Water flows through a pipe of radius 4 cm at a rate of 3.5 m/s. Suppose each water molecule is singly ionized (highly unrealistic). What is the electric current density associated with the flow of these ions?
What is the electric current in the pipe?
To find the electric current density associated with the flow of ions, you need to calculate the charge per unit area. Here's how you can do it:
1. Start by finding the number of ions passing through a cross-sectional area of the pipe per unit time.
Since water is incompressible, the volume of water passing through the pipe per unit time is constant. The volume flow rate (Q) is given by the formula Q = A * v, where A is the cross-sectional area of the pipe and v is the velocity of the water flow. In this case, the cross-sectional area is the area of a circle given by A = π * r^2, where r is the radius of the pipe, which is 4 cm or 0.04 m. The velocity of water flow is 3.5 m/s. Therefore, the volume flow rate Q = (π * 0.04^2) * 3.5.
2. Next, determine the number of water molecules in the volume flow rate.
The molar mass of water is approximately 18 g/mol. The density of water is roughly 1000 kg/m^3. Using these values, you can calculate the number of water molecules passing through the pipe per unit time. Convert the mass flow rate to moles by dividing the mass flow rate by the molar mass (Q in kg/s divided by 0.018 kg/mol). Then, calculate the number of moles using Avogadro's number (6.022 x 10^23 molecules/mol).
3. Now, take into account that each water molecule is singly ionized, meaning it carries one elementary charge (1.6 x 10^(-19) C). Multiply the number of moles calculated in the previous step by Avogadro's number and the elementary charge to find the total charge passing through the pipe per unit time.
4. Finally, divide the charge per unit time by the cross-sectional area of the pipe to find the electric current density. The unit of electric current density is amperes per square meter (A/m^2).
To find the electric current in the pipe, multiply the current density by the cross-sectional area of the pipe.
Keep in mind that this exercise assumes each water molecule is singly ionized, which is highly unrealistic. Additionally, the calculation ignores other factors that can affect the flow of ions, such as ion mobility and the presence of ions other than water.