To establish the relationship between the magnetic moment \( y \) and the angular momentum \( l \) of a uniformly charged rotating solid sphere, we need to delve into the fundamental concepts of electromagnetism and rotational dynamics. Let's break this down step by step.

Understanding the System

We have a solid sphere with:

• Radius: \( r \)
• Mass: \( m \)
• Charge: \( q \) (uniformly distributed)
• Angular speed: \( x \) (rotating about a diameter)

Angular Momentum of the Sphere

The angular momentum \( l \) of a solid sphere rotating about an axis can be calculated using the formula:

l = I \cdot x

where \( I \) is the moment of inertia of the sphere. For a solid sphere, the moment of inertia is given by:

I = \frac{2}{5} m r^2

Substituting this into the angular momentum formula, we get:

l = \frac{2}{5} m r^2 \cdot x

Magnetic Moment of the Sphere

Next, we need to find the magnetic moment \( y \) generated by the rotating charge. The magnetic moment for a rotating charge can be expressed as:

y = \frac{1}{2} q v r

where \( v \) is the linear velocity of the charge at the surface of the sphere. The linear velocity \( v \) can be related to the angular speed \( x \) by:

v = r \cdot x

Substituting this into the magnetic moment formula gives:

y = \frac{1}{2} q (r \cdot x) r = \frac{1}{2} q r^2 x

Relating Magnetic Moment and Angular Momentum

Now we have expressions for both the angular momentum \( l \) and the magnetic moment \( y \). We can substitute the expression for \( l \) into the equation for \( y \).

From our earlier calculation, we have:

l = \frac{2}{5} m r^2 x

Rearranging this gives:

x = \frac{5l}{2mr^2}

Now, substituting this value of \( x \) back into the equation for \( y \):

y = \frac{1}{2} q r^2 \left(\frac{5l}{2mr^2}\right)

Upon simplifying, we find:

y = \frac{5ql}{4m}

Final Relationship

To express \( y \) in terms of \( l \), we can rearrange the equation to show the relationship:

y = \frac{ql}{2m}

This final equation demonstrates that the magnetic moment \( y \) is indeed proportional to the angular momentum \( l \) of the sphere, with the charge \( q \) and mass \( m \) acting as constants of proportionality.

In summary, we have derived the relationship between the magnetic moment and angular momentum of a uniformly charged rotating sphere through a systematic approach involving the definitions of angular momentum and magnetic moment, leading to the conclusion that \( y = \frac{ql}{2m} \).