Problem 59
Question
In one experiment the electric field is measured for points at distances \(r\) from a uniform line of charge that has charge per unit length \(\lambda\) and length \(l\), where \(l \gg r\). In a second experiment the electric field is measured for points at distances \(r\) from the center of a uniformly charged insulating sphere that has volume charge density \(\rho\) and radius \(R =\) 8.00 mm, where \(r > R\). The results of the two measurements are listed in the table, but you aren't told which set of data applies to which experiment: For each set of data, draw two graphs: one for \(Er^2\) versus r and one for \(Er\) versus \(r\). (a) Use these graphs to determine which data set, A or B, is for the uniform line of charge and which set is for the uniformly charged sphere. Explain your reasoning. (b) Use the graphs in part (a) to calculate \(\lambda\) for the uniform line of charge and \(\rho\) for the uniformly charged sphere.
Step-by-Step Solution
Verified Answer
Data set with constant \( Er^2 \) is the sphere, and the one with constant \( Er \) is the line of charge. Calculate \( \lambda \) and \( \rho \) using their respective constant values.
1Step 1: Understand the Relationship between E, r, and the Charge Configuration
For a line of charge, the electric field E at a distance r is given by \( E = \frac{\lambda}{2\pi\varepsilon_0 r} \), suggesting that \( E \times r \) is constant. For a uniformly charged sphere, at a distance \( r \, (r > R) \), \( E \) behaves like a point charge, \( E = \frac{\rho V}{4\pi\varepsilon_0 r^2} \), implying that \( E \times r^2 \) is constant.
2Step 2: Plot Er^2 and Er against r for Each Data Set
Graphically plot \( E \times r^2 \) and \( E \times r \) for both data sets. If the plot of \( E \times r^2 \) against \( r \) is constant, it corresponds to the uniformly charged sphere. Conversely, if \( E \times r \) against \( r \) is constant, it belongs to the line of charge.
3Step 3: Identify which Data Set is which Experiment
Based on the plots, identify: Data set with constant \( E \times r \) indicates the uniform line of charge, and the data set with constant \( E \times r^2 \) corresponds to the sphere.
4Step 4: Calculate the Charge Density and Charge per Unit Length
For the line of charge, solve \( \lambda \) using the constant value from the \( E \times r \) plot: \( \lambda = 2\pi\varepsilon_0 (E \times r) \). For the sphere, solve for \( \rho \) using the constant value from the \( E \times r^2 \) plot: \( \rho = \frac{3\varepsilon_0 (E \times r^2)}{R} \) with \( R = 8.00 \, \text{mm} \).
Key Concepts
Line of ChargeUniformly Charged SphereCharge DensityGauss's Law
Line of Charge
A line of charge consists of electric charge distributed along a straight line. Imagine an infinitely long rod that has charges evenly spread out. This is a common theoretical model that's used to understand the properties of electric fields. One key characteristic of a line of charge is its linear charge density, often represented by \( \lambda \). This quantity tells us how much charge there is per unit length. Imagine \( \lambda \) as a way to measure how much charge is packed into each centimeter of the rod.
When we're dealing with a line of charge, the electric field \( E \) at any distance \( r \) from the line behaves in a specific manner. The formula \( E = \frac{\lambda}{2\pi\varepsilon_0 r} \) describes this behavior, where \( \varepsilon_0 \) is the permittivity of free space. This equation tells us that the electric field decreases as \( r \) increases, highlighting an inverse relationship where the field becomes weaker further away from the line.
It's important to note here that when you plot \( E \times r \) against \( r \), you typically find a constant line if dealing with a uniformly charged line. This characteristic can help in identifying a line of charge within an experiment.
When we're dealing with a line of charge, the electric field \( E \) at any distance \( r \) from the line behaves in a specific manner. The formula \( E = \frac{\lambda}{2\pi\varepsilon_0 r} \) describes this behavior, where \( \varepsilon_0 \) is the permittivity of free space. This equation tells us that the electric field decreases as \( r \) increases, highlighting an inverse relationship where the field becomes weaker further away from the line.
It's important to note here that when you plot \( E \times r \) against \( r \), you typically find a constant line if dealing with a uniformly charged line. This characteristic can help in identifying a line of charge within an experiment.
Uniformly Charged Sphere
A uniformly charged sphere is a three-dimensional object with charge evenly distributed throughout its volume. Imagine a solid ball where every part carries the same amount of charge. This model helps understand electric fields not only at the surface of the sphere but also outside of it.
When looking outside a uniformly charged sphere, the electric field at a distance \( r \) from the center of the sphere can be thought of behaving like a point charge, especially if \( r \) is greater than the radius \( R \) of the sphere. The formula \( E = \frac{\rho V}{4\pi\varepsilon_0 r^2} \) captures this behavior, where \( \rho \) represents the volume charge density, \( V \) is the volume, and \( \varepsilon_0 \) is the permissivity of free space.
In experiments, if you plot \( E \times r^2 \) against \( r \), a constant line suggests you are dealing with a uniformly charged sphere. This finding allows differentiation from other charge distributions, such as a line of charge. This model simplifies calculations and predictions in electrostatics, especially for fields outside large spherical objects.
When looking outside a uniformly charged sphere, the electric field at a distance \( r \) from the center of the sphere can be thought of behaving like a point charge, especially if \( r \) is greater than the radius \( R \) of the sphere. The formula \( E = \frac{\rho V}{4\pi\varepsilon_0 r^2} \) captures this behavior, where \( \rho \) represents the volume charge density, \( V \) is the volume, and \( \varepsilon_0 \) is the permissivity of free space.
In experiments, if you plot \( E \times r^2 \) against \( r \), a constant line suggests you are dealing with a uniformly charged sphere. This finding allows differentiation from other charge distributions, such as a line of charge. This model simplifies calculations and predictions in electrostatics, especially for fields outside large spherical objects.
Charge Density
Charge density is a measure of electric charge per unit area, length, or volume. It’s an important concept in understanding how charge is spread in a given space.
In the context of a line of charge, we talk about linear charge density, \( \lambda \), which is the amount of charge per unit length. If you imagine stretching a wire with charge along its length, \( \lambda \) would tell you how much charge is present per meter of that wire.
On the other hand, for a uniformly charged sphere, the relevant parameter is the volume charge density, \( \rho \), which gives the amount of charge per unit volume. It’s like asking how many charges are scattered throughout one cubic meter of a charged sphere.
Understanding these parameters helps in predicting and calculating electric fields in physics problems. Charge density directly influences the electric field’s strength and distribution, making it a pivotal aspect of electrostatics.
In the context of a line of charge, we talk about linear charge density, \( \lambda \), which is the amount of charge per unit length. If you imagine stretching a wire with charge along its length, \( \lambda \) would tell you how much charge is present per meter of that wire.
On the other hand, for a uniformly charged sphere, the relevant parameter is the volume charge density, \( \rho \), which gives the amount of charge per unit volume. It’s like asking how many charges are scattered throughout one cubic meter of a charged sphere.
Understanding these parameters helps in predicting and calculating electric fields in physics problems. Charge density directly influences the electric field’s strength and distribution, making it a pivotal aspect of electrostatics.
Gauss's Law
Gauss's Law is a fundamental principle in electromagnetism. It relates the electric flux flowing out of a closed surface to the charge enclosed within that surface. Mathematically, it is expressed as \( \Phi = \frac{Q}{\varepsilon_0} \), where \( \Phi \) is the electric flux, \( Q \) is the enclosed charge, and \( \varepsilon_0 \) is the permittivity of free space.
This law is incredibly useful for calculating electric fields, particularly in situations with high symmetry such as spheres, cylinders, or planes. For example, when dealing with a uniformly charged sphere, Gauss's Law helps derive the fact that outside of the sphere, the electric field behaves as though all the charge were concentrated at a point in the center.
In the case of a line of charge, Gauss's Law assists in showing that the electric field depends inversely on the distance from the line. This makes it an essential tool not only for understanding theoretical concepts but also for solving practical electrostatic problems.
This law is incredibly useful for calculating electric fields, particularly in situations with high symmetry such as spheres, cylinders, or planes. For example, when dealing with a uniformly charged sphere, Gauss's Law helps derive the fact that outside of the sphere, the electric field behaves as though all the charge were concentrated at a point in the center.
In the case of a line of charge, Gauss's Law assists in showing that the electric field depends inversely on the distance from the line. This makes it an essential tool not only for understanding theoretical concepts but also for solving practical electrostatic problems.
- Helps calculate fields in a complex charge distribution.
- Simplifies problems with symmetrical shapes.
- Crucial for understanding electric fields beyond introductory physics levels.
Other exercises in this chapter
Problem 54
A slab of insulating material has thickness 2d and is oriented so that its faces are parallel to the yz-plane and given by the planes \(x = d\) and \(x = -d\).
View solution Problem 56
A nonuniform, but spherically symmetric, distribution of charge has a charge density \(\rho(r)\) given as follows: $$\rho(r) = \rho_0 \bigg(1 - \frac{4r}{3R}\bi
View solution Problem 62
A region in space contains a total positive charge \(Q\) that is distributed spherically such that the volume charge density \(\rho(r)\) is given by $$\rho(r) =
View solution Problem 63
Suppose that to repel electrons in the radiation from a solar flare, each sphere must produce an electric field \(\overrightarrow{E}\) of magnitude 1 \(\times\)
View solution