Question

A line charge of uniform charge density \(\rho_{0} \mathrm{C} / \mathrm{m}\) and of length \(\ell\) is oriented along the \(z\) axis at \(-\ell / 2

Short answer

Question: Given a line charge of uniform charge density ρ_0 C/m and length ℓ oriented along the z-axis from -ℓ/2 to ℓ/2, find the electric field strength at any position along the x-axis. In part (b), determine the force acting on an identical line charge oriented along the x-axis at ℓ/2 < x < 3ℓ/2. Answer: To find the electric field strength at any position along the x-axis, we need to integrate the electric field contribution from each small charge segment within the line charge. This is done by finding the x-component of the electric field and integrating it over the full length of the line charge from -ℓ/2 to ℓ/2. After performing the integration, we will find the electric field strength at any position along the x-axis, E_x. For part (b), to find the force acting on an identical line charge oriented along the x-axis between ℓ/2 and 3ℓ/2, we need to multiply the electric field found in part (a) with the charge density ρ_0 and integrate it over the given length. After performing the integration, we will find the force acting on the line charge.

Step-by-step solution

Define the line charge and its position along the z-axis

We're given a line charge of uniform charge density ρ_0 C/m and of length ℓ. The line charge is situated along the z-axis from -ℓ/2 to ℓ/2.

Determine the electric field contribution from a small charge segment dQ

To find the electric field contribution from a small charge segment, dQ, we need to apply Coulomb's law. The law states that the electric field E due to a charge Q is given by E = kQ/r², where k is the electrostatic constant and r is the distance between the charge and the point we're measuring the electric field. For a small charge element dQ, located at z, the electric field will be given by dE = k*dQ/r². In this case, r is the distance from the dQ to the point on the x-axis.

Compute the charge element dQ

We know that the charge density ρ_0 = dQ/dℓ, so we can write dQ = ρ_0*dℓ. Here, dℓ is the infinitesimal length of the line charge along the z-axis.

Relate the charge element, dQ, to the electric field, dE

Substituting the expression for dQ into our formula for dE, we obtain the following relation: dE = k*ρ_0*dℓ/r².

Determine the angle between the electric field contribution and the x-axis

We need to determine the angle, θ, between the electric field contribution, dE, and the x-axis, so that we can integrate it along the x-axis. By using the right triangle formed by r, the position along the x-axis, and the position along the z-axis, we can find θ. We have tanθ = z/x, so θ = arctan(z/x).

Calculate the x-component of the electric field contribution

To find the x-component of the electric field contribution, dE_x, we multiply dE by cosθ. Using the angle θ found in step 5, we have dE_x = dE*cos(arctan(z/x)). Simplifying and substituting the expression for dE, we get: dE_x = k*ρ_0*dℓ*cos(arctan(z/x))/r².

Integrate the x-component of the electric field over the length of the line charge

To find the total electric field along the x-axis, we must integrate the x-component of the electric field, dE_x, over the length of the line charge. The limits of integration are from -ℓ/2 to ℓ/2. The resulting integral is E_x = ∫(k * ρ_0 * cos(arctan(z/x)) * dℓ)/(r^2) as z goes from -ℓ/2 to ℓ/2.

Solve the integral

Solving this integral can be a bit tricky and may require some specialized techniques or software. After performing the integration, we'll find the electric field strength at any position along the x-axis, E_x. For part (b):

Determine the force acting on an identical line charge along the x-axis

We're given an identical line charge oriented along the x-axis between ℓ/2 and 3ℓ/2. The force acting on this line charge will be given by the product of the electric field and the charge density, ρ_0, multiplied by its infinitesimal length dℓ_x.

Integrate the force over the length of the line charge along the x-axis

To find the total force acting on the line charge, integrate the force found in step 9 over the length of the line charge between ℓ/2 and 3ℓ/2: F_x = ∫(ρ_0 * E_x * dℓ_x) as x goes from ℓ/2 to 3ℓ/2.

Solve the integral

Just like in step 8, solving this integral can be a bit tricky and may require some specialized techniques or software. After performing the integration, we will find the force acting on an identical line charge that is oriented along the x-axis between ℓ/2 and 3ℓ/2.

Key concepts

Line Charge

In the realm of electromagnetism, a line charge refers to a configuration where electric charge is distributed uniformly along a line, rather than concentrated at a single point or spread over a surface. This line of charge can extend infinitely or be finite, depending on the specific problem. Line charges have a uniform charge density, meaning the amount of charge per unit length remains constant over the entire length of the line.
A line charge is quite different from point charges or surface charges as it requires integration along the length to find resultant electric fields or forces. In our particular exercise, the line charge is centered along the z-axis, extending from \(-\ell/2\) to \(\ell/2\). By understanding line charges, we can delve deeper into calculations involving electric fields produced by these distributed charges.

Coulomb's Law

Coulomb's law is a fundamental principle for understanding electric forces and fields. It dictates that the electric force \(F\) between two point charges is directly proportional to the product of the magnitudes of charges, and inversely proportional to the square of the distance \(r\) separating them. Mathematically, it is expressed as \[ F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \] where \(k\) is Coulomb’s constant, approximately \(8.99 \times 10^9 \, \text{Nm}^2\text{/C}^2\).
When it comes to line charges, we consider small charge elements \(dQ\) and apply Coulomb’s law to each of these elements. We then integrate these infinitesimal elements to calculate the entire electric field or force produced by the charge distribution. This iterative process requires understanding of calculus as well as a mastery of the concepts underlying Coulomb's law.

Charge Density

Charge density is a measure of how much electric charge exists per unit length, area, or volume in a particular region. In the context of line charges, we focus on linear charge density, often denoted by \( \rho \), with units typically in \(\text{C/m}\). For a line charge along a specific axis, such as the z-axis, the charge density is the charge per unit length along this line.
The formula used to describe charge density in line charges is \(\rho = \frac{Q}{L}\), where \(Q\) is the total charge distributed along the length \(L\). By knowing the charge density, we can find the amount of charge \(dQ\) in any small segment \(d\ell\) of the line, expressed as \(dQ = \rho \cdot d\ell\). This is crucial for solving problems related to electric fields as it allows us to apply integration to derive the net effects due to distributed charges.

Integration in Electromagnetics

Integration is a powerful mathematical tool employed in electromagnetics to handle cases involving distributed charges, be they line, surface, or volume charges. In our exercise, we employ integration techniques to sum up contributions from different charge segments to find the total electric field produced by a line charge.
When determining the electric field resulting from a line charge, we integrate across the entire length of the line. For example, the strength of the electric field along the x-axis due to a line charge along the z-axis involves integrating the field contribution of each small charge segment \(d\ell\). The integral is typically expressed as: \[ E_x = \int \frac{k \cdot \rho_0 \cdot \cos(\text{arctan}(z/x)) \cdot d\ell}{r^2} \] from \(-\ell/2\) to \(\ell/2\).
This technique allows for precise calculation of effects due to charge distributions, providing essential insights into the behavior of electric fields and forces in various configurations.