Question

A total of \(3.05 \cdot 10^{6}\) electrons are placed on an initially uncharged wire of length \(1.33 \mathrm{~m}\). a) What is the magnitude of the electric field a perpendicular distance of \(0.401 \mathrm{~m}\) away from the midpoint of the wire? b) What is the magnitude of the acceleration of a proton placed at that point in space? c) In which direction does the electric field force point in this case?

Short answer

Answer: To find the acceleration of the proton, follow these steps: 1. Calculate the total charge on the wire. 2. Use Coulomb's Law to derive an expression for the electric field produced by the wire at a certain point. 3. Calculate the electric field at a distance of 0.401 m away from the midpoint of the wire. 4. Calculate the force on the proton based on its charge and the electric field. 5. Calculate the acceleration of the proton using Newton's second law. The direction of the electric field force and the force on the proton will point towards the wire, as the wire has a negative charge (electrons), and the proton has a positive charge.

Step-by-step solution

Calculate the total charge on the wire.

The total number of electrons on the wire is \(3.05 \cdot 10^{6}\), and the charge of each electron is \(1.6 \cdot 10^{-19} \mathrm{C}\). Therefore, the total charge on the wire can be calculated as follows: $$ Q = n \cdot e $$ Where \(Q\) is the total charge, \(n\) is the number of electrons, and \(e\) is the elementary charge. In this case: $$ Q = (3.05 \cdot 10^{6})(1.6 \cdot 10^{-19} \mathrm{C}) $$

Expression for the electric field from the wire at a certain point.

The electric field produced by a charged wire can be calculated using Coulomb's Law, which states that the electric field, \(E\), due to a charge \(q\), at a distance \(r\) is given by: $$ E = \frac{kQ}{r^{2}} $$ Where \(k\) is the electrostatic constant, \(k=8.99 \cdot 10^{9} \frac{\mathrm{N\cdot m^2}}{\mathrm{coul^2}}\).

Calculate the electric field at a 0.401 m distance.

To find the electric field at a distance of 0.401 m away from the midpoint of the wire, we need to find the electric field produced by each small segment of the wire and then integrate over the straight line charge. However, because of the symmetry in this particular case, we only need to multiply the electric field calculated using Coulomb's Law at 0.401 m from the midpoint of the wire by 2. First, let's calculate the charge at the midpoint of the wire by dividing the total charge by the length of the wire: $$ \lambda = \frac{Q}{L} $$ Where \(\lambda\) is the linear charge density and \(L\) is the length of the wire. In this case: $$ \lambda = \frac{Q}{1.33 \mathrm{~m}} $$ Now let's calculate the electric field at a distance of 0.401 m away from the midpoint of the wire: $$ E = \frac{2k\lambda}{r^{2}} $$ Where \(E\) is the electric field, \(k\) is the electrostatic constant, \(\lambda\) is the linear charge density, and \(r\) is the distance from the wire.

Calculate the acceleration of a proton placed at that point in space.

The force on a charged particle in an electric field is given by: $$ F = qE $$ Where \(F\) is the force, \(q\) is the charge of the particle, and \(E\) is the electric field. In this case, the particle is a proton with a charge of \(q = +1.6 \cdot 10^{-19} \mathrm{C}\). The force on the proton is: $$ F = (1.6 \cdot 10^{-19} \mathrm{C})(E) $$ The acceleration of a particle is given by Newton's second law: $$ a = \frac{F}{m} $$ Where \(a\) is the acceleration, \(F\) is the force, and \(m\) is the mass of the particle. In this case, the mass of a proton is \(m = 1.67 \cdot 10^{-27} \mathrm{kg}\). The acceleration of the proton is: $$ a = \frac{F}{1.67 \cdot 10^{-27} \mathrm{kg}} $$

Direction of the electric field force.

Since the charges on the wire are negative (electrons), the electric field will point toward the wire. Additionally, because the proton has a positive charge, the force on the proton will be in the direction of the electric field. Therefore, the force on the proton and the direction of the electric field will point toward the wire.

Key concepts

Coulomb's Law

Understanding Coulomb's Law is essential to predicting the behavior of charged particles. It describes the force between two point charges, which is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. Mathematically, Coulomb's Law is expressed as \(F = k\frac{|q_1\cdot q_2|}{r^2}\), where \(F\) is the force between the charges, \(q_1\) and \(q_2\) are the amounts of the charges, \(r\) is the distance between the centers of the two charges, and \(k\) is the Coulomb’s constant, approximately equal to \(8.99 \cdot 10^9 \frac{N\cdot m^2}{coul^2}\).
This fundamental law not only helps calculate forces but is also used indirectly in calculating electric fields, as shown in the textbook problem where we use it to determine the electric field at a point away from a charged wire.

Electric Charge

The concept of Electric Charge is fundamental to the study of electromagnetism. Charges come in two types — positive and negative. Like charges repel each other whereas opposite charges attract. The unit of electric charge is the Coulomb, represented by \(C\). Electrons carry a negative charge, each with a charge of approximately \( -1.6 \cdot 10^{-19} C\), while protons carry the equivalent positive charge. The total charge \(Q\) on an object can be determined by counting the number of excess or deficit electrons and multiplying by the elementary charge \(e\).
In our example, the wire becomes charged by the addition of electrons and therefore carries a net negative electric charge.

Linear Charge Density

Linear Charge Density, given the symbol \(\lambda\), is a measure of charge per unit length along a line. It is used when dealing with extended objects like wires, where charge is distributed along a line rather than at a point. The linear charge density is calculated by dividing the total charge \(Q\) of the wire by its length \(L\), expressed as \(\lambda = \frac{Q}{L}\).
This calculation allows us to model the behavior of the electric field around the wire, which is useful for understanding the electric field's influence at specific points in space, such as calculating the electric field a specific distance away from the wire, as required in the exercise.

Electric Force

The Electric Force experienced by a charge in an electric field is a vector quantity that can be calculated using the expression \(F = qE\), where \(F\) is the electric force, \(q\) is the charge of the particle, and \(E\) is the electric field. This force can be attractive or repulsive depending on the sign of the charge relative to the source of the electric field.
For example, in the given solution, we calculate the force on a proton, which is positive, positioned in the electric field created by the negatively charged wire. This force is what would cause the proton to accelerate towards the wire.

Proton Acceleration

The principle of Proton Acceleration pertains to the change in velocity that a proton experiences when subjected to an electric force. The acceleration of the proton \(a\) is derived from Newton's second law of motion, which states that \(a = \frac{F}{m}\), where \(F\) is the force applied to the proton, and \(m\) is the mass of the proton. To find the acceleration, we apply the electric force that we've calculated using the electric field and proton's charge, and divide it by the mass of the proton, which is \(1.67 \cdot 10^{-27} kg\).
In our exercise, we use this formula to calculate how rapidly a proton would speed up when placed a certain distance from a charged wire – a demonstration of electric forces at work on fundamental particles.