Problem 60 - Fundamentals of Physics

Question

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 \(r = R\), the pipe's outer edge.

Step 1: Understand the Problem

We are given a scenario where charged powder flows through a cylindrical pipe, potentially causing sparks. The objective is to determine the electric field as a function of the radial distance, analyze how the electric field behaves with varying distance, and verify if the electric field reaches a threshold for sparking.

Step 2: Apply Gauss's Law for Electric Field

According to Gauss's Law, the integral form is \( \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\epsilon_0} \). In a cylinder with a radius \(r\) (where \(r < R\)), the charge enclosed is \(Q_{\text{enc}} = \rho \pi r^2 L\) . The electric field \(\vec{E}\) is directed radially outward and is the same at all points at a distance \(r\) from the axis. The surface area of a cylinder of radius \(r\) and length \(L\) is \(2 \pi r L\). Thus, \(E\cdot 2\pi r L = \frac{\rho \pi r^2 L}{\epsilon_0} \). Solve for \(E\): \[ E = \frac{\rho r}{2 \epsilon_0} \].

Step 3: Analyze Electric Field Dependence on r

The expression \( E = \frac{\rho r}{2 \epsilon_0} \) shows that the electric field \(E\) increases linearly with \(r\).

Step 4: Determine Electric Field Direction

The powder is negatively charged, hence the electric field vector \(\vec{E}\) is directed radially inward.

Step 5: Calculate the Maximum Electric Field

To find the maximum \(E\), use the pipe's radius, \(R=5.0\, \text{cm} = 0.05\, \text{m}\). Substitute \(r=R\) and \(\rho=1.1 \times 10^{-3} \mathrm{C} / \mathrm{m}^{3}\) into the expression \(E = \frac{\rho r}{2 \epsilon_0} \), giving \[ E = \frac{(1.1 \times 10^{-3}) (0.05)}{2 (8.85 \times 10^{-12})} \approx 3.11 \times 10^{6} \mathrm{N/C} \].

Step 6: Assess Sparking Conditions

The maximum electric field at \(r=R\) is \(3.11 \times 10^{6} \mathrm{N/C}\), which exceeds the threshold of \(3.0 \times 10^{6} \mathrm{N/C}\). Therefore, sparking is possible, and it would occur at the outer edge \(r=R\) of the cylindrical pipe.

Key Concepts

Electric Field

The electric field is a fundamental concept in understanding how electric charges interact with each other. It represents the force field surrounding a charged particle. This force field determines how other charges will move when placed within it. In simpler words, the electric field helps us predict the direction and magnitude of the force exerted on a charge in its presence. When dealing with a cylindrical pipe, like in our exercise, the electric field can be calculated using Gauss's Law. As seen in our example, when the charge is evenly distributed through a pipe, the electric field

• increases linearly with distance from the center, expressed as \( E = \frac{\rho r}{2 \epsilon_0} \).
• is stronger the further you move away from the center, up until you reach the pipe's boundary.
• is affected by factors like charge density (\(\rho\)) and radius (\(r\)).

Understanding how the electric field operates around different shapes, like cylinders, is crucial in predicting phenomena like sparking.

Electrostatic Discharge

Electrostatic discharge (ESD) is the sudden flow of electricity between two electrically charged objects. This discharge can occur when the electric field in a material exceeds a critical limit. In industrial settings, like the biscuit factory mentioned in our exercise, ESD can ignite flammable particles, leading to explosive situations. To prevent this, manufacturers must ensure

• charges do not accumulate to levels that exceed safe electric field thresholds.
• proper grounding of equipment is maintained to minimize charge buildup.
• materials used, like pipes, consistently maintain contact with the ground.

In our specific example, the threshold electric field for sparking is \(3.0 \times 10^{6} \text{ N/C}\), and the calculated maximum field suggests a risk for sparking at this boundary. This underlines the importance of monitoring and controlling ESD in environments dealing with fine particles.

Cylindrical Symmetry

Cylindrical symmetry in physics refers to a system where something has symmetrical properties around a linear axis. Imagine a long cylindrical pipe, like the ones transporting powder in our exercise. Charges distributed uniformly within such a cylinder affect the electric field in a symmetrical manner.
In the exercise,

• this symmetry helps simplify calculations, as the electric field's behavior becomes uniform along the length of the pipe.
• We assume the electric field is radial, spreading outward evenly from the center toward the edges of the pipe.
• All points equidistant from the axis experience the same magnitude of electric field.

This concept allows for using Gauss's Law effectively, providing a straightforward path to understanding how charges behave in such configurations.

Charge Density

Charge density, denoted as \(\rho\), represents how much electric charge is present in a given volume. It is crucial for calculating electric fields using Gauss's Law. A uniform charge density, as assumed in our exercise, implies each section of the powder-filled pipe holds an equal amount of charge per unit volume. This simplifies our calculations and predictions by assuming a constant spread of charge.

• The charge density directly influences the magnitude of the electric field within the pipe.
• Higher charge densities result in stronger electric fields.
• Understanding charge density helps in controlling electrostatic hazards, preventing scenarios that lead to sparks and potential explosions.

In practical terms, maintaining an even charge distribution in industrial processes helps manage risks related to electrostatic discharges.