Problem 70 - University Physics with Modern Physics

Question

Falling Square Loop. A vertically oricnted, square loop of copper wire falls from a region where the field \(\overrightarrow{\boldsymbol{B}}\) is horizontal. uniform. where the field is zero. The loop is released from rest and initially is entirely within the magnetic-field region. Let the side length of the loop be \(s\) and let the diameter of the wire be \(d\) . The resistivity of copper is \(\rho_{R}\) and the density of copper is \(\rho_{m}\) . If the loop reaches its terminal speed while its upper segment is still in the magnetic- field region, find an expression for the terminal speed.

Step-by-Step Solution

Answer

The terminal velocity \( v_t \) is given by: \[ v_t = \frac{\rho_m \cdot g \cdot d \cdot \pi (d/2)^2}{B^2 \cdot \rho_R} \].

Step 1: Understanding the Problem

We are dealing with a square loop of copper wire falling through a magnetic field, inducing a current due to changing magnetic flux. Resistive forces will counteract gravitational forces until a terminal velocity is reached.

Step 2: Magnetic Flux Calculation

The magnetic flux through the loop is given by \( \Phi = B \cdot s^2 \), where \( B \) is the magnetic field strength and \( s \) is the side length of the square loop. As the loop falls, the change in flux will induce an electromotive force (emf) according to Faraday's law of induction.

Step 3: Electromotive Force and Induced Current

Faraday's law states that the emf \( \epsilon \) is equal to the change in magnetic flux over time: \( \epsilon = -\frac{d\Phi}{dt} \). For a moving loop, \( \epsilon = B \cdot s \cdot v_t \), where \( v_t \) is the terminal velocity. The induced current \( I \) can be calculated using Ohm's law: \( I = \frac{\epsilon}{R} \).

Step 4: Resistive Force on the Loop

The current flowing in the magnetic field experiences a force given by \( F = B \cdot I \cdot s \). This force opposes the motion of the loop and balances the gravitational force at terminal velocity.

Step 5: Expression for Resistance

The resistance of the loop, \( R \), can be determined using the resistivity of copper: \( R = \frac{\rho_R \cdot 4s}{\pi (d/2)^2} \). This accounts for the entire loop being a circle made of copper wire.

Step 6: Balancing Forces at Terminal Velocity

At terminal velocity, the gravitational force \( mg \), where \( m = \rho_m \cdot s^2 \cdot d \), is balanced by the magnetic force \( F = B \cdot I \cdot s \). Setting \( mg = B \cdot I \cdot s \) gives us \( \rho_m \cdot s^2 \cdot d \cdot g = B \cdot \left( \frac{B \cdot s \cdot v_t}{R} \right) \cdot s \).

Step 7: Solve for Terminal Velocity

Substitute the expression for resistance and solve for \( v_t \):\[ v_t = \frac{\rho_m \cdot g \cdot d \cdot \pi (d/2)^2}{B^2 \cdot \rho_R} \]. This is the terminal velocity of the falling loop.

Key Concepts

Faraday's Law of Induction

Faraday's Law of Induction is a fundamental principle in electromagnetism. It describes how a changing magnetic field can induce an electromotive force (emf) in a conductor. This law is the cornerstone of electromagnetic induction and has practical applications in many electrical devices.

The law states that the induced emf in a closed loop is equal to the negative rate of change of magnetic flux through the loop. Mathematically, it can be expressed as:

• \(\epsilon = -\frac{d\Phi}{dt}\)

Where:

• \(\epsilon\) is the induced emf,
• \(\Phi\) is the magnetic flux,
• \(t\) is time.

In the context of the falling square loop in a magnetic field, as the loop moves, the flux through it changes. This changing flux is what causes an emf to be generated according to Faraday's law. This induced emf is crucial because it leads to the generation of an induced current, which interacts with the magnetic field to counteract the loop's motion at terminal velocity.

Ohm's Law

Ohm's Law is a foundational concept in electronics and physics, establishing the relationship between voltage, current, and resistance in an electrical circuit. It is represented by the formula:

• \(V = I \cdot R\)

Where:

• \(V\) is voltage,
• \(I\) is current,
• \(R\) is resistance.

In our square loop of copper wire, Ohm's Law helps us to determine the current flowing through the loop as a result of the induced emf from Faraday's law. The formula to find the current is:

• \(I = \frac{\epsilon}{R}\)

This tells us that the current \(I\) is equal to the induced emf \(\epsilon\) divided by the resistance \(R\) of the loop. The resistance can be calculated from the resistivity of the copper wire, its length, and cross-sectional area. Ohm's Law is used to find how much current flows due to the induced emf, affecting the dynamics as the loop reaches terminal velocity.

Magnetic Flux

Magnetic Flux is a measure of the quantity of magnetism, representing the total number of magnetic field lines passing through a given area. It is a critical concept for understanding electromagnetic induction.

Magnetic flux through a surface is given by:

• \(\Phi = B \cdot A \cdot \cos \theta\)

Where:

• \(\Phi\) is the magnetic flux,
• \(B\) is the magnetic field strength,
• \(A\) is the area the field lines pass through,
• \(\theta\) is the angle between the magnetic field and the normal to the surface.

In the falling loop scenario, the loop initially experiences a uniform magnetic field. As it moves, the area through which magnetic field lines pass—effectively making up the magnetic flux—changes. This change results in an induced emf as described by Faraday's law. By understanding magnetic flux, the nuances of how the loop interacts with the magnetic field as it falls can be better comprehended.