An infinite line of charge has a linear charge density of 10-7 c/m. What will be the magnitude of th

An infinite line of charge has a linear charge density of 10-7 c/m. What will be the magnitude of the force acting on an alpha particle placed at a distance of 4 cm from the line of charge?

The magnitude of the force acting on the alpha particle placed 4 cm away from the line of charge is approximately 7.20 x 10^-12 N.

Explanation

Gather the information:

  • Linear charge density of the line (λ) = 10^-7 C/m
  • Distance of the alpha particle from the line (r) = 4 cm (convert to meters: r = 0.04 m)
  • Charge of an alpha particle (q_alpha) = 2 * elementary charge (e) ≈ 3.2 x 10^-19 C (since an alpha particle has two protons)

Recognize the limitations of using a point charge approximation:

While an infinite line of charge can be thought of as a collection of infinitely many point charges, directly summing their individual contributions using Coulomb's law becomes cumbersome and impractical.

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Employ integration to account for the distributed charge:

Instead, we can leverage the concept of linear charge density (λ), which represents the charge per unit length along the line. This allows us to treat the line charge as a continuous distribution and use integral calculus to find the net force.

Set up the integral:

Imagine dividing the infinite line charge into infinitesimal segments of length dl. Each segment carries a charge dq given by:

dq = λ * dl 

The electric field dE produced by this segment at the alpha particle's position can be calculated using Coulomb's law:

dE = k * dq / r^2 

where k is Coulomb's constant (k ≈ 9 x 10^9 Nm^2/C^2).

To find the total electric field (E) at the alpha particle's position, we need to integrate dE over the entire length of the line charge (from negative infinity to positive infinity):

E = ∫ (k * λ * dl) / r^2 

Solve the integral:

This integral has a known solution, resulting in:

E = λ / (2πε₀ * r) 

where ε₀ is the permittivity of free space (ε₀ ≈ 8.85 x 10^-12 F/m).

Calculate the electric field:

Substitute the given values:

E ≈ (10^-7 C/m) / (2π * 8.85 x 10^-12 F/m * 0.04 m) ≈ 4.50 x 10^4 N/C 

Calculate the force on the alpha particle:

Finally, use the electric field and the alpha particle's charge to find the force:

F = q_alpha * E ≈ (3.2 x 10^-19 C) * (4.50 x 10^4 N/C) ≈ 7.20 x 10^-12 N 

Therefore, the magnitude of the force acting on the alpha particle placed 4 cm away from the line of charge is approximately 7.20 x 10^-12 N.

What is Electrostatics?

Electrostatics is a branch of physics that deals with the study of stationary electric charges and the forces and fields associated with them. It primarily focuses on the behavior of electric charges at rest, without the involvement of any current flow. The fundamental concept in electrostatics is the electric charge, which can be positive or negative.

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Key principles and phenomena in electrostatics include:

  • Electric Charge: Electric charge is a fundamental property of matter, and it comes in two types: positive and negative. Like charges repel each other, while opposite charges attract.

  • Coulomb's Law: Named after Charles-Augustin de Coulomb, this law describes the force between two point charges. The force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.

  • Electric Field: An electric field is a region around a charged object where a force would be exerted on other charged objects. The electric field is a vector quantity, having both magnitude and direction.

  • Electric Potential and Voltage: Electric potential is a measure of the electric potential energy per unit charge at a point in space. Voltage, often used interchangeably with electric potential, is the electric potential difference between two points.

  • Gauss's Law: Formulated by Carl Friedrich Gauss, this law relates the electric flux through a closed surface to the charge enclosed by that surface. It provides a convenient way to calculate electric fields for symmetric charge distributions.

  • Electric Potential Energy: The electric potential energy of a system of charges is the work done in assembling the system. The potential energy is associated with the configuration of the charges.

  • Conductors and Insulators: Materials can be classified as conductors (allow the easy flow of electric charge) or insulators (restrict the flow of electric charge).

  • Electrostatics plays a crucial role in various practical applications, such as the operation of electronic devices, the behavior of materials in the presence of static charges, and the design of components in electrical systems. It forms the foundation for understanding more complex electrical phenomena and is a key aspect of classical electromagnetism.

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