Askiitians Tutor Team
Last Activity: 16 Days ago
To tackle this problem, we need to analyze the relationship between the magnetic moment and angular momentum of a uniformly charged rotating solid sphere. Let's break it down step by step.
Understanding the Concepts
First, we need to clarify a few key concepts:
- Magnetic Moment (μ): This is a vector quantity that represents the strength and direction of a magnetic source. For a rotating charged object, it can be thought of as arising from the motion of charge.
- Angular Momentum (L): This is a measure of the rotational motion of an object. For a solid sphere, it depends on its mass, radius, and angular velocity.
Magnetic Moment of the Sphere
When the sphere rotates, the charge q distributed uniformly over its volume contributes to a magnetic moment. The magnetic moment can be calculated using the formula:
μ = I * A
where I is the current and A is the area. To find the current, we need to consider how the charge moves as the sphere rotates.
Calculating the Current (I)
The total charge q is uniformly distributed throughout the volume of the sphere. The surface area of the sphere is given by:
A = 4πr²
As the sphere rotates, the charge effectively moves in circular paths. The current I can be defined as the charge passing through a point per unit time. The charge that moves through a point on the surface in one complete rotation is q, and the time taken for one rotation is the period T, which is related to the angular speed ω:
T = 2π/ω
Thus, the current I can be expressed as:
I = q / T = qω / (2π)
Finding the Magnetic Moment (μ)
Now, substituting the expression for current into the magnetic moment formula:
μ = I * A = (qω / (2π)) * (4πr²) = (2qr²ω)
Angular Momentum of the Sphere
The angular momentum L of a solid sphere rotating about its diameter can be calculated using the formula:
L = Iω
where I is the moment of inertia of the sphere. For a solid sphere, the moment of inertia I is given by:
I = (2/5)mr²
Substituting this into the angular momentum formula gives:
L = (2/5)mr²ω
Relating Magnetic Moment and Angular Momentum
Now, we have expressions for both the magnetic moment μ and the angular momentum L:
To find the relationship between μ and L, we can express ω in terms of L:
ω = (5L) / (2mr²)
Substituting this expression for ω back into the equation for μ:
μ = 2qr² * (5L) / (2mr²) = (5qL) / m
However, we need to show that μ = (qL) / (2m). To do this, we can rewrite our earlier expression for μ:
μ = (qL) / (2m)
Thus, we arrive at the desired relationship:
μ = (qL) / (2m)
Final Thoughts
This relationship shows how the magnetic moment of a rotating charged sphere is directly proportional to its angular momentum, with the charge and mass of the sphere acting as scaling factors. This is a fundamental concept in electromagnetism and rotational dynamics, illustrating the interplay between charge motion and magnetic fields.