Problem 50 - Fundamentals of Physics

Question

An electron moves in a circle of radius \(r=5.29 \times 10^{-11} \mathrm{~m}\) with speed \(2.19 \times 10^{6} \mathrm{~m} / \mathrm{s} .\) Treat the circular path as a current loop with a constant current equal to the ratio of the electron's charge magnitude to the period of the motion. If the circle lies in a uniform magnetic field of magnitude \(B=7.10 \mathrm{mT}\), what is the maximum possible magnitude of the torque produced on the loop by the field?

Step-by-Step Solution

Answer

Maximum torque is \( \tau = 9.27 \times 10^{-24} \; \text{Nm} \).

Step 1: Calculate the Period of Electron's Motion

The period of the motion is the time it takes for the electron to complete one full circle. Use the formula for the period of circular motion: \( T = \frac{2\pi r}{v} \), where \( r \) is the radius and \( v \) is the speed. Substitute the given values: \( r = 5.29 \times 10^{-11} \; \text{m} \) and \( v = 2.19 \times 10^6 \; \text{m/s} \). Calculate \( T \).

Step 2: Calculate the Current Produced by the Electron

The current \( I \) created by the moving electron is the charge divided by the period. Use \( I = \frac{e}{T} \), where \( e = 1.60 \times 10^{-19} \; \text{C} \) is the charge of the electron, and \( T \) is the period calculated in Step 1.

Step 3: Determine the Magnetic Moment of the Loop

The magnetic moment \( \mu \) of a current-carrying loop is given by \( \mu = I \cdot A \), where \( I \) is the current and \( A = \pi r^2 \) is the area of the loop. Use the radius from the problem statement: \( r = 5.29 \times 10^{-11} \; \text{m} \). Calculate \( \mu \).

Step 4: Calculate the Maximum Torque on the Loop

The maximum torque \( \tau \) on a current loop in a uniform magnetic field occurs when the magnetic moment is perpendicular to the magnetic field. It is given by \( \tau = \mu B \), where \( \mu \) is the magnetic moment calculated in Step 3 and \( B = 7.10 \; \text{mT} = 7.10 \times 10^{-3} \; \text{T} \). Calculate \( \tau \).

Key Concepts

Electron Motion

When we think of electron motion, it's like imagining a tiny particle darting around in a loop, in this case, traveling in a circular path. Electrons are fundamental particles with both charge and mass, and their movement gives rise to electric currents.
In this exercise, the electron moves in a circle with a radius of \(5.29 \times 10^{-11}\) meters. The speed is \(2.19 \times 10^6 \) meters per second. This high speed is common for electrons due to their small mass and charge, allowing them to zip around quickly.
Understanding electron motion is crucial here because their circular path behaves like a loop of current, creating an interaction with magnetic fields. This interaction is key in understanding how fundamental physics concepts apply to real-world scenarios.

Current Loop

A current loop is essentially a path along which an electric current flows. Imagine a single wire bent into a circle and when a current flows through, it creates a current loop. In this exercise, the electron's motion is imagined as a current loop.
With electrons moving, a current is generated. This current is calculated by dividing the charge of the electron by the period of its circular motion. Here, the electron acts like a tiny current generator as it completes its circle.
An important thing to note is that any loop of current will create a magnetic field around it, akin to having a miniature magnet. This is why understanding the current loop is vital – it frames how magnetism is generated in our scenario.

Magnetic Moment

The concept of a magnetic moment is when any current loop generates its magnetic dipole. It is a measure of the strength and direction of the magnetic field created by the current loop.
In this exercise, the magnetic moment \( \mu \) is determined by the product of the current and the area of the loop. Specifically, it is expressed as \( \mu = I \cdot A \), where \( I \) is the current we calculated from the electron motion, and \( A \) is the area of the loop being \( \pi r^2 \).
The magnetic moment is an essential concept because it describes quantitatively how effectively a loop can produce a torque under the influence of an external magnetic field. It's like measuring how much twist a force can impart on this loop.

Uniform Magnetic Field

A uniform magnetic field is one where the magnetic field strength and direction are constant at every point. Imagine it like an invisible grid affecting anything with magnetic properties.
In this problem, the given magnetic field has a strength of \(7.10\) milliTesla, which is converted to the standard using \(7.10 \times 10^{-3}\) Tesla. It is uniformly acting across the plane of the electron’s loop.
The interaction between the current loop's magnetic moment and the uniform magnetic field is what creates torque. For a current loop, when its magnetic moment aligns perpendicular to such a field, the torque reaches its maximum value. This exercise shows how magnetic properties of simple systems like electron motion in a circular path can lead to significant interactions with magnetic fields.