Problem 9 - Fundamentals of Physics

Question

The magnitude \(J(r)\) of the current density in a certain cylindrical wire is given as a function of radial distance from the center of the wire's cross section as \(J(r)=B r\), where \(r\) is in meters, \(J\) is in amperes per square meter, and \(B=2.00 \times 10^{5} \mathrm{~A} / \mathrm{m}^{3}\). This function applies out to the wire's radius of \(2.00 \mathrm{~mm}\). How much current is contained within the width of a thin ring concentric with the wire if the ring has a radial width of \(10.0 \mu \mathrm{m}\) and is at a radial distance of \(1.20 \mathrm{~mm} ?\)

Step-by-Step Solution

Answer

The current in the ring is \(5.76 \times 10^{-6} \, \text{A}\).

Step 1: Understand the Problem

We need to calculate the current flowing through a thin ring within a cylindrical wire where the current density varies with the radial distance according to the function \(J(r) = B r\). The ring is located at a certain radial distance and has a small radial width.

Step 2: Define Variables and Constants

Let the radial distance of the ring be given as \(r = 1.20\, \text{mm} = 1.20 \times 10^{-3} \, \text{m}\) and the radial width \(\Delta r = 10.0 \, \mu\text{m} = 10.0 \times 10^{-6} \, \text{m}\). The constant \(B\) is \(2.00 \times 10^{5} \, \text{A}/ \text{m}^{3}\).

Step 3: Determine Current Density at Radial Distance

Calculate the current density \(J(r)\) at the distance \(r = 1.20 \times 10^{-3} \, \text{m}\). \[ J(r) = B r = (2.00 \times 10^{5} \, \text{A}/ \text{m}^{3})(1.20 \times 10^{-3} \, \text{m}) = 2.40 \times 10^{2} \, \text{A}/\text{m}^{2}. \]

Step 4: Calculate the Area of the Ring

The cross-sectional area of the ring is the difference between the outer and inner surfaces. So, \(A = 2 \pi r \Delta r\). Substituting the given values:\[ A = 2 \pi (1.20 \times 10^{-3} \, \text{m})(10.0 \times 10^{-6} \, \text{m}) = 2.40 \times 10^{-8} \, \text{m}^2.\]

Step 5: Calculate Current Through the Ring

The current \(I\) through the ring is given by the product of current density \(J(r)\) and the cross-sectional area \(A\) of the ring. \[ I = J(r) \cdot A = (2.40 \times 10^{2} \, \text{A}/\text{m}^{2})(2.40 \times 10^{-8} \, \text{m}^{2}) = 5.76 \times 10^{-6} \, \text{A}. \]

Step 6: Conclude with the Final Answer

Therefore, the total current contained within the width of the thin ring at the specified radial distance is \(5.76 \times 10^{-6} \, \text{A}\).

Key Concepts

Cylindrical Wire

A cylindrical wire is a common component in electrical circuits and devices. The typical structure of this wire is a long, circular cylinder, and its main characteristic is its uniform shape along its length. Normally built from conductive materials, like copper or aluminum, the cylindrical shape is significant because it allows for predictable and uniform distribution of current across its cross-section.
Understanding how the current flows within this type of wire is crucial, especially when the wire itself does not consist of insulated segments like in specific applications.

• The current density in a wire indicates how much current is flowing through a unit area of the wire.
• In cylindrical wires, current density may vary depending on the radial distance from the center, as they can sometimes behave asymmetrically.

The importance of the cylindrical structure is essential in calculating the distribution of current, such as in this example where the current density changes radially.

Radial Distance

Radial distance is a crucial concept when dealing with problems involving circular geometries, like cylindrical wires. It refers to the distance from the central axis of the wire to a specific point within the material. In our case, the properties of the wire or any analysis dealing with the wire change as you move further away from the center.
Consider this as the radius for the section of the circle we're interested in analyzing.

• Understanding how radial distance affects current density is key; for instance, in this example, current density increases linearly with the radial distance.
• Often given in millimeters or meters to calculate physical properties accurately in physics exercises.

Being able to map out how properties like current density and even temperature vary with radial distance is foundational when working with electrical components.

Cross-Sectional Area

The cross-sectional area is the "slice" of the wire that shows the inner components and is critical in calculating various physical properties, such as resistance and current. Specifically, in this example, it's the area through which the current flows.
For a ring or annular structure, which is what we have in a cylindrical wire, the cross-sectional area is determined by:

• Finding the differential area between two radii: using the formula \( A = 2 \pi r \Delta r \), where \( \Delta r \) is the ring's thickness.
• This area directly influences how much current can pass through the ring.

The careful calculation of this area helps determine how much current is present within a segment of the wire, considering the current density profile.

Current Calculation

In electrical engineering, the calculation of current is fundamental, as it tells us how much charge is flowing through a wire over time. To find the current through our specific circular segment of the wire, we use the relationship between current density and the cross-sectional area.
The flow of electrical charge within a material is described by current density, and hence, to find the total current, multiplying the density by the area is necessary.

• The formula used is \( I = J(r) \cdot A \), which connects current density and area to yield the total current.
• Accurate current calculations ensure that components can handle the flux of electricity without overheating or failing.

Understanding this concept ensures proper sizing and selection of materials for an electrical application, leading to efficient and safe designs.