The correct order in which the dimensions of ''time'' decreases in the following physical quantities is
(A) Stefan's constant
(B) Coeffecient of volume expansion
(C ) Work
(D) Velocity gradient

Options

A. D, B, C, A
B. B, D, C, A
C. A, B, C, D
D. B, C, D, A

Answer

To solve the problem of determining the correct order in which the dimensions of "time" decrease in the given physical quantities, we will analyze each option step by step.

### Step 1: Analyze Stefan's Constant (Option A)
Stefan's constant is given by the formula:
\[ \sigma = \frac{E}{t \cdot A \cdot T^4} \]
Where:
- \(E\) is energy (dimensions: \(M L^2 T^{-2}\))
- \(t\) is time (dimensions: \(T\))
- \(A\) is area (dimensions: \(L^2\))
- \(T\) is temperature (dimensions: \(K\))

Now, substituting the dimensions:
\[ \sigma = \frac{M L^2 T^{-2}}{T \cdot L^2 \cdot K^4} = M^1 L^0 T^{-3} K^{-4} \]
Thus, the dimensions of Stefan's constant are:
\[ [M^1 L^0 T^{-3} K^{-4}] \]
Here, the exponent of \(T\) is \(-3\).

### Step 2: Analyze Coefficient of Volume Expansion (Option B)
The coefficient of volume expansion is defined as:
\[ \beta = \frac{\Delta V / V_0}{\Delta T} \]
Where:
- \(\Delta V\) is the change in volume (dimensions: \(L^3\))
- \(V_0\) is the original volume (dimensions: \(L^3\))
- \(\Delta T\) is the change in temperature (dimensions: \(K\))

Thus, the dimensions are:
\[ \beta = \frac{L^3 / L^3}{K} = K^{-1} \]
So, the dimensions of the coefficient of volume expansion are:
\[ [M^0 L^0 T^0 K^{-1}] \]
Here, the exponent of \(T\) is \(0\).

### Step 3: Analyze Work (Option C)
Work is defined as:
\[ W = F \cdot d \]
Where:
- \(F\) is force (dimensions: \(M L T^{-2}\))
- \(d\) is displacement (dimensions: \(L\))

Thus, the dimensions of work are:
\[ W = M L T^{-2} \cdot L = M L^2 T^{-2} \]
Here, the exponent of \(T\) is \(-2\).

### Step 4: Analyze Velocity Gradient (Option D)
The velocity gradient is defined as:
\[ \text{Velocity Gradient} = \frac{\Delta v}{\Delta x} \]
Where:
- \(\Delta v\) is change in velocity (dimensions: \(L T^{-1}\))
- \(\Delta x\) is change in distance (dimensions: \(L\))

Thus, the dimensions are:
\[ \text{Velocity Gradient} = \frac{L T^{-1}}{L} = T^{-1} \]
So, the dimensions of the velocity gradient are:
\[ [M^0 L^0 T^{-1}] \]
Here, the exponent of \(T\) is \(-1\).

### Step 5: Summarize the Dimensions
Now we summarize the dimensions of \(T\) for each option:
- **Stefan's Constant (A)**: \(T^{-3}\)
- **Coefficient of Volume Expansion (B)**: \(T^{0}\)
- **Work (C)**: \(T^{-2}\)
- **Velocity Gradient (D)**: \(T^{-1}\)

### Step 6: Order the Dimensions
Now we can order the dimensions of time from highest to lowest:
1. Coefficient of Volume Expansion (B): \(T^{0}\)
2. Velocity Gradient (D): \(T^{-1}\)
3. Work (C): \(T^{-2}\)
4. Stefan's Constant (A): \(T^{-3}\)

### Final Answer
The correct order in which the dimensions of "time" decreases is:
**B, D, C, A** (or **Option 1: DBCA**)

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