Problem from Asia's Toughest Physics Exam

ZPhysics

6 Aug 202320:46

EducationalLearning

32 Likes 10 Comments

TLDRThe video delves into a challenging physics problem from the Asian Physics Olympiad, exploring the behavior of a magnet falling through a copper tube. The speaker explains the mathematical and physical principles involved, including terminal velocity, differential equations, and Faraday's law. The problem requires deriving the magnet's position over time and calculating the induced electromotive force (EMF) in a ring surrounding the tube. The speaker methodically solves the equations and provides insights into advanced concepts, making it a compelling watch for those interested in complex physics challenges.

Takeaways

🔍 The problem involves analyzing the motion of a magnet falling through a copper tube, with the primary force considered being electromagnetic in nature.
📉 The magnet does not fall at the standard acceleration due to gravity (9.81 m/s²) but rather falls much slower due to electromagnetic forces.
📚 The first step is to find the terminal velocity (Vr) of the magnet by setting acceleration to zero, yielding the equation Vr = mg/k.
🔢 The problem requires solving a differential equation to find the position Z(t) of the magnet over time, with initial conditions set to zero.
✏️ The solution involves using an integrating factor to simplify the differential equation, ultimately deriving an expression for Z(t).
🔄 The expression for Z(t) is found using the product rule and integration by parts, incorporating constants determined by boundary conditions.
🌐 The problem transitions to cylindrical coordinates, focusing on the magnet's dipole moment and magnetic field components in the Z direction.
🧲 The induced electromotive force (EMF) in a ring surrounding the tube is calculated using Faraday's law, specifically through the loop integral of the cross product of velocity and magnetic field.
📏 The magnitude of the induced EMF is proportional to the velocity and the magnetic field component in the radial direction, as well as the loop's circumference.
📐 The force opposing the magnet's fall is calculated as proportional to the radial component of the magnetic field, which acts to slow down the magnet.

Q & A

What is the terminal velocity of the magnet as it falls through the copper tube?
-The terminal velocity \( v_R \) of the magnet can be found by setting the acceleration to zero in the governing equation. This gives \( v_R = \frac{mg}{k} \), where \( m \) is the mass of the magnet, \( g \) is the acceleration due to gravity, and \( k \) is a coefficient proportional to the speed of the magnet.
How is the governing equation of the magnet's motion derived?
-The governing equation is derived by considering the forces acting on the magnet. The equation is \( m\ddot{z} = mg - k\dot{z} \), where \( m\ddot{z} \) is the mass times the acceleration, \( mg \) is the gravitational force, and \( k\dot{z} \) is the force proportional to the velocity, representing resistance due to Lenz's law.
What method is used to solve the differential equation for the position \( z(t) \) of the magnet?
-The differential equation is solved using an integrating factor. This involves multiplying both sides of the equation by a function \( f(t) \) that simplifies the left-hand side into a derivative of a product, allowing for easier integration.
What is the integrating factor used to solve the differential equation?
-The integrating factor \( f(t) \) is found to be \( e^{\frac{k}{m}t} \), which simplifies the differential equation by turning the left-hand side into a derivative of the product \( f(t)z(t) \).
What are the boundary conditions used to solve for the constants in the solution?
-The boundary conditions are \( z(0) = 0 \) (the initial position is zero) and \( \dot{z}(0) = 0 \) (the initial velocity is zero). These conditions are used to solve for the constants \( C_1 \) and \( C_2 \) in the general solution.
What is the final expression for the position \( z(t) \) of the magnet over time?
-The final expression for the position \( z(t) \) is \( z(t) = \frac{gm}{k} \left( t - \frac{m}{k} \right) \left( 1 - e^{-\frac{k}{m}t} \right) \), which describes the magnet's position as a function of time, considering both gravitational force and resistive force.
How is the induced electromotive force (EMF) in the ring calculated?
-The induced EMF is calculated using Faraday's Law, which states that the EMF is the negative rate of change of magnetic flux through the ring. An alternative form, suitable when the magnet and ring are moving, uses the loop integral of the cross product of the velocity and magnetic field vectors.
What components of the magnetic field are considered for calculating the induced EMF?
-The radial and axial components of the magnetic field are considered. The cross product of the velocity (in the Z direction) with the radial magnetic field component gives the induced EMF.
What role does Newton's Third Law play in the problem?
-Newton's Third Law indicates that the force exerted by the induced current in the ring on the magnet has the same magnitude but opposite direction as the force exerted by the magnet on the ring. This force opposes the motion of the magnet, slowing it down.
Why is cylindrical coordinate system used in this problem?
-Cylindrical coordinates are used because the magnet is falling through a cylindrical tube, making this coordinate system natural and convenient for analyzing the problem, especially when considering the radial and axial components of the magnetic field.

Outlines

00:00

🧲 Introduction to the Magnet Problem in the Asian Physics Olympiad

In this segment, the speaker introduces a challenging problem from the Asian Physics Olympiad, involving a magnet falling through a copper tube. The problem does not follow standard free-fall rules due to magnetic effects. The initial task is to find the terminal velocity of the magnet by setting the acceleration to zero and deriving the expression for terminal velocity as mg/k. This serves as an entry point into a more complex discussion of the physics involved.

05:01

📈 Deriving the Magnet's Position Over Time

Here, the speaker tackles the problem of determining the magnet's position, Z(t), as a function of time, starting from an initial speed and position of zero. The differential equation m(z'') + k(z') = mg is simplified and solved using an integrating factor approach, leading to a complex solution involving exponential functions. This part involves multiple steps, including integration by parts and product rule applications, to arrive at the final expression for the position as a function of time.

10:03

🧮 Applying Initial Conditions to Determine Constants

The third part focuses on applying boundary conditions to refine the solution obtained for the magnet's position. By setting the initial velocity and position to zero, the speaker derives the constants involved in the solution. The detailed steps involve differentiating the position function, solving for constants like C2 and C, and eventually concluding that one of the constants is zero. The final expression for the position as a function of time is then presented.

15:03

🔄 Switching to Cylindrical Coordinates for Further Calculations

This segment shifts focus to cylindrical coordinates, which are more suitable for analyzing the motion of the magnet through the cylindrical tube. The speaker explains the magnetic field components and sets up Faraday's law to calculate the induced EMF in the ring due to the magnet's motion. The discussion includes evaluating a cross product and performing line integrals, emphasizing the complexity of the problem and the advanced level of understanding required to solve it.

20:04

🔁 Determining the Induced Electromotive Force and Concluding Thoughts

In the final segment, the speaker derives the formula for the electromotive force (EMF) induced in the ring, using the radial component of the magnetic field and the current I. The discussion also covers the force exerted by the ring on the magnet, following Newton's third law, where the forces are equal in magnitude but opposite in direction. The speaker concludes by inviting viewers to tackle another challenging problem, reinforcing the advanced nature of the content.

Mindmap

Keywords

💡Terminal Velocity

Terminal velocity refers to the constant speed that a freely falling object eventually reaches when the resistance of the medium prevents further acceleration. In the video, the terminal velocity of a magnet falling through a copper tube is discussed, calculated as the point where the forces of gravity and resistance balance out.

💡Lenz's Law

Lenz's Law states that the direction of an induced current is always such that it will oppose the change in magnetic flux that caused it. The video references this law to explain why the magnet falls slower through the copper tube, as the induced current in the tube creates a magnetic field that opposes the motion of the magnet.

💡Differential Equation

A differential equation is a mathematical equation that relates a function to its derivatives, used to describe various phenomena like motion, heat, and waves. The video involves solving a differential equation to find the position of the magnet over time, which is central to understanding the magnet's behavior as it falls.

💡Integration by Parts

Integration by parts is a technique used to integrate products of functions. In the video, this method is applied to solve the differential equation governing the magnet's motion, illustrating a step-by-step process for deriving the position function Z(t).

💡Faraday's Law

Faraday's Law of Induction states that a change in magnetic flux through a loop induces an electromotive force (EMF) in the loop. The video uses this law to calculate the induced EMF in a ring surrounding the falling magnet, showing how the motion of the magnet generates a current in the copper tube.

💡Magnetic Flux

Magnetic flux is a measure of the quantity of magnetism, considering the strength and extent of a magnetic field. In the video, the change in magnetic flux as the magnet moves through the copper tube is key to understanding the induced EMF and the resultant forces that slow the magnet's fall.

💡Electromotive Force (EMF)

Electromotive force (EMF) is the voltage generated by a changing magnetic field. The video explains how the motion of the magnet through the tube generates an EMF in the surrounding ring, and how this EMF relates to the current induced in the ring according to Faraday's Law.

💡Product Rule

The product rule is a calculus rule used to differentiate the product of two functions. In the video, the product rule is utilized to simplify the differential equation, which is essential for deriving the function that describes the magnet's position over time.

💡Newton's Third Law

Newton's Third Law states that for every action, there is an equal and opposite reaction. The video applies this principle to explain how the force exerted by the induced current in the copper tube acts in the opposite direction to the magnet's motion, thereby slowing it down.

💡Magnetic Dipole Moment

The magnetic dipole moment is a vector quantity that represents the strength and direction of a magnetic source. The video discusses the magnet's dipole moment as it falls through the tube, assuming that it remains constant and contributes to the magnetic interactions that influence the magnet's motion.

Highlights

The problem involves the behavior of a magnet falling through a copper tube, where it does not fall at the usual acceleration due to gravity.

The first task is to determine the terminal velocity of the magnet, which is found by setting the acceleration to zero, resulting in a terminal velocity of mg/k.

The second task requires solving a differential equation to obtain the position of the magnet as a function of time, starting with an initial speed and coordinate of zero.

The governing equation is a second-order differential equation that describes the motion of the magnet with respect to time.

The solution involves integrating both sides of the equation and using an integrating factor, which is found to be e^(k/m * t).

The product rule is applied to simplify the equation, leading to a new equation that is then integrated again to find the position as a function of time.

The integration by parts method is used to solve the integral that appears in the equation, leading to a final expression for the position of the magnet.

Boundary conditions are applied to find the constants of integration, resulting in the final solution for the position as a function of time.

The problem becomes more complex as it transitions to cylindrical coordinates, where the magnet's dipole moment and magnetic field components are considered.

Faraday's law is used to calculate the induced EMF in the ring, with a specific formulation considering the movement of the conductor and the magnetic field.

The cross product of velocity and the magnetic field is computed to find the induced EMF, focusing on the radial and axial components of the magnetic field.

The integral of the cross product over the loop is evaluated, resulting in a final expression for the induced EMF, which is proportional to the strength of the magnetic field and the radius of the loop.

The force on the magnet due to the induced current in the ring is analyzed using Newton's third law, emphasizing the opposing direction of the force.

The problem is recognized as extremely challenging, as indicated by the author's detailed walkthrough and the application of advanced mathematical techniques.

The problem highlights the importance of understanding both theoretical concepts and practical applications in physics, particularly in electromagnetism and mechanics.

Transcripts

Browse More Related Video

What happens if we DON'T ignore Air Resistance

HOW is the speed ZERO when rolling? Bulgarian Physics Olympiad

LAST Question from the British Physics Olympiad

Faradays Law in 3 minutes

Faraday's Law of Electromagnetic Induction, Magnetic Flux & Induced EMF - Physics & Electromagnetism

michael faraday | law of electromagnetic induction | faraday's law of induction

Related Tags

Physics Olympiad Magnetism Differential Equations Electromagnetism Faraday's Law Lenz's Law Advanced Physics Mathematical Modeling Projectile Motion STEM Education