Problem 46 - University Physics with Modern Physics

Question

A very long, rectangular loop of wire can slide without friction on a horizontal surface. Initially the loop has part of its area in a region of uniform magnetic field that has magnitude \(B =\) 2.90 T and is perpendicular to the plane of the loop. The loop has dimensions 4.00 cm by 60.0 cm, mass 24.0 g, and resistance \(R =\) 5.00 \(\times\) 10\(^{-3} \Omega\). The loop is initially at rest; then a constant force \(F_{ext}\) = 0.180 N is applied to the loop to pull it out of the field (Fig. P29.46). (a) What is the acceleration of the loop when \(v =\) 3.00 cm/s? (b) What are the loop's terminal speed and acceleration when the loop is moving at that terminal speed? (c) What is the acceleration of the loop when it is completely out of the magnetic field?

Step-by-Step Solution

Answer

(a) Acceleration is 2.08 m/s². (b) Terminal speed is 36.3 m/s, acceleration is 0 m/s². (c) Acceleration is 7.50 m/s².

Step 1: Analyze the Forces on the Loop

When the loop is moving in the magnetic field, it experiences a magnetic force given by Lorentz force principles. The magnetic force can be calculated using Faraday's Law of Induction and Lenz's Law. Calculate the electromotive force (emf) using \( \varepsilon = vBL \), where \( v \) is the speed, \( B \) is the magnetic field, and \( L \) is the length of the side perpendicular to \( v \).

Step 2: Calculate the Induced Current

The induced current \( I \) in the loop is due to the emf: \( I = \frac{\varepsilon}{R} = \frac{vBL}{R} \). With \( v = 0.03 \) m/s, \( B = 2.90 \) T, \( L = 0.04 \) m, and \( R = 5.00 \times 10^{-3} \Omega \), substitute to find \( I \).

Step 3: Determine the Magnetic Force

The magnetic force opposing the motion of the loop is given by \( F_B = ILB \), where \( I \) is the current calculated in Step 2. Substitute to find \( F_B \).

Step 4: Calculate the Net Force

The net force \( F_{net} \) on the loop is the difference between the external force and the magnetic force: \( F_{net} = F_{ext} - F_B \). With \( F_{ext} = 0.180 \) N, substitute to find \( F_{net} \).

Step 5: Find the Acceleration

The acceleration \( a \) of the loop can be determined using Newton's second law: \( a = \frac{F_{net}}{m} \). Recall that the mass \( m = 0.024 \) kg. Substitute to find \( a \).

Step 6: Determine the Terminal Speed

At terminal speed, the net force is zero, so \( F_{ext} = F_B \). Solve for speed \( v_t \) using \( F_{ext} = \frac{v_t B L^2}{R} \).

Step 7: Verify Acceleration at Terminal Speed

Repeat the process from Step 5 to verify the acceleration at terminal speed; it should be zero because forces are balanced.

Step 8: Calculate the Acceleration When Out of the Field

Once the loop is completely out of the magnetic field, there is no magnetic force, so \( F_{net} = F_{ext} \). Use this in Newton's second law to find the new acceleration, which is simply \( a = \frac{F_{ext}}{m} \).

Key Concepts

Faraday's Law of Induction

One of the fascinating principles in electromagnetism is Faraday's Law of Induction. This law explains how a changing magnetic field can induce an electromotive force (emf) in a coil or loop. In this exercise, a metallic loop is within a magnetic field, and as it moves, this change in position causes the magnetic flux through the loop to change. When the loop is pulled out of a magnetic field horizontally, the movement itself changes the magnetic environment around the loop, which results in an induced emf. You can calculate this induced emf using the formula \( \varepsilon = vBL \), where \( v \) is the velocity of the loop, \( B \) is the magnetic field strength, and \( L \) is the length of the loop perpendicular to the motion. This induced emf is what gives rise to the current that is responsible for the magnetic force opposing the loop's motion.

Lorentz Force

The Lorentz Force is a fundamental concept that explains how charged particles such as electrons are affected by electric and magnetic fields. In our exercise, the Lorentz force plays a critical role in understanding the behavior of the loop as it slides in and out of the magnetic field. When the loop experiences an induced current due to the emf from Faraday’s Law, a Lorentz force is exerted perpendicular to both the current and the magnetic field. The magnitude of this force can be calculated with the equation \( F_B = ILB \), where \( I \) is the induced current in the loop, \( L \) is the length of the side of the loop perpendicular to velocity, and \( B \) is the magnetic field strength. This force opposes the external force applied to the loop, which helps us understand the motion and acceleration of the loop.

Lenz's Law

Lenz's Law is an extension of Faraday's Law of Induction, providing the direction of the induced emf and current. It states that the induced current will flow in such a way that its magnetic field opposes the change in magnetic flux that produced it. This principle ensures the conservation of energy. In our scenario with the moving wire loop, the current generated by the induced emf creates a magnetic field that resists the loop's change in position—specifically, it opposes the motion that is pulling the loop out of the magnetic field. Thus, when calculating forces, Lenz's Law informs why the magnetic force \( F_B \) works against the external force pulling the loop, leading to a net force that affects the loop's acceleration.

Terminal Velocity

Terminal velocity refers to the constant speed that an object achieves when the net force acting on it becomes zero, causing no further acceleration. In the context of our exercise, the loop reaches terminal velocity when the magnetic force opposing the pull of the external force equals the applied force \( F_{ext} \). This balance results in \( F_{ext} = F_B \). When solving for terminal velocity \( v_t \), the formula \( F_{ext} = \frac{v_t B L^2}{R} \) is used. At terminal velocity, the loop continues to move at this constant speed without further acceleration as the opposing forces have reached equilibrium.

Newton's Second Law

Newton's Second Law is foundational to understanding motion, explaining how the velocity of an object changes when it is subjected to an external force. It states that the acceleration \( a \) of an object is directly proportional to the net force \( F_{net} \) acting on it and inversely proportional to its mass \( m \). In this exercise, the net force is influenced by both the external force \( F_{ext} \) and the magnetic force \( F_B \) that opposes the motion. This means \( F_{net} = F_{ext} - F_B \). Using this understanding, the acceleration can be calculated as \( a = \frac{F_{net}}{m} \). Whenever the external environment changes, such as the loop exiting the magnetic field, only \( F_{ext} \) remains, and this adjustment changes the acceleration of the loop. Moreover, once the loop is fully out of the magnetic field, the only force acting is the external force, simplifying the calculation of acceleration.