Problem 60 - Fundamentals of Physics

Question

. The chocolate crumb mystery. Explosions ignited by electrostatic discharges (sparks) constitute a serious danger in facilities handling grain or powder. Such an explosion occurred in chocolate crumb powder at a biscuit factory in the 1970 s. Workers usually emptied newly delivered sacks of the powder into a loading bin, from which it was blown through electrically grounded plastic pipes to a silo for storage. Somewhere along this route, two conditions for an explosion were met: (1) The magnitude of an electric field became \(3.0 \times 10^{6} \mathrm{~N} / \mathrm{C}\) or greater, so that electrical breakdown and thus sparking could occur. (2) The energy of a spark was \(150 \mathrm{~mJ}\) or greater so that it could ignite the powder explosively. Let us check for the first condition in the powder flow through the plastic pipes. Suppose a stream of negatively charged powder was blown through a cylindrical pipe of radius \(R=5.0 \mathrm{~cm}\). Assume that the powder and its charge were spread uniformly through the pipe with a volume charge density \(\rho\). (a) Using Gauss' law, find an expression for the magnitude of the electric field \(\vec{E}\) in the pipe as a function of radial distance \(r\) from the pipe center. (b) Does \(E\) increase or decrease with increasing \(r ?\) (c) Is \(\vec{E}\) directed radially inward or outward? (d) For \(\rho=1.1 \times 10^{-3} \mathrm{C} / \mathrm{m}^{3}\) (a typical value at the factory), find the maximum \(E\) and determine where that maximum field occurs. (e) Could sparking occur, and if so, where? (The story continues with Problem 70 in Chapter 24.)

Step-by-Step Solution

Answer

Yes, sparking could occur at the pipe surface, as the electric field can exceed the threshold.

Step 1: Define the Scenario and Principles

We have a cylindrical pipe with negative powder having a charge density \( \rho \). We need to use Gauss's law to find the electric field \( E \) as a function of radial distance \( r \) from the pipe center. The goal is to find the conditions for sparking.

Step 2: Apply Gauss's Law

For a cylindrical symmetry, Gauss's law is applied as \( \oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\varepsilon_0} \). The Gaussian surface is a coaxial cylinder of radius \( r \leq R \) and length \( L \). The surface area of this cylinder is \( 2 \pi r L \).

Step 3: Calculate Enclosed Charge

The charge enclosed \( Q_{enc} \) is the charge density \( \rho \) times the volume inside the cylinder: \( Q_{enc} = \rho \cdot \pi r^2 L \).

Step 4: Solve for Electric Field

Substitute \( Q_{enc} \) into Gauss's law: \[ E \cdot 2 \pi r L = \frac{\rho \pi r^2 L}{\varepsilon_0} \]. Simplifying gives: \[ E = \frac{\rho r}{2 \varepsilon_0} \].

Step 5: Analyze Electric Field Dependence on Radius

The expression \( E = \frac{\rho r}{2 \varepsilon_0} \) shows that the electric field \( E \) increases linearly with radial distance \( r \).

Step 6: Determine Direction of Electric Field

Since the powder is negatively charged, the electric field \( \vec{E} \) will be directed radially inward towards the charged area.

Step 7: Find Maximum Electric Field

For \( r = R \), the maximum electric field is \[ E_{max} = \frac{\rho R}{2 \varepsilon_0} \]. Substituting \( \rho = 1.1 \times 10^{-3} \mathrm{C/m^3} \), \( R = 0.05 \mathrm{m} \) and \( \varepsilon_0 = 8.85 \times 10^{-12} \mathrm{C^2/(N \, m^2)} \), calculate \( E_{max} \).

Step 8: Calculate Maximum Electric Field

Using the expression:\[ E_{max} = \frac{(1.1 \times 10^{-3})(0.05)}{2(8.85 \times 10^{-12})} \approx 3.11 \times 10^6 \mathrm{N/C} \].

Step 9: Evaluate Sparking Potential

Since the maximum electric field \( E_{max} \approx 3.11 \times 10^6 \mathrm{N/C} \) is greater than \( 3.0 \times 10^6 \mathrm{N/C} \), sparking could indeed occur at the surface \( r = R \) of the pipe.

Key Concepts

Gauss's Law

Gauss's Law is a powerful tool for analyzing electric fields. Especially when the charge distribution in question exhibits symmetrical properties, Gauss's Law provides a simpler method of calculation compared to other methods.
At its core, Gauss's Law states that the electric flux through a closed surface is directly proportional to the enclosed charge. Mathematically, this is expressed as \( \oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\varepsilon_0} \), where \( \vec{E} \) is the electric field, \( d\vec{A} \) is an infinitesimal area of the closed surface, \( Q_{enc} \) is the enclosed charge, and \( \varepsilon_0 \) is the permittivity of free space.
By choosing an appropriate Gaussian surface, typically a shape that complements the symmetry of the problem, one can efficiently compute the electric field. In the case of cylindrical geometries, such as the scenario of the chocolate powder, a cylindrical Gaussian surface is ideal. This matches the symmetry of the problem and simplifies the mathematics involved.
Utilizing Gauss's Law not only simplifies the process of understanding electric fields in symmetrical charge distributions but also underscores the relationship between field, flux, and charge, providing a comprehensive view on how charges influence space.

Electric Field

An electric field is a vector field that associates a force with every point in space due to the presence of charge. In this scenario, the electric field arises from charged particles within the chocolate crumb powder moving through a cylindrical pipe.
The electric field is characterized by its magnitude and direction. In our problem, the formula derived from Gauss's Law, \( E = \frac{\rho r}{2 \varepsilon_0} \), tells us that the electric field's magnitude within the pipe depends linearly on the radial distance \( r \) from the center. Hence, it increases as one moves farther from the center.
The direction of the electric field is influenced by the nature of the charge. Since the powder is negatively charged, the electric field vectors point inward towards the axis of the cylindrical pipe. Understanding the behavior of the electric field is crucial for determining the potential for sparking, as it reaches a maximum at the surface of the pipe.

Cylindrical Symmetry

Cylindrical symmetry is an essential concept when dealing with physical systems like pipes or cables, where dimensions or properties are uniform along the length and around the circumference. In the given situation with the cylindrical pipe, this symmetry dramatically simplifies the process of analyzing the electric field.
With cylindrical symmetry, it is enough to focus on how properties change with the radial distance, as the behavior along the length and around the circumference remains constant. This allows us to assume that the electric field at any distance \( r \) from the center is the same around the entire circumference, making calculations more straightforward.
To calculate the electric field using Gauss's Law, a Gaussian surface that matches the system's symmetry—a coaxial cylinder in this case—is taken. This approach reduces a complex 3D problem into one-dimensional evaluation along the radial direction, emphasizing how the lines of symmetry guide the application of physical laws.

Volume Charge Density

Volume charge density \( \rho \) refers to the amount of electric charge per unit volume at a given point in space. It is an important parameter in electrostatics as it describes how charge is distributed in a region.
In the context of the chocolate crumb powder flowing through a pipe, the volume charge density helps determine how much charge is moving through the volume of the pipe and predicts the resultant electric field. With uniform distribution, \( \rho \) represents a constant value throughout the pipe, enabling more straightforward analysis using Gauss's Law.
The value of \( \rho = 1.1 \times 10^{-3} \mathrm{C/m^3} \) was used in calculations to find that a maximum electric field \( E_{max} \approx 3.11 \times 10^6 \mathrm{N/C} \) occurs at the pipe's surface. This information is critical in assessing the risk of sparking, which could ignite explosively sensitive materials, underscoring the role of volume charge density in ensuring safety and managing electrostatic risks.