Presentation: magnetic forces from fields

Look at those railguns. Just look at them. In the very capable hands of Arnold Schwarzenegger. Of course, with this being an Arnold Schwarzenegger movie, mayhem ensues. The best kind of mayhem. Movie mayhem.

While a previous presentation discussed the connection between magnets and magnetic fields, in this presentation we will consider magnetic fields exerting forces on things that aren't magnets: current-carrying wires and moving charges.

First, right-hand rules (or at least, the first right-hand rule).

There are three right-hand rules, with the two distinct hand shapes shown here (which are slightly different than shown in the textbook). The shape on the right was known to millennial UC-Davis students as "Crouching Tiger" (resembling a tiger claw) while the shape on the left was "Hidden Dragon" (resembling, uh, a dragon--do you see it? No? That's why it's "hidden"). Practice your right-hand rule drills. Hidden Dragon. Crouching Tiger. Hidden Dragon. Crouching Tiger. These will now be our physics gang signs, so you better represent.

We will be using only "Hidden Dragon," the first right-hand rule (RHR1) in this presentation, to determine the direction of a magnetic force F exerted by a magnetic field B. While it is not necessary to write these letters on your fingers, this will be only time you will be able to write on your hands for physics exams this semester, so enjoy this while you can.

Second, the magnitudes and directions of magnetic forces on currents and charges.

Instead of objects directly exerting magnetic forces on objects, in this two-step model, a source of a magnetic field such as a magnet or a current-carrying wire creates a magnetic B field everywhere around them, and it is this magnetic field that exerts a force on a test object, such as a magnet (as discussed in the previous presentation), or a current-carrying wire, or a moving charge.

Further discussion on the first step in this two-step process, specifically how a current-carrying wire creates a magnetic field around itself will be discussed in a subsequent presentation. So here we take the existence of a magnetic field for granted, and describe the second step in this two-step process, on how it exerts forces on current-carrying wires and moving charges.

So a magnetic field will exert a force on a current-carrying wire, where the magnitude is given by the amount of current I (in units of coulombs per second, or amps), the (straight) segment of wire L within this magnetic field (in units of meters), the strength of the magnetic field B (in units of teslas), and the sine of the angle θ measured between the magnetic field line and the direction of current. (Many of the applications we will consider this semester have current-carrying wires that are perpendicular to the magnetic field lines, in which case θ = 90°). The product of all these factors is the magnitude of the magnetic force on the current-carrying wire, so the compound units on the right-hand side of this equation (coulombs·meters·teslas per second) are equivalent to the units of force (newtons) on the left-hand side of the equation.

While the magnitude of the magnetic force on a current-carrying wire is straightforward to calculate, the direction of the magnetic force is determined by the first right-hand rule RHR1 ("Hidden Dragon").

Consider a working prototype of Arnold Schwarzenegger's railgun shown at the start of this presentation. (Video link: "railgun project.")

A metal projectile is free to slide along two high voltage "rails." Because it is in contact with both rails, current passes through the projectile itself.

In this perspective view, the right-to-left direction of current passing through the projectile is shown as a blue vector, while the downwards magnetic field lines are shown as green vectors. Using RHR1, with your thumb pointing along the blue current vector, and index finger pointing along the blue magnetic field lines, your third finger (if your "Hidden Dragon" is properly done) predicts that the magnetic force on the projectile points down along the rails, out towards you. (Using RHR1 here may be slightly awkward for your wrist--this is not unusual.) But as long as current continues to be conducted through the projectile, the magnetic force on it keeps accelerating it down between the rails.

In practice, this requires very large currents and very strong magnetic fields, and often results in melting the projectile (or worse yet, arc welding it in place to the rails). Stay tuned for further developments.

Instead of a magnetic field exerting a force on a current-carrying wire, a magnetic field can also exert a force on a individual moving charge, where the magnitude is given by the amount of charge q (in units of coulombs), the speed of the charge v (in units of meters per second), the strength of the magnetic field B (in units of teslas), and the sine of the angle θ measured between the magnetic field line and the direction of the charge's velocity. The product of all these factors is the magnitude of the magnetic force on the moving charge, so the compound units on the right-hand side of this equation (coulombs·meters·teslas per second) are again equivalent to the units of force (newtons) on the left-hand side of the equation.

While the magnitude of the magnetic force on a charge is straightforward to calculate, the direction of the magnetic force is determined by the first right-hand rule RHR1 ("Hidden Dragon"). RHR1 predicts the direction of the magnetic force on a positive moving charge, however, the magnetic force on a negative moving charge will be opposite that given by RHR1. We'll hear more on that later, from Shakira, of all people.

Have any of you held a magnet up to the screen of a old-school cathode-ray television? If you have never done this, or have suppressed that traumatic childhood memory, we'll sweep a strong magnet across the screen of a TV. (Note that unlike the bare black and white TV glass screen, the metal mask just behind a color TV glass screen will be permanently magnetized.) Normally electrons stream down the TV to hit the screen at certain places to build up an image, but the magnet's magnetic field exerts a force on these moving charges, deflecting their paths and distorting the image. As before, RHR1 ("Hidden Dragon") is used to find the specific direction of this force, and we'll systematically practice using it more in class. (Video link: "080913-1050522.")

Let's close this presentation with Shakira's use of the "Hidden Dragon" right-hand rule. But what is up with that left-hand rule? RHR1 is used to determine the force of a magnetic field on a moving positive particle, but you must reverse that direction from RHR1 if the moving charge is actually negative. However, for a negative charge you can use your left hand's thumb (for the velocity direction of the moving negative charge) and index finger (for the direction of the magnetic field lines) to determine the direction of the force on that negative charge, as given by your left middle finger. So use RHR1 for magnetic forces on positive charges, and use LHR1 for magnetic forces on negative charges.

Yes, Shakira is so down with magnetic forces from fields.