Which of the following physical quantities have the same dimensions?
(1) Energy density (2) Refractive index
(3) Magnetic induction (4) Pressure

Options

A. 1 and 2
B. 2 and 3
C. 1 and 4
D. 1 and 3

Explanation

To determine which of the given physical quantities have the same dimensions, we will analyze each quantity step by step.

### Step 1: Energy Density
Energy density is defined as energy per unit volume.

- **Energy (E)** has the dimension of work, which is given by:
\[ [E] = [W] = [F \cdot d] = [M \cdot L \cdot T^{-2} \cdot L] = [M \cdot L^2 \cdot T^{-2}] \]
- **Volume (V)** has the dimension:
\[ [V] = [L^3] \]
- Therefore, the dimension of energy density is:
\[ [\text{Energy Density}] = \frac{[E]}{[V]} = \frac{[M \cdot L^2 \cdot T^{-2}]}{[L^3]} = [M \cdot L^{-1} \cdot T^{-2}] \]

### Step 2: Refractive Index
The refractive index (n) is defined as the ratio of the speed of light in vacuum to the speed of light in a medium.

- Since it is a ratio of two speeds, it is dimensionless:
\[ [n] = [M^0 \cdot L^0 \cdot T^0] \]

### Step 3: Magnetic Induction
Magnetic induction (B) can be derived from the formula relating force (F), current (I), and length (L):
\[ B = \frac{F}{I \cdot L} \]
- The dimension of force (F) is:
\[ [F] = [M \cdot L \cdot T^{-2}] \]
- The dimension of current (I) is:
\[ [I] = [A] \]
- The dimension of length (L) is:
\[ [L] = [L] \]
- Therefore, the dimension of magnetic induction is:
\[ [B] = \frac{[F]}{[I] \cdot [L]} = \frac{[M \cdot L \cdot T^{-2}]}{[A] \cdot [L]} = [M \cdot L^0 \cdot T^{-2} \cdot A^{-1}] \]

### Step 4: Pressure
Pressure (P) is defined as force per unit area.

- The dimension of area (A) is:
\[ [A] = [L^2] \]
- Therefore, the dimension of pressure is:
\[ [P] = \frac{[F]}{[A]} = \frac{[M \cdot L \cdot T^{-2}]}{[L^2]} = [M \cdot L^{-1} \cdot T^{-2}] \]

### Conclusion
Now we summarize the dimensions we have calculated:
1. Energy Density: \([M \cdot L^{-1} \cdot T^{-2}]\)
2. Refractive Index: \([M^0 \cdot L^0 \cdot T^0]\) (dimensionless)
3. Magnetic Induction: \([M \cdot L^0 \cdot T^{-2} \cdot A^{-1}]\)
4. Pressure: \([M \cdot L^{-1} \cdot T^{-2}]\)

From the above analysis, we can see that:
- Energy Density and Pressure have the same dimensions: \([M \cdot L^{-1} \cdot T^{-2}]\).

Thus, the correct answer is that Energy Density (1) and Pressure (4) have the same dimensions.

### Final Answer
The quantities that have the same dimensions are:
**(1) Energy Density and (4) Pressure.**