The dimensions of Planck's constant are

Options

A. (a)`ML^(2)T^(-1)`
B. (b)`ML^(2)T^(2)`
C. (c)`M^(-1)L^(3)T^(-2)`
D. (d)`M^(-3)L^(1)T^(-2)`

Answer

To find the dimensions of Planck's constant, we can follow these steps:

### Step 1: Understand the relationship involving Planck's constant
We start with the relationship between wavelength (λ), Planck's constant (h), mass (m), and velocity (v). The formula is given by:

\[ \lambda = \frac{h}{mv} \]

### Step 2: Rearranging the formula
From the above equation, we can express Planck's constant (h) as:

\[ h = \lambda \cdot mv \]

### Step 3: Identify the dimensions of each component
- **Wavelength (λ)** has dimensions of length, which is denoted as [L].
- **Mass (m)** has dimensions of mass, denoted as [M].
- **Velocity (v)** has dimensions of length per time, denoted as [LT^{-1}].

### Step 4: Substitute the dimensions into the equation
Now, substituting the dimensions into the equation for Planck's constant:

\[ h = \lambda \cdot mv \]

This can be expressed in terms of dimensions as:

\[ [h] = [L] \cdot [M][LT^{-1}] \]

### Step 5: Simplify the dimensions
Now, we can simplify the expression:

\[ [h] = [L] \cdot [M] \cdot [L][T^{-1}] = [M][L^2][T^{-1}] \]

### Step 6: Write the final dimensional formula
Thus, the dimensions of Planck's constant (h) can be expressed as:

\[ [h] = [M^1 L^2 T^{-1}] \]

### Step 7: Conclusion
The final answer for the dimensions of Planck's constant is:

\[ M^1 L^2 T^{-1} \]