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Magazine Chapter 24: Problem 31
Charge Is Distributed Uniformly Charge is distributed uniformly throughout the
volume of an infinitely long cylinder of
radius \(R\). (a) Show that, at a distance \(r\) from the cylinder axis (for
\(r
Short Answer
Expert verified
Inside ( r < R): \(|\vec{E}| \frac{|\rho| r}{2 \varepsilon_{0}}\). Outside ( r > R): \(|\vec{E}| \frac{|\rho| R^2}{2 \varepsilon_{0} r}\).
Step by step solution
01
Understanding the problem
Charge is distributed uniformly throughout the volume of an infinitely long cylinder of radius R. Need to find the electric field at a distance r from the axis for both r < R and r > R.
02
Using Gauss's Law Inside the Cylinder (r
Gauss's Law states \(\oint\vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\varepsilon_{0}}\). For an infinitely long cylinder, choose a cylindrical Gaussian surface of radius r<R and length L. The enclosed charge is given by \(Q_{enc} = \rho \pi r^2 L\), where \(\rho\) is the volume charge density. Since the electric field \(\vec{E}\) is radially symmetrical, \(\vec{E} \) is constant over the surface area which is \(2 \pi r L\).
03
Calculating the Enclosed Charge Inside the Cylinder
Integrate Gauss鈥檚 Law: \(E (2 \pi r L) = \frac{\rho \pi r^2 L}{\varepsilon_{0}}\). Divide both sides by \(2 \pi r L\) to solve for \(E\): \(E = \frac{\rho r}{2 \varepsilon_{0}}\). Thus, \(|\vec{E}| = \frac{|\rho| r}{2 \varepsilon_{0}}\).
04
Using Gauss's Law Outside the Cylinder (r>R)
For the case r>R, choose a cylindrical Gaussian surface of radius r>R and length L. The enclosed charge remains constant and is given by the total charge of the cylinder of radius R and length L, \(Q_{enc} = \rho \pi R^2 L\).
05
Calculating the Electric Field Outside the Cylinder
Using Gauss鈥檚 Law: \(E (2 \pi r L) = \frac{\rho \pi R^2 L}{\varepsilon_{0}}\). Divide both sides by \(2 \pi r L\) to solve for \(E\): \(E = \frac{\rho R^2}{2 \varepsilon_{0} r}\). Thus, \(|\vec{E}| = \frac{|\rho| R^2}{2 \varepsilon_{0} r}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Electric Field
The electric field, denoted as \(\text{E}\), is a vector field that represents the force per unit charge exerted on a positive test charge at any point in space. In simpler terms, it shows how a charge would 'feel' the force at a particular location. The electric field due to a distribution of charges can be found using Gauss's Law, making calculations more manageable, especially in systems with symmetrical charge distributions.
The electric field inside and outside a charged cylinder depends on the distance from the cylinder's axis. The field inside the cylinder increases linearly with distance from the center, while outside, it decreases with distance.
The electric field inside and outside a charged cylinder depends on the distance from the cylinder's axis. The field inside the cylinder increases linearly with distance from the center, while outside, it decreases with distance.
Uniform Charge Distribution
A uniform charge distribution means that charge is spread evenly throughout a given volume. In the context of this exercise, the charge is distributed uniformly along the volume of an infinitely long cylinder. This means the volume charge density, called \(\rho\), remains constant throughout the cylinder. This simplifies calculations as \(\rho\) doesn't change at different points within the cylinder.
For uniform charge distribution:
For uniform charge distribution:
- The charge per unit volume is constant.
- The total charge can be calculated by multiplying the volume charge density \(\rho\) by the volume.
Cylindrical Symmetry
Cylindrical symmetry means that a system looks the same when rotated around a particular axis. In our exercise, the cylinder's charge distribution remains unchanged irrespective of how we rotate around its central axis. This symmetry simplifies the use of Gauss's Law to find the electric field because:
- The electric field depends only on the radial distance \(\text{r}\) from the axis, not the angle or height.
- A cylindrical Gaussian surface matches the symmetry, making the calculations straightforward.
Volume Charge Density
Volume charge density, denoted as \(\rho\), is a measure of the amount of electric charge per unit volume at a given location. For our problem, the volume charge density is constant throughout the cylinder, indicating a uniform charge distribution. Volume charge density is crucial because:
- It helps in calculating the enclosed charge for Gauss's Law.
- The relationship between \(\rho\) and the electric field is direct, simplifying the determination of the field.
Gaussian Surface
A Gaussian surface is an imaginary closed surface used in Gauss's Law to simplify the calculation of electric fields due to symmetrical charged objects. In this exercise, we use cylindrical Gaussian surfaces to leverage the cylindrical symmetry of the problem. When choosing a Gaussian surface:
- It should match the symmetry of the charge distribution.
- For points inside the cylinder \(r
- For points outside the cylinder \(r>R\), the Gaussian surface is a larger coaxial cylinder enclosing the entire charge.
