One type of ink-jet printer, celled an electrostatic ink-jet...

One type of ink-jet printer, celled 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 berizonully across the page. The ink jet forms 30 - $\mu \mathrm{m}$-diameter drope of ink, charges them by spraying 800,000 electrons on the surface, and shoots them toward the page at a speed of $20 \mathrm{~m} / \mathrm{s}$. Along the way. the drops pass through two parallel electrodes that are $6.0 \mathrm{~mm}$ long, $4.0 \mathrm{~mm}$ wide, and spaced $1.0 \mathrm{~mm}$ apart. The distance from the center of the plates to the paper is $2.0 \mathrm{~cm}$. To form the letters, which have a maximum height of $6.0 \mathrm{~mm}$, the drops aeed to be deflected up or down a maximum of $3.0 \mathrm{~mm}$. Ink, which consists of dye particles suspended in alcohol, has a density of $800 \mathrm{~kg} / \mathrm{m}^3$. a. Estimate the maximum electric field strength needed in the space between the electrodes. b. What amount of charge is needed on each electrode to produce this electric field?

One type of ink-jet printer, celled 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 berizonully across the page. The ink jet forms 30 - $\mu \mathrm{m}$-diameter drope of ink, charges them by spraying 800,000 electrons on the surface, and shoots them toward the page at a speed of $20 \mathrm{~m} / \mathrm{s}$. Along the way. the drops pass through two parallel electrodes that are $6.0 \mathrm{~mm}$ long, $4.0 \mathrm{~mm}$ wide, and spaced $1.0 \mathrm{~mm}$ apart. The distance from the center of the plates to the paper is $2.0 \mathrm{~cm}$. To form the letters, which have a maximum height of $6.0 \mathrm{~mm}$, the drops aeed to be deflected up or down a maximum of $3.0 \mathrm{~mm}$. Ink, which consists of dye particles suspended in alcohol, has a density of $800 \mathrm{~kg} / \mathrm{m}^3$.
a. Estimate the maximum electric field strength needed in the space between the electrodes.
b. What amount of charge is needed on each electrode to produce this electric field?

Key Concepts

Charge Quantization

Charge quantization refers to the fact that charge exists in discrete amounts, typically as multiples of the elementary charge. In problems involving charged droplets, knowing the number of electrons transferred allows one to calculate the net charge, which is critical in determining the magnitude of the electric force acting on the droplet.

Uniform Electric Field and Parallel Plate Geometry

In many practical applications, such as in electrostatic deflectors, a uniform electric field is generated between two parallel electrodes. The field in this configuration depends on the potential difference and the separation distance between the plates. Understanding the relationship between the field, the charge density on the electrodes, and the resulting force on a charged particle is crucial in designing systems that precisely control particle trajectories.

Kinematics of Uniform Acceleration

This concept involves studying the motion of objects under constant acceleration. It is particularly relevant when analyzing how charged particles, accelerated by a steady electric force, move over a given distance. By applying kinematic equations, one can relate the acceleration, time, and displacement of a particle, which is essential for predicting the deflection needed in applications like ink-jet printing.

Newton's Second Law

Newton’s second law (F = ma) is fundamental in relating the force acting on an object to its mass and the acceleration produced. In the context of charged droplets in an electric field, it is used to determine the acceleration due to the electric force, which in turn helps predict the droplet’s deflection over a known distance and time.

Electrostatic Force

This concept involves the force that a charged particle experiences when placed in an electric field. The force is given by F = qE, where q is the charge of the particle and E is the electric field strength. It forms the basis of understanding how charged droplets can be deflected by applying an electric field in devices such as electrostatic ink-jet printers.

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