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:

• μ = 2qr²ω
• L = (2/5)mr²ω

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.