Explanations 
Textbooks 
Exams 
GCSE 
IGCSE 
AS 
A Level 
International A Level 
University Admissions Tests 
Flashcards 
Vaia AI 
Notes 
Study Plans 
Study Sets 
Find a job 
Find a degree 
Student Deals 
Magazine 
Mobile App Chapter 9: Problem 32
A circular loop rotates around a horizontal axis coming out of the page (side view shown, below). The loop is rotating in a uniform \(B\)-field, pointed downward. For every two full rotations, how often does the induced current change direction? (A) Twice (B) Four times (C) Eight times (D) Twelve time (E) Sixteen times
Short Answer
Expert verified
Answer: (B) Four times.
Step by step solution
01
Understand Faraday's law and induced current
According to Faraday's law, the induced electromotive force (EMF) is given by the rate of change of magnetic flux through a loop. The induced current flows through the loop in a direction that opposes the change in magnetic flux, as per Lenz's law. This means that whenever the magnetic flux through the loop changes, the induced current will change its direction in response.
In our case, the external magnetic field is uniform and downward pointing, and the circular loop is rotating around a horizontal axis. Let's analyze the change in magnetic flux for each quarter rotation and determine the corresponding change in induced current direction.
02
Analyzing rotation and changes in magnetic flux
We can divide the rotation angle into quarters for convenience:
1. Initial position (0 degrees): The magnetic flux through the loop is maximum as the plane of the loop is perpendicular to the magnetic field. Let's name the induced current direction as "A".
2. Quarter rotation (90 degrees): The magnetic flux through the loop becomes zero, as the plane of the loop becomes parallel to the magnetic field. At this point, the induced current changes direction to "B", opposing the decreasing magnetic flux.
3. Half rotation (180 degrees): The magnetic flux reaches its maximum again but with the opposite sign, as the plane of the loop is now flipped and still perpendicular to the magnetic field. Induced current direction is back to "A", opposing the increase in magnetic flux.
4. Three-quarter rotation (270 degrees): The magnetic flux is zero again, with the plane of the loop parallel to the magnetic field. Induced current changes direction back to "B" again, opposing the decreasing magnetic flux.
So, in one full rotation, the induced current changes direction twice.
03
Determine direction changes for two full rotations
Since the induced current changes direction twice in one full rotation, for two full rotations, the induced current will change direction 2 times 2, which is 4. Therefore, the answer is:
(B) Four times
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Faraday's Law
When approaching any problem related to electromagnetism, it is paramount to start with Faraday's Law. This fundamental law indicates that a change in magnetic flux through a conducting loop induces an electromotive force (EMF) in the loop. Mathematically, Faraday's Law is expressed as \( EMF = -\frac{d\Phi_B}{dt} \), where \( \Phi_B \) represents the magnetic flux. It is key to understand that the EMF generated leads to an induced current, provided that the circuit is closed. The negative sign, introduced by Lenz's Law, signifies that the induced EMF acts in a direction to oppose the change in flux that produced it. This concept serves as a critical foundation when solving questions similar to the exercise provided, and it should be kept in mind for all such physics SAT review topics and AP Physics questions.
Magnetic Flux
Magnetic flux is an essential concept to grasp in understanding induced currents. Think of it as the quantity that measures how much of the magnetic field passes through a given area. It is denoted by \( \Phi_B \) and is calculated by the dot product of the magnetic field \( \vec{B} \) and the area vector \( \vec{A} \), which is \( \Phi_B = \vec{B} \cdot \vec{A} \). For a uniform magnetic field and a flat loop, this translates to \( \Phi_B = B \cdot A \cdot \cos(\theta) \), where \( \theta \) is the angle between the magnetic field and the normal to the loop's area. During the loop's rotation in a uniform magnetic field, as displayed in the exercise, the magnetic flux changes, which is directly linked to the occurrence of the induced EMF and the resultant current change.
Lenz's Law
Lenz's Law is like nature's own insurance policy, ensuring that nothing changes too abruptly. It provides the direction of the induced current resulting from a change in magnetic flux. According to Lenz's Law, the induced current will flow in such a manner that it opposes the change in magnetic flux that produced it. This law explains the negative sign in Faraday's Law, and its conceptual understanding is critical when one has to determine the direction of the induced current. Applying Lenz's Law is a vital skill in physics SAT review sessions and aids in solving a multitude of AP Physics questions where electromagnetic induction is involved.
Uniform Magnetic Field
In a uniform magnetic field, the strength and direction of the magnetic field are constant at every point within the field. This condition simplifies many physics problems because the magnetic flux through a loop only depends on the orientation of the loop with respect to the magnetic field. Understanding a uniform magnetic field is crucial, as it often represents the external magnetic field involved in school-level physics problems, including the scenario we explored in the exercise.
Physics SAT Review
For students preparing for the physics SAT, it's important to familiarize oneself with concepts such as electromagnetic induction and the laws of Faraday and Lenz. Reviewing induced current and magnetic flux with practical exercises, like the one provided, helps solidify these principles. Engaging with various problems that require calculating the EMF and the induced current direction can greatly improve your understanding and performance in the physics SAT or any high school level physics exam.
AP Physics Questions
The AP Physics exams often include questions about electromagnetic induction, which require a deep understanding of Faraday's and Lenz's laws. It's essential for students to learn how to determine the direction of induced currents and how they change with time, particularly when faced with situations involving a uniform magnetic field. Practice problems, like the exercise we discussed, are great resources for AP students to master the topic, as they encapsulate the core principles and require the application of both conceptual knowledge and mathematical skills.
