Question

A negatively charged droplet has a mass of \(5 \times 10^{-16} \mathrm{~kg}\) and carries a charge of \(8 \times 10^{-18} \mathrm{C}\). The droplet falls through the hole in the upper plate when the electric field is \(0 \mathrm{~V} / \mathrm{m}\). How does the drop move within the plates as the electric field is increased slowly from \(0 \mathrm{~V} / \mathrm{m}\) to \(800 \mathrm{~V} / \mathrm{m}\) ? (Note: Assume that the drop remains between the plates at all times.) 1\. It moves downwards, stops, then moves upwards. 2\. It moves downwards, accelerating all the time. 3\. It moves downwards, stops, then moves downwards again. 4\. It moves downwards, stops, and remains stationary.

Short answer

1. It moves downwards, stops, and then moves upwards.

Step-by-step solution

Identify the Forces Acting on the Droplet

There are two main forces acting on the droplet: 1. Gravitational force (downward direction): F_gravity = m * g. Where m is the mass of the droplet and g is the acceleration due to gravity (approximately 9.81 m/s²). 2. Electric force (upward direction, since the droplet is negatively charged): F_electric = q * E. Where q is the charge of the droplet and E is the electric field strength.

Determine the Droplet's Motion at Zero Electric Field

When the electric field is zero, there is no electric force acting on the droplet. Therefore, the droplet will only experience a downward gravitational force, which will cause it to fall through the hole in the upper plate.

Calculate the Gravitational Force Acting on the Droplet

To calculate the gravitational force, we use the formula: F_gravity = m * g Given, m = \(5 \times 10^{-16} kg\) g = \(9.81 m/s^2\) F_gravity = (5 x \(10^{-16} kg\)) * (9.81 \(m/s^2\)) F_gravity = \(4.905 \times 10^{-15} N\)

Compute the Maximum Electric Force

To compute the maximum electric force when the electric field is 800 V/m, we use the formula: F_electric = q * E Given, q = \(8 \times 10^{-18} C\) E = \(800 V/m\) F_electric = (8 x \(10^{-18} C\)) * (800 \(V/m\)) F_electric = \(6.4 \times 10^{-15} N\)

Compare Gravitational and Electric Forces and Determine the Droplet's Motion

Now that we have the gravitational force (F_gravity = \(4.905 \times 10^{-15} N\)) and the maximum electric force (F_electric = \(6.4 \times 10^{-15} N\)), we can determine the droplet's motion. Initially, the droplet will be influenced by gravity and move downwards. As the electric field increases, the upward electric force becomes stronger. When Electric force is equal to gravitational force (F_electric = F_gravity), the droplet will stop moving (no net force on it). Finally, as the electric force keeps increasing and exceeds the gravitational force (F_electric > F_gravity), the droplet will experience a net upward force and move upwards.

Select the Correct Answer

From the analysis above, we can conclude that the droplet moves downwards, stops, and then moves upwards, which corresponds to option 1: 1. It moves downwards, stops, and then moves upwards.

Key concepts

Electromagnetism

Electromagnetism is a fundamental branch of physics that deals with the interaction of electric charges and magnetic fields. It plays a crucial role in understanding the behavior of charged particles like our negatively charged droplet in the exercise. Electromagnetism can describe how electric fields affect the motion of charges. In a uniform electric field, as in our exercise where the field strength increases from 0 to 800 V/m, charged particles experience a force that impacts their motion. This force, known as the electric force, is pivotal in determining the droplet's path. Its direction is influenced by the charge nature – an important clue for solving the problem at hand.

Gravitational Force

Gravitational force is the attractive force experienced by objects with mass. According to Newton's law of universal gravitation, any two masses exert a force of attraction on each other, proportional to the product of their masses and inversely proportional to the square of the distance between their centers. In physics problems like our droplet's motion, the gravitational force is given by the formula: \( F_{\text{gravity}} = m \cdot g \) - Here, \( m \) is the mass of the object (5 x 10^{-16} kg), and \( g \) is the acceleration due to gravity (approx. 9.81 m/s²). For our droplet, this constant downward force needs to be overcome by any upward electric force resulting from the electric field to change its direction or halt its fall. Understanding this concept clarifies why gravity initially dictates the droplet's motion downward.

Electric Force

The electric force in electromagnetism acts on charged particles within an electric field. The force on a particle is quantified by the equation: \( F_{\text{electric}} = q \cdot E \) - Here, \( q \) represents the charge (8 x 10^{-18} C), and \( E \) is the electric field strength (which varies in this exercise). For a negatively charged particle, like our droplet, the electric force's direction is opposite the field direction. As the field's strength increases, the upward electric force increases too, eventually balancing out the gravitational force causing the droplet to stop. When the electric force surpasses the gravitational pull, the droplet experiences a net upward force, causing it to rise. This interplay between forces is vital in understanding the motion analysis of the droplet.

Motion Analysis

Motion analysis involves studying the forces acting on an object and predicting its movement. In this exercise, by analyzing the gravitational and electric forces on the droplet, we can make informed predictions about its trajectory:

• Initially, the only force acting is gravity, making the droplet fall downward.
• As the electric field strength increases, so does the electric force upwards.
• Motion halts when the electric force equals the gravitational force, as the net force is zero.
• Further increase in the electric field leads to an upward net force, reversing the droplet's motion direction.

This step-by-step understanding exemplifies how physics enables the prediction of motion based on forces. It illustrates Newton’s first and second laws, describing objects in motion and explaining how force imbalances cause motion changes.