If M is the magnetic moment, B is the magnetic induction and T is the time then `MBT^(2)` has the dimensions of

Options

A. Intensity of magnetization
B. Intensity of magnetic field
C. Moment of Inertia
D. Magnetic permeability

Answer

To solve the problem of finding the dimensions of the expression \( MBT^2 \), where \( M \) is the magnetic moment, \( B \) is the magnetic induction, and \( T \) is the time, we will follow these steps:

### Step 1: Identify the dimensions of each quantity
1. **Magnetic Moment (M)**: The unit of magnetic moment is ampere meter squared (A·m²).
- In terms of dimensions, we have:
\[ [M] = [I][L^2] = M^0 L^2 T^0 I^1 \]
where \( I \) is electric current, \( L \) is length, and \( T \) is time.

2. **Magnetic Induction (B)**: The unit of magnetic induction is Tesla (T), which can also be expressed as:
\[ [B] = \frac{kg}{A \cdot s^2} \]
- In terms of dimensions, we have:
\[ [B] = [M][L][T^{-2}][I^{-1}] = M^1 L^1 T^{-2} I^{-1} \]

3. **Time (T)**: The dimension of time is simply:
\[ [T] = T^1 \]

### Step 2: Combine the dimensions
Now we can combine the dimensions of \( M \), \( B \), and \( T^2 \):
\[ MBT^2 = [M][B][T^2] \]
Substituting the dimensions we found:
\[ MBT^2 = (M^0 L^2 T^0 I^1) \cdot (M^1 L^1 T^{-2} I^{-1}) \cdot (T^2) \]

### Step 3: Simplify the expression
Now, we will simplify the combined dimensions:
\[ MBT^2 = M^{0+1} L^{2+1} T^{0-2+2} I^{1-1} \]
This simplifies to:
\[ MBT^2 = M^1 L^3 T^0 I^0 \]
Thus, the dimensions of \( MBT^2 \) are:
\[ [M][L^3][T^0] \]

### Step 4: Identify the physical quantity
The dimensions \( M^1 L^2 T^0 \) correspond to the dimensions of moment of inertia, which is given by:
\[ \text{Moment of Inertia} = M L^2 \]

### Conclusion
Thus, the expression \( MBT^2 \) has the dimensions of moment of inertia.

### Final Answer
The dimensions of \( MBT^2 \) are \( M^1 L^2 T^0 \), which corresponds to the moment of inertia.

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