Built on Facts

Many of my students had their final exams today. In the weeks leading up to the exam, we reviewed by doing homework problems and working final exams from previous years. Here’s one of the questions:

Protons of velocity 5×10⁶ m/s are moving perpendicular to electric and magnetic fields which are crossed at right angles to each other. (The velocity is perpendicular to both fields). The protons pass through the region of crossed fields undeflected. If the electric field strength is 15,000 V/m, what is the strength of the magnetic field?

It’s not a difficult question if you’ve taken an introductory physics class, so if you have, try it out before looking at the solution. Good luck!Solution: Since the proton is undeflected, the electric and magnetic fields must be producing equal and opposite forces. So what are those forces? There’s the magnetic force

where v is the velocity, B is the magnetic field, and q is of course the charge Then there’s the electric force

These forces must be equal, since there’s no deflection and thus no acceleration.

Canceling the q’s and solving for B gives

Which with the numbers given in our problem gives B = 0.003 Tesla, which is 30 Gauss. This is stronger than the Earth’s magnetic field by about a factor of 60, which is nevertheless not all that strong. It’s roughly the strength of a good refrigerator magnet.