Find the ratio of magnetic dipole moment and magnetic field at the centre of a disc. Radius of disc is `R` and it is rotating at constant angular speed o about its axis. The disc is insulating and uniformly charged

Explanation

To find the ratio of the magnetic dipole moment (M) to the magnetic field (B) at the center of a uniformly charged disc rotating with a constant angular speed (ω), we can follow these steps:

### Step 1: Define the Magnetic Dipole Moment (M)
The magnetic dipole moment (M) for a rotating charged disc can be expressed as:
\[ M = I \cdot A \]
where \( I \) is the current and \( A \) is the area of the disc.

### Step 2: Calculate the Area (A) of the Disc
The area \( A \) of a disc with radius \( R \) is given by:
\[ A = \pi R^2 \]

### Step 3: Determine the Current (I)
The current \( I \) can be defined in terms of the charge \( Q \) and the frequency \( f \) of rotation. The frequency \( f \) is related to the angular speed \( \omega \) by:
\[ f = \frac{\omega}{2\pi} \]
Thus, the current \( I \) can be expressed as:
\[ I = \frac{Q}{T} = Q \cdot f = Q \cdot \frac{\omega}{2\pi} \]

### Step 4: Substitute Current into the Magnetic Dipole Moment Equation
Substituting the expression for current into the equation for magnetic dipole moment:
\[ M = I \cdot A = \left(Q \cdot \frac{\omega}{2\pi}\right) \cdot (\pi R^2) \]
This simplifies to:
\[ M = \frac{Q \omega R^2}{2} \]

### Step 5: Calculate the Magnetic Field (B) at the Center of the Disc
The magnetic field \( B \) at the center of a rotating disc can be derived from the Biot-Savart law. For a small disc, the magnetic field \( dB \) can be expressed as:
\[ dB = \frac{\mu_0}{2} \cdot \frac{dI}{R} \]
where \( dI \) is the differential current.

### Step 6: Express the Differential Current (dI)
The differential current \( dI \) can be expressed in terms of the charge density:
\[ dI = \frac{dQ}{dt} = \frac{Q}{\pi R^2} \cdot 2\pi r \cdot f \]
where \( r \) is the radius from the center to the differential area.

### Step 7: Integrate to Find Total Magnetic Field
Integrating \( dB \) from 0 to \( R \):
\[ B = \int_0^R dB = \frac{\mu_0}{2} \cdot \int_0^R \frac{Q \cdot \frac{\omega}{2\pi}}{R} \cdot 2\pi r \, dr \]
This leads to:
\[ B = \frac{\mu_0 \omega Q}{4\pi R} \]

### Step 8: Find the Ratio \( \frac{M}{B} \)
Now, we can find the ratio of the magnetic dipole moment to the magnetic field:
\[ \frac{M}{B} = \frac{\frac{Q \omega R^2}{2}}{\frac{\mu_0 \omega Q}{4\pi R}} \]
This simplifies to:
\[ \frac{M}{B} = \frac{2\pi R^3}{\mu_0} \]

### Final Result
Thus, the ratio of the magnetic dipole moment to the magnetic field at the center of the disc is:
\[ \frac{M}{B} = \frac{\pi R^3}{2 \mu_0} \]

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