{2\Delta{y}}{t^2}## = 42666.7 m/s^2 (I was a bit hesitant to continue with this number as it seemed too large to be correct).

(3) Found charge on the drop of ink (e is the charge of an electron): ##800000\times{e} = 1.28176\times{10^{-13}}##

(4) ##\vec{F}_{net} = \vec{F}_E+\vec{F}_G = m\vec{a}##
As there is no force in the x direction, we can solve for forces in y-direction:
##F_y = F_E-F_G = ma_y##
##qE-mg = ma_y##
##E = \frac{ma_y+mg}{q}##
##E = 3.4\times10^6 \frac{N}{C}##

I'd appreciate it if someone could guide me to the proper solution as my answer is incorrect.

Sidenote - the mass of the drop was found with the following calculation:

Volume of drop = ## V = \frac{4}{3}\pi{r^3}## and radius = ##r = \frac{29}{2} \mu{m} = 1.45\times10^{-5} m##
##V = \frac{4}{3}\pi{r^3} = 1.277\times10^{-14}##
##m=\rho{V} = 800V = 1.0216\times10^{-11} kg##

Last edited:
Hi pbj_sweg,

I don't think that you should assume that the ink drop trajectory just grazes the edge of the plate. That is, the deflection of the ink does not have to be 0.5 mm (your Δy/2). Instead, I think you should make the Δy at the end of the plates an unknown.

Start by assuming that there is some electric field E and write an expression for the acceleration of the ink drop using it.

## What is electric field strength?

Electric field strength is a measure of the electric force per unit charge at a given point in space. It is represented by the symbol E and is measured in units of volts per meter (V/m).

## How does electric field strength affect an ink-jet printer?

In an ink-jet printer, electric field strength is used to control the movement of charged ink droplets as they are ejected from the print head onto the paper. By varying the electric field strength, the printer can precisely control the placement and size of the droplets, resulting in high-quality prints.

## What factors can affect the electric field strength in an ink-jet printer?

The electric field strength in an ink-jet printer can be affected by the voltage applied to the print head, the distance between the print head and the paper, and the properties of the ink being used (such as its conductivity and viscosity).

## How is electric field strength measured in an ink-jet printer?

Electric field strength in an ink-jet printer is typically measured using a voltmeter. The voltage is measured between the print head and the paper, and the distance between the two is also taken into account to calculate the electric field strength.

## Can high electric field strength cause any issues in an ink-jet printer?

Yes, high electric field strength can cause issues in an ink-jet printer. It can lead to excessive ink splattering, resulting in poor print quality and potential damage to the printer's components. It can also cause excessive heat buildup, which can affect the printer's performance and longevity.

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