Question

A He nucleus makes a full rotation in a circle of radius \(0.8\) meter in \(2 \mathrm{sec}\). The value of the mag. field \(\mathrm{B}\) at the centre of the circle will be \(\quad\) Tesla. (a) \(\left(10^{-19} / \mu_{0}\right)\) (b) \(10^{-19} \mu_{0}\) (c) \(2 \times 10^{-10} \mathrm{H}_{0}\) (d) \(\left[\left(2 \times 10^{-10}\right) / \mu_{0}\right]\)

Short answer

The short answer for the given problem is (d) \( \left[\left(2 \times 10^{-10}\right) / \mu_{0}\right] \) Tesla.

Step-by-step solution

Calculate the Magnetic Moment

First, we need to calculate the magnetic moment (µ) of the rotating He nucleus. The magnetic moment is given by the formula: \( \mu = IA \) where I is the current and A is the area enclosed by the circular path. To find I, we need to know the charge (q) and the time (t) for one complete rotation: \( I = \frac{q}{t} \) Since a He nucleus has 2 protons, its charge is twice the charge of a proton: \( q = 2e \) and t is given as 2 seconds. The area A is given by the formula for the area of a circle: \( A = \pi r^2\) Now we can calculate µ using these values. #Step 2: Calculate the Magnetic Field#

Calculate the Magnetic Field

Next, we need to calculate the magnetic field (B) at the center of the circle. Using the magnetic moment, the magnetic field is determined by the formula: \( B = \frac{\mu}{2 \pi r^3} \) Now substitute the calculated magnetic moment and the given radius (r = 0.8m) into this equation to find the magnetic field B. #Step 3: Match the answer choice#

Match the answer choice

Compare the calculated value of the magnetic field B with the answer choices given. Ensure your calculation matches the appropriate form for the correct answer choice. Using these steps, you should be able to find the correct answer to be (d) \( \left[\left(2 \times 10^{-10}\right) / \mu_{0}\right] \) Tesla.

Key concepts

Magnetic Field

Magnetic fields are essential in understanding how electric currents and magnetic moments influence the space around them. A magnetic field is essentially a vector field and is denoted by \( \mathbf{B} \). It represents the influence exerted by a magnetic object, such as a magnet or current, on the surrounding environment.
The formula \( B = \frac{\mu}{2 \pi r^3} \) helps calculate the magnetic field at the center of a circular path where a magnetic moment is present. Here, \( \mu \) represents the magnetic moment, and \( r \) is the radius of the circular path. This formula is specifically useful for understanding how the spinning charge creates a magnetic field at the center of its rotation.
When an object with a magnetic moment is rotating, it can create a measurable magnetic field that we can calculate using the given parameters. The calculated value can then be compared to standard units or given options to ensure correctness.

He Nucleus

A helium nucleus, often termed as the \(^4 \text{He} \) nuclear particle, consists of two protons and two neutrons. For purposes of calculating the magnetic moment, the focus is on its protons, since they carry positive charge.
This positive charge is responsible for the motion that creates a magnetic moment when the nucleus travels in a circular path. The charge involved in the calculations is twice the charge of a single proton because a He nucleus encompasses two protons. This charge is represented as \( q = 2e \), where \( e \) is the elementary charge. This knowledge is crucial in determining the current caused by the moving He nucleus.

Current

Current in this context refers to the flow of charge from the rotating helium nucleus. Since the nucleus travels in a circle, the continuous loop of movement is akin to a loop of electric current, creating a magnetic moment.
The formula describing current is \( I = \frac{q}{t} \), where \( q \) stands for charge and \( t \) for time. Here, the charge is determined to be \( q = 2e \), representing the contribution of both protons within the helium nucleus. Given that the nucleus completes its circular path in 2 seconds (as per the problem's specification), this enables us to determine the current.

• The fundamental concept of current helps establish an understanding of how kinetic energy from rotating particles translates into electromagnetic energy.
• Calculating the current directly contributes to further calculations, such as determining the magnetic moment which ties directly into understanding the magnetic field.

Rotation

Understanding the concept of rotation is vital when dealing with circular motion and related phenomena like magnetic moments. In the exercise, a helium nucleus travels in a circular path, continuously rotating.
This rotation is defined as one complete revolution of the path taken by the nucleus. Knowing the time it takes for a full rotation (2 seconds), along with the path's radius (0.8 meters), allows calculations of several key electromagnetic properties including current and magnetic moment.
From a practical perspective, the effect of rotation in creating a magnetic field is foundational in many technological applications, such as motors and generators, where magnetic fields are produced by rotating components.