A uniformly charged disc of radius r and having charge q rotates with constant angular velocity `omega`. The magnetic dipole moment of this disc is `(1)/(n)qomegar^(2)`. Find the value of n.

Answer

To find the value of \( n \) in the expression for the magnetic dipole moment of a uniformly charged rotating disc, we can follow these steps:

### Step 1: Understand the problem
We have a uniformly charged disc of radius \( r \) with total charge \( q \) rotating with a constant angular velocity \( \omega \). We need to find the magnetic dipole moment \( \mu \) and relate it to the given expression \( \frac{1}{n} q \omega r^2 \).

### Step 2: Recall the formula for magnetic dipole moment
The magnetic dipole moment \( \mu \) for a rotating charge distribution can be expressed in terms of angular momentum \( L \) and total charge \( q \):
\[ \mu = \frac{L \cdot q}{2m} \]
where \( m \) is the mass of the object.

### Step 3: Calculate the angular momentum \( L \)
The angular momentum \( L \) of the disc can be calculated using the moment of inertia \( I \) and the angular velocity \( \omega \):
\[ L = I \omega \]
For a disc, the moment of inertia \( I \) is given by:
\[ I = \frac{1}{2} m r^2 \]
Thus, substituting this into the expression for angular momentum:
\[ L = \left(\frac{1}{2} m r^2\right) \omega \]

### Step 4: Substitute \( L \) into the magnetic dipole moment formula
Now substituting \( L \) into the formula for \( \mu \):
\[ \mu = \frac{\left(\frac{1}{2} m r^2 \omega\right) q}{2m} \]
Simplifying this gives:
\[ \mu = \frac{q r^2 \omega}{4} \]

### Step 5: Compare with the given expression
We are given that:
\[ \mu = \frac{1}{n} q \omega r^2 \]
Now we can set the two expressions for \( \mu \) equal to each other:
\[ \frac{q r^2 \omega}{4} = \frac{1}{n} q \omega r^2 \]

### Step 6: Solve for \( n \)
Since \( q \), \( \omega \), and \( r^2 \) are common on both sides, we can cancel them out (assuming they are not zero):
\[ \frac{1}{4} = \frac{1}{n} \]
Taking the reciprocal gives:
\[ n = 4 \]

### Final Answer
Thus, the value of \( n \) is \( 4 \).

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