Exercise 22.36What force is exerted on the water in an MHD drive utilizing a 25.0-cm-diameter tube, if 100-A current is passed across the tube that is perpendicular to a 2.00-T magnetic field? (The relatively small size of this force indicates the need for very large currents and magnetic fields to make practical MHD drives.)
This is College Physics Answers with Shaun Dychko. We're going to calculate the food pressure created by this magneto hydro-dynamic drive. And so we have a tube with diameter 25 centimeters. And a current will flow from the top to the bottom of the tube, whereas, the magnetic field will flow from the left to the right. And so the field and current are right angles and so the force between them will be current times the length of this so called wire that's in the magnetic field multiplied by the field strength and there's no need to multiply by sine theta because theta is 90 here, in which case sine of 90 is one. Well this isn't really a wire here, this is the fluid. But the fluid has this length equal to the diameter of the pipe. And that's why I've substituted the letter d in place of <i>l</i> here. Now, the pressure will be the force divided by the cross-sectional area of the tube. And so that's <i>IdB</i> from here divided by the area which is Pi times diameter squared over four. And you can take the four to the numerator and cancel one of the diameters and you get four <i>IB</i> over Pi <i>d</i>. And so that's four times a hundred Amps times two Tesla divided by Pi times 25 times ten to the minus two meters, giving us 1020 Pascals. Now it turns out that even though we have a extraordinarily high current here, nevertheless, this is a very small pressure in comparison to atmospheric pressure. So this number divided by 1.013 times ten to the five Pascals atmospheric pressure is 0.01. And so that is not a significant fraction of atmospheric pressure.