Question

One of the greatest physics experiments in history measured the charge-to-mass ratio of an electron, \(q / m .\) If a uniform potential difference is created between two plates, atomized particles - each with an integral amount of charge-can be suspended in space. The assumption is that the particles of unknown mass, \(M,\) contain a net number, \(n\), of electrons of mass \(m\) and charge \(q .\) For a plate separation of \(d,\) what is the potential difference necessary to suspend a particle of mass \(M\) containing \(n\) net electrons? What is the acceleration of the particle if the voltage is cut in half? What is the acceleration of the particle if the voltage is doubled?

Short answer

Answer: The potential difference necessary to suspend the particle is given by \(V = \frac{M * g * d}{q * n}\). The accelerations of the particle when the potential difference is halved and doubled are given by \(a_1 = \frac{F_g - F_{e1}}{M}\) and \(a_2 = \frac{F_{e2} - F_g}{M}\), respectively.

Step-by-step solution

Calculate the gravitational force on the particle

The gravitational force acting on the particle is given by: \(F_g = M * g\) where \(M\) is the mass of the particle, \(g\) is the acceleration due to gravity (approximately 9.81 m/s²).

Calculate the electric force on the particle

The electric force acting on the particle is given by: \(F_e = q * n * E\) where \(q\) is the charge of an electron, \(n\) is the net number of electrons, \(E\) is the electric field. We know that electric field, \(E\), is given by \(V / d\) where \(V\) is the potential difference and \(d\) is the distance between the plates. Hence, \(F_e = q * n * (V / d)\).

Balance the gravitational force with the electric force

In order for the particle to remain suspended in the electric field, the gravitational force must be equal to the electric force: \(F_g = F_e\) Using the expressions for gravitational and electric force from steps 1 and 2: \(M * g = q * n * (V / d)\) Now, we can solve for the potential difference, \(V\): \(V = \frac{M * g * d}{q * n}\)

Calculate acceleration if voltage is halved

When the potential difference is cut in half, the electric force will also be cut in half. We will denote the revised force as \(F_{e1}\). \(F_{e1} = \frac{1}{2} * F_e = \frac{1}{2} * q * n * (V / d)\) The net force acting on the particle will be the difference between gravitational and electric forces: \(F_{net} = F_g - F_{e1}\) The acceleration of the particle when voltage is halved will be: \(a_1 = \frac{F_{net}}{M} = \frac{F_g - F_{e1}}{M}\)

Calculate acceleration if voltage is doubled

When the potential difference is doubled, the electric force will also be doubled. We will denote the revised force as \(F_{e2}\). \(F_{e2} = 2 * F_e = 2 * q * n * (V / d)\) In this case, the net force acting on the particle will be the difference between electric and gravitational forces: \(F_{net} = F_{e2} - F_g\) The acceleration of the particle when voltage is doubled will be: \(a_2 = \frac{F_{net}}{M} = \frac{F_{e2} - F_g}{M}\)

Key concepts

Gravitational Force

Gravitational force is a fundamental interaction that every mass exerts on every other mass, drawing them together. It's described by Isaac Newton's universal law of gravitation, which states that any two objects pull on each other with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them.

In the context of the charge-to-mass ratio of an electron, gravitational force becomes significant when we consider the weight of a particle containing electrons. The force of gravity on the particle, symbolically represented as \(F_g = M * g\), where \(M\) is the mass of the particle and \(g\) is the acceleration due to gravity (approximately 9.81 m/s² on Earth's surface), is what opposes the electric force in our scenario to suspend the particle between two plates.

Electric Force

The electric force is part of the electromagnetic force, one of the four fundamental forces of nature, and arises from the interaction between charged particles. Coulomb's Law quantifies this force, stating it is directly proportional to the product of the charges and inversely proportional to the square of the distance separating them.

When analyzing the charge-to-mass ratio of an electron, the electric force comes into play as it is used to counteract the gravitational force on charged particles. In the exercise, the electric force is computed using \(F_e = q * n * E\), where \(E\) represents the electric field strength. This force is the mechanism that can potentially suspend a charged particle between two plates by balancing out the gravitational pull.

Potential Difference

Potential difference, often referred to as voltage, is a measure of the work done to move a charge from one point to another in an electric field. It's the driving force that causes electric charges to move in a conductor and is quantified in volts (V).

Within our exercise on the charge-to-mass ratio of an electron, the potential difference between two plates creates an electric field that exerts a force on electrons. By finding the necessary potential difference that equates the electric force with gravitational force, you can determine the voltage needed to suspend a particle in space. This potential difference, solved as \(V = \frac{M * g * d}{q * n}\), can be adjusted to manipulate the electric force for various experimental outcomes.

Electric Field

An electric field is a region in space where an electric charge experiences a force. The strength of an electric field is defined as the force experienced per unit charge and is expressed in volts per meter (V/m). The direction of the field corresponds with the direction a positive charge would move within it.

In the problem under discussion, the electric field \(E\) is created by the potential difference \(V\) across the plates, separated by a distance \(d\), such that \(E = \frac{V}{d}\). It is this field that exerts the electric force on the particles, enabling either their suspension in the field or dictating their subsequent acceleration when the voltage is altered.