Water in magnetic field · RMPH 2021 #D

Physics · electromagnetism

Problem

All situations take place in an enclosure thermostated at 70°C with air and saturated water vapor. The pressure is constant at p0=1.0×105 N m−2p_0 = 1.0 \times 10^5 \, \text{N m}^{-2}. Physical constants: g=9.8 m s−2g = 9.8 \, \text{m s}^{-2}, μ0=4×10−7 H m−1\mu_0 = 4 \times 10^{-7} \, \text{H m}^{-1}, ε0=8.9×10−12 F m−1\varepsilon_0 = 8.9 \times 10^{-12} \, \text{F m}^{-1} Water properties: Density ρ=1.0×103 kg⋅m−3\rho = 1.0 \times 10^3 \, \text{kg} \cdot \text{m}^{-3}, Saturated vapor pressure at 70°C ps=3.1×104 N⋅m−2p_s = 3.1 \times 10^4 \, \text{N} \cdot \text{m}^{-2}, Surface tension coefficient at 70°C σ=6.5×10−2 N⋅m−1\sigma = 6.5 \times 10^{-2} \, \text{N} \cdot \text{m}^{-1}, Magnetic susceptibility χ=−9.0×10−6\chi = -9.0 \times 10^{-6} Water is a diamagnetic substance. Its introduction into a magnetic field leads to the appearance of magnetic dipoles that are equivalent to magnets having the opposite polarity to that of the applied magnetic field. As a result, when in an inhomogeneous magnetic field, water tends to leave the place with a more intense magnetic field. A region of the vacuum is situated in the magnetic field with induction BB. The density of energy in that region has the value w0w_0. When there is water in that region, the density of energy is www_w. D.1. Write the expression Δw=ww−w0\Delta w = w_w - w_0 for the difference in energy densities as a function of BB, μ0\mu_0, μr\mu_r. The water in a glass vessel is subjected to the action of a non-uniform magnetic field with a vertical direction. The point MM is on the same vertical as the point NN, below it. The induction of the magnetic field has the modulus BB at the point MM and the modulus 00 at the point NN. Imagine that a small volume of water vv is moved from point MM to point NN and then brought back from point NN to point MM. D.2. Comparing the magnetic potential energy for the initial and final states of the water volume vv determines the expression of the pressure difference (due to the existence of the inhomogeneous magnetic field) between the points NN and MM. A vertical graduated cylinder of glass (a mensure) with length ℓ=4.0×10−1 m\ell = 4.0 \times 10^{-1} \, \text{m} is filled in three quarters with water. A solenoid is wound on its lower half. The solenoid generates inside it a uniformly vertical magnetic field having the induction modulus B=KIB = KI. In expression II is the intensity of the electric current flowing through the solenoid and K=5.0×10−2 T A−1K = 5.0 \times 10^{-2} \, \text{T A}^{-1} is a constant of device. It assumes that due to the scattering of field lines the magnetic field outside the solenoid is negligible. D.3. Determine the expression of intensity of the electrical current through the solenoid so that the water in the vessel would start to boil. Calculate the value of this intensity of the electrical current.

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