Question

There is a uniform charge distribution of $\lambda =$ $8.00\cdot {10}^{-8}\mathrm{C}/\mathrm{m}$ along a thin wire of length $L=6.00\mathrm{cm}$ The wire is then curved into a semicircle that is centered about the origin, so the radius of the semicircle is $R=L/\pi .$ Find the magnitude of the electric field at the center of the semicircle.

Short answer

Answer: The magnitude of the electric field at the center of the semicircle is $0\text{N/C}$.

Step-by-step solution

Define the variables and constants

Let's list the given values: - Linear charge density: $\lambda =8.00\cdot {10}^{-8}\text{C/m}$ - Length of the wire: $L=6.00\text{cm}=0.060\text{m}$ - Radius of the semicircle $R=\frac{L}{\pi }$ We also have the constant: - Coulomb's constant: $k=8.988\cdot {10}^9\text{N}\cdot {\text{m}}^2/{\text{C}}^2$

Set up the integral for the electric field

We will divide the semicircular wire into small segments of length $\mathrm{\Delta }s$ and charge $dQ=\lambda \mathrm{\Delta }s$. We will find the electric field $d\overrightarrow{E}$ at the center of the semicircle due to each segment, and then integrate to find the total electric field. As all segments have the same distance $R$ to the center, their electric fields will differ only in their angles. Let's consider a small segment at an angle $\theta$ with respect to the $x$-axis. The electric field due to this segment can be decomposed into $x$ and $y$ components: $d\overrightarrow{E}=d\overrightarrow{E_x}+d\overrightarrow{E_y}$ Since the segments are symmetric about the $x$-axis, the $y$-component of the electric field will cancel out. Therefore, we only need to consider the $x$-component of the electric field: $d{E}_x=\frac{kdQ}{R^2}\cos \theta$ Now, we can replace $dQ$ with $\lambda \mathrm{\Delta }s=\lambda Rd\theta$: $d{E}_x=\frac{k\lambda R}{R^2}\cos \theta d\theta$ We will integrate this expression over the semicircle ($\theta$ ranges from $0$ to $\pi$): $E_x={\int }_0^{\pi }\frac{k\lambda R}{R^2}\cos \theta d\theta$

Calculate the electric field

Now, we can evaluate the integral: $E_x=\frac{k\lambda }{R}{\int }_0^{\pi }\cos \theta d\theta$ $E_x=\frac{k\lambda }{R}[\sin \theta {]}_0^{\pi }$ $E_x=\frac{k\lambda }{R}(\sin \pi -\sin 0)$ $E_x=0$ Since the $x$-component of the electric field is zero, the total electric field at the center of the semicircle is effectively zero. In conclusion, the magnitude of the electric field at the center of the semicircle is $0\text{N/C}$.

Key concepts

Electric Field

An electric field is a fundamental concept in electromagnetism, representing the invisible force field around electric charges. It describes the forces exerted on positive test charges placed in the vicinity of other charges. In the case of a static charge distribution such as the semicircle charge distribution in our exercise, each tiny segment of charge contributes to the overall electric field at a point.

To visualize an electric field, imagine it as arrows pointing away from positive charges and towards negative ones. The strength and direction of the electric field at any given point are determined by the sources of the electric field (charges) and the relative position of the point. In the uniform charge distribution on a semicircle, all the small charged segments produce electric fields which, due to the symmetry, only have an uncancelled component along the axis of symmetry.

Linear Charge Density

Linear charge density is a measure used to describe how much electric charge is distributed along a line. In our problem, the wire is uniformly charged, meaning its charge is evenly distributed along its length. The linear charge density, denoted by $\lambda$, can be thought of as the amount of charge per unit length.

For a continuous charge distribution, the charge $dQ$ for a tiny segment of length $\mathrm{\Delta }s$ can be found by multiplying the linear charge density by this small length. What's significant about linear charge density is that it allows us to break down complex charge distributions into small, manageable pieces to calculate the resultant electric field using integration.

Coulomb's Law

Coulomb's Law is a cornerstone of electrostatics, formulated by Charles-Augustin de Coulomb in the 18th century. It describes how the force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance separating them. Expressed mathematically, $F=k\frac{|q_1{q}_2|}{r^2}$, with $F$ being the force, $q_1$ and $q_2$ the charges, $r$ the distance between them, and $k$ is Coulomb's constant.

In the context of the semicircular charge distribution, Coulomb's law helps us determine the electric field contribution from each infinitesimal charge segment. By integrating these contributions, which vary depending on their position around the semicircle, we can attempt to calculate the total electric field at the center. Although Coulomb's law often applies to point charges, its principles also apply to continuous charge distributions like our semicircle when integrated over the charge distribution.

Integration in Physics

Integration is a mathematical tool that is widely used in physics to combine the effects of varying quantities over a certain range. It allows us to calculate quantities like the electric field of a continuous charge distribution by summing up infinitely small contributions from each part of the distribution.

In the semicircle charge distribution problem, we set up an integral to find the total electric field at the center of the semicircle. Each segment produces an infinitesimally small field $d{E}_x$, and integration sums these to calculate the entire field. A key aspect of using integration in such problems is understanding the symmetry of the situation; for example, components of the electric field perpendicular to the axis of symmetry will cancel out due to the semicircular distribution. As a result, while the individual charged segments exert forces in various directions, their cumulative effect in a symmetrical arrangement can be efficiently computed through integration.