Suppose the pressure at the surface of mercury in as barometer tube is `P_1` and the pressure at the surface of mercury in the cup is `P_2`.

Options

A. `P_1=0, P_2=` atgmospheric pressure
B. `P_1`= atmosphereic pressure `P_2=0`
C. `P_1=P_2=`atmospheric pressure
D. `P_1=P_2=0`

Answer

To solve the problem, we need to analyze the situation involving a barometer and the pressures at two different points: the surface of mercury in the barometer tube (P1) and the surface of mercury in the cup (P2).

### Step-by-Step Solution:

1. **Understanding the Barometer Setup**:
- A barometer consists of a tube filled with mercury, which is inverted and placed in a dish (or cup) containing mercury. The height of the mercury column in the tube (H) is determined by the atmospheric pressure.

2. **Pressure at the Surface of Mercury in the Barometer Tube (P1)**:
- At the top of the mercury column in the barometer tube, there is a vacuum. This means that there is no air pressure acting on the mercury at that point.
- Therefore, the pressure at the surface of mercury in the barometer tube (P1) is equal to 0.

\[ P_1 = 0 \, \text{(since it is a vacuum)} \]

3. **Pressure at the Surface of Mercury in the Cup (P2)**:
- The mercury in the cup is open to the atmosphere. Thus, the pressure at the surface of the mercury in the cup is equal to the atmospheric pressure (P_atmospheric).
- Therefore, the pressure at the surface of mercury in the cup (P2) is equal to the atmospheric pressure.

\[ P_2 = P_{\text{atmospheric}} \]

4. **Conclusion**:
- We have determined that:
- \( P_1 = 0 \)
- \( P_2 = P_{\text{atmospheric}} \)

### Final Answer:
- \( P_1 = 0 \)
- \( P_2 = P_{\text{atmospheric}} \)