The Dimensional formula of magnetic energy density is ?

Options

A. `MLT ^(-2) A ^(-1)`
B. `ML T^(-2) A ^(-1)`
C. `ML ^(2) T ^(-3) A ^(-2)`
D. `ML ^(-1) T ^(-2)`

Explanation

To find the dimensional formula of magnetic energy density, we can follow these steps:

### Step 1: Understand the Concept of Magnetic Energy Density
Magnetic energy density is defined as the amount of magnetic energy stored in a unit volume. Therefore, it can be expressed mathematically as:
\[ \text{Magnetic Energy Density} = \frac{\text{Magnetic Energy}}{\text{Volume}} \]

### Step 2: Determine the Dimensional Formula of Magnetic Energy
Magnetic energy is a form of energy, and the dimensional formula for energy is the same as that for work. The dimensional formula for work (or energy) is given by:
\[ \text{Energy} = \text{Force} \times \text{Displacement} \]
The dimensional formula for force is:
\[ \text{Force} = \text{Mass} \times \text{Acceleration} = MLT^{-2} \]
Thus, the dimensional formula for energy becomes:
\[ \text{Energy} = \text{Force} \times \text{Displacement} = MLT^{-2} \times L = ML^2T^{-2} \]

### Step 3: Determine the Dimensional Formula of Volume
The volume of an object is given by:
\[ \text{Volume} = \text{Length}^3 = L^3 \]

### Step 4: Combine the Formulas to Find Magnetic Energy Density
Now, substituting the dimensional formula for magnetic energy and volume into the equation for magnetic energy density, we have:
\[ \text{Magnetic Energy Density} = \frac{ML^2T^{-2}}{L^3} \]
This simplifies to:
\[ \text{Magnetic Energy Density} = ML^{-1}T^{-2} \]

### Step 5: Final Result
Thus, the dimensional formula of magnetic energy density is:
\[ \text{Dimensional Formula} = ML^{-1}T^{-2} \]

### Conclusion
From the given options, the correct answer is:
**Option 4: \( ML^{-1}T^{-2} \)**

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