Consider a person taking a bath in a bathtub who is under death sentence.Consider the density of salinity in blood to be ∂_b and the maximum permissible pressure in the blood vessels to be ‘P’.The water can hold a density of salinity of max ∂_w .The Capital Punishment is to be carried out by elecrifying the water.

1. For a DC supply to the water is ‘V’.Considering the water salinity of ∂_w .Find a relation for Potential ‘V’ at which the blood vessels just ruptures.
2. Find the relation for the same effect when the AC supply is used.
3. From the point of application of the potential to the water in 1.Find the time at which the blood vessels rupture.
4. When an AC supply is given to the water ’V’.Considering the water salinity of ∂_w.Find the time of which the blood vessels ruptures.

To tackle this complex scenario, we need to break it down into manageable parts, focusing on the physics of electric fields, pressure, and the effects of salinity on blood density. We will derive the relationships for both DC and AC supplies and explore the time it takes for blood vessels to rupture under these conditions.

Understanding the Basics

First, let's clarify some key concepts. The density of salinity in blood, denoted as ∂b, affects how the blood behaves under pressure. The maximum permissible pressure in the blood vessels is represented by 'P'. The water's salinity density is given as ∂w. When an electric potential is applied to the water, it creates an electric field that can affect the blood vessels.

Electric Potential and Pressure Relationship

When a DC voltage 'V' is applied to the water, it generates an electric field that can exert pressure on the blood vessels. The relationship between the electric potential and the pressure can be derived from the concept of electrohydrodynamics. The pressure exerted by the electric field can be expressed as:

• P = k * E

Here, 'k' is a constant that depends on the properties of the medium (in this case, the water and blood), and 'E' is the electric field strength, which can be calculated from the voltage and the distance between electrodes. The electric field 'E' can be expressed as:

• E = V/d

Substituting this into the pressure equation gives:

• P = k * (V/d)

Finding the Critical Voltage for Rupture

To find the critical voltage at which the blood vessels rupture, we set the pressure equal to the maximum permissible pressure:

• P = P_max

Thus, we have:

• P_max = k * (V/d)

Rearranging this gives us the voltage:

• V = (P_max * d) / k

AC Supply Considerations

When using an AC supply, the situation changes slightly due to the alternating nature of the current. The effective voltage (or RMS voltage) can be used to represent the AC supply. The relationship for the pressure remains similar, but we must consider the frequency of the AC current, which affects how the electric field interacts with the blood.

AC Voltage and Pressure

For an AC voltage 'V_ac', the effective voltage can be expressed as:

• V_ac = V_peak / √2

Substituting this into our pressure equation gives:

• P_max = k * (V_ac/d)

Thus, the critical voltage for AC becomes:

• V_ac = (P_max * d) / k

Time Until Rupture

The time until the blood vessels rupture under both DC and AC conditions can be influenced by several factors, including the rate of pressure increase and the physical properties of the blood vessels. For a DC supply, the pressure builds steadily until it reaches the rupture point. The time 't_dc' can be modeled as:

• t_dc = (V / R) * C

Where 'R' is the resistance of the water and 'C' is the capacitance of the system. For AC, the time until rupture can be more complex due to the oscillating nature of the current. The effective time 't_ac' can be approximated as:

• t_ac = (V_ac / R) * C

Conclusion

In summary, we derived the relationships for the critical voltage at which blood vessels rupture under both DC and AC conditions, as well as the time until rupture occurs. The key takeaway is that the pressure exerted by the electric field, influenced by the salinity of the water and the properties of the blood, plays a crucial role in determining the outcome. Understanding these relationships can help in assessing the risks associated with such scenarios.