Question #42248

An incompressible, electrically conductive fluid of density $\rho = 1000 \text{ kg/m}^3$ and electrical conductivity $\sigma = 10^4 \text{ S/m}$ flows steadily through a horizontal rectangular pipe. The pipe narrows smoothly from a cross-sectional area $A_1 = 20 \text{ cm}^2$ to $A_2 = 10 \text{ cm}^2$. In the narrow section, a uniform vertical magnetic field $B = 0.5 \text{ T}$ is applied across the fluid over a length $L = 2 \text{ m}$. The side walls of the narrow section are perfectly conducting and short-circuited, allowing a uniform transverse induced current. If the fluid velocity in the wide section is $v_1 = 2 \text{ m/s}$, find the total pressure drop $\Delta P = P_1 - P_2$ between the wide section and the exit of the magnetic field region. (Assume the fluid is otherwise inviscid)