Problem 79 - Physics for Scientists and Engineers with Modern Physics

Question

In a certain region of space near Earth's surface, a uniform horizontal magnetic field of magnitude \(B\) exists above a level defined to be \(y=0 .\) Below \(y=0,\) the field abruptly becomes zero (Fig. \(29-54\) ). A vertical square wire loop has resistivity \(\rho,\) mass density \(\rho_{m},\) diameter \(d,\) and side length \(\ell .\) It is initially at rest with its lower horizontal side at \(y=0\) and is then allowed to fall under gravity, with its plane perpendicular to the direction of the magnetic field. ( \(a\) ) While the loop is still partially immersed in the magnetic field (as it falls into the zero-field region), determine the magnetic "drag" force that acts on it at the moment when its speed is \(v .(b)\) Assume that the loop achieves a terminal velocity \(v_{\mathrm{T}}\) before its upper horizontal side exits the field. Determine a formula for \(v_{\mathrm{T}}\). \((c)\) If the loop is made of copper and \(B=0.80 \mathrm{~T},\) find \(v_{\mathrm{T}}\).

Step-by-Step Solution

Answer

The terminal velocity \(v_T = \frac{\rho_m g \rho \pi d^2}{4B^2}\). For copper, substitute values to compute \(v_T\).

Step 1: Understand the problem

We have a square loop of wire with resistivity \(\rho\) and mass density \(\rho_m\), falling through a magnetic field \(B\). As it falls, it experiences an induced emf and an associated opposing magnetic drag force. We will compute this force as it falls and determine its terminal velocity.

Step 2: Calculate the induced emf

When a portion of a conducting loop moves through a magnetic field at velocity \(v\), an emf is induced due to the change in magnetic flux. The emf \(\varepsilon\) is given by Faraday's law: \[\varepsilon = B\ell v\]where \(\ell\) is the length of the side of the square loop that is moving through the field. This occurs while part of the loop is exiting the magnetic region.

Step 3: Determine the current in the loop

Using the equation \(\varepsilon = B\ell v\) and Ohm's law \(I = \frac{\varepsilon}{R}\), where the resistance \(R\) of the loop can be found using \(R = \frac{\rho \ell}{A}\), with \(A\) being the cross-sectional area of the wire, calculated as \(A = \frac{\pi d^2}{4}\):\[I = \frac{B \ell v}{\frac{\rho \ell}{\frac{\pi d^2}{4}}}=\frac{4B\ell v}{\rho \pi d^2}\]

Step 4: Find the magnetic drag force

The magnetic drag force \(\mathbf{F}_{\text{drag}}\) on the loop is due to the interaction between the current and the magnetic field. This force is calculated as:\[F_{\text{drag}} = BI\ell = B\left(\frac{4B\ell v}{\rho \pi d^2}\right)\ell= \frac{4B^2 \ell^2 v}{\rho \pi d^2}\]

Step 5: Derive the terminal velocity

At terminal velocity \(v_T\), the drag force equals the gravitational force acting on the loop. The gravitational force \(F_g\) is given by:\[F_g = \rho_m g \ell^2\]Setting \(F_{\text{drag}} = F_g\) gives:\[\frac{4B^2 \ell^2 v_T}{\rho \pi d^2} = \rho_m g \ell^2\]Solving for \(v_T\), we get:\[v_T = \frac{\rho_m g \rho \pi d^2}{4B^2}\]

Step 6: Calculate the terminal velocity for copper

Substitute the values \(B = 0.80 \text{ T}\), \(\rho_m = 8,920 \text{ kg/m}^3\), \(\rho = 1.68 \times 10^{-8} \text{ ohm}\cdot\text{m}\), and assume a value for \(d\). Plug these into the derived formula for \(v_T\):\[v_T = \frac{8,920 \times 9.81 \times 1.68 \times 10^{-8} \times \pi d^2}{4 \times (0.80)^2}\]Solve this equation once \(d\) is known to find the final numerical value of \(v_T\).

Key Concepts

Magnetic Field

Magnetic fields are invisible fields that exert a force on moving charges, and they exist around magnetic materials and electric currents. In this exercise, a uniform horizontal magnetic field exists in a specified region of space. A magnetic field is characterized by its strength, often denoted as \( B \), which represents the magnitude of the field in teslas (T). The direction of the magnetic field is crucial to understanding the forces that act on objects moving within it. Here, the field is horizontal, and the square wire loop falls perpendicularly through this field, indicating that the magnetic forces experienced by the wire are maximized due to the perpendicular orientation. As the loop falls, it partially exits the magnetic field, creating changes in forces and inducing currents, which are vital to the occurrence of electromagnetic induction.

Faraday's Law

Faraday's Law of electromagnetic induction is a fundamental principle that describes how electric currents are induced in conductors. When a conductor like our wire loop moves through a magnetic field, a change in magnetic flux through the loop occurs. This change induces an electromotive force (emf), which can drive a current if the surface is closed.

• The magnitude of the induced emf is given by \( \varepsilon = B\ell v \), where \( B \) is the magnetic field strength, \( \ell \) is the length of the side of the loop moving within the field, and \( v \) is the velocity of the loop.
• This relationship shows that a faster-moving loop or a stronger magnetic field results in a greater emf.

Ohm’s Law relates this emf to current \( I \), with resistance \( R \) using \( I = \frac{\varepsilon}{R} \). Faraday's Law highlights the interplay between motion, magnetic fields, and electricity, explaining how energy is transformed in dynamically changing environments.

Terminal Velocity

Terminal velocity is the constant speed that a falling object reaches when the force of gravity is balanced by the drag force opposing its motion. In our scenario, as the wire loop falls, it is subject to two primary forces: the gravitational force pulling it downwards and the magnetic drag force acting upward.

• The magnetic drag force \( F_{\text{drag}} \) arises due to the interaction of the induced current with the magnetic field.
• This force is calculated as \( F_{\text{drag}} = \frac{4B^2 \ell^2 v}{\rho \pi d^2} \).
• At terminal velocity \( v_T \), these forces are equal: \( F_{\text{drag}} = F_g \), with \( F_g \) representing the gravitational force.

Solving the equations provides \( v_T = \frac{\rho_m g \rho \pi d^2}{4B^2} \), illustrating that this constant speed depends on various factors, including the properties of the material and dimensions of the loop. Terminal velocity is an essential concept in understanding how objects settle into steady states of motion where applied forces are balanced.

Copper Properties

Copper is a highly significant material in this exercise, valued for its distinctive physical properties. As a good conductor of electricity, copper features a low resistivity of \( 1.68 \times 10^{-8} \text{ ohm} \cdot \text{m} \). This property greatly influences the current induced in the loop due to the emf generated by the falling loop through the magnetic field.

• Its low resistivity ensures minimal resistance to the induced current, allowing efficient energy transformation.
• The mass density of copper is \( 8,920 \text{ kg/m}^3 \), affecting the gravitational force exerted on the loop and, subsequently, its terminal velocity.

In practice, these attributes make copper an ideal choice for applications requiring significant electrical conductivity and manageable weight. Understanding copper's properties allows us to predict and calculate crucial parameters such as terminal velocity with real-world materials, demonstrating its essential role in physics and engineering tasks.