Just The Facts: Magnetism for AP Physics B

I'm running an AP Physics B prep session in Lynchburg, Virginia in a couple of weeks.  I've been asked to discuss magnetism.

Of course I'll bring my demonstrations and practice questions.  But these will be merely confusing without students knowing the background facts.  So, I'll hand out a "just the facts" sheet to everyone for their reference during the demos and practice questions.  Afterwards, they can use this sheet for their own review.

What do you need to know about magnetism for the AP B exam?

I.                  Magnetic force on a charge or wire

The amount of force provided by a magnetic field on a charge is qvBsinq.

·         q is the charge of the particle in the field, measured in coulombs

·         v is the speed of the particle

·         B is the magnetic field, measured in tesla

·         q is the angle between the velocity and magnetic field, usually 90^o

The amount of force by a magnetic field on a current-carrying wire is ILBsinq.

·         same definitions as above for B, q

·         I is the amount of current in the wire, measured in amps

·         L is the length of the wire inside the magnetic field

The direction of force provided by a magnetic field is given by the first right hand rule

·         Do NOT plug in negative signs to qvBsinq.

·         Point toward v (or I), curl fingers toward B, thumb is direction of force on a positive charge or a current-carrying wire

·         (flip the direction for a negative charge)

·         Magnetic force is always perpendicular to velocity; this generally gives circular motion for a charged particle in the magnetic field.

II.               What can produce a magnetic field?

A bar magnet can produce a magnetic field.

·         Its field points out of the north end, and into the south end.

A current carrying wire can produce a magnetic field.

·         Its field wraps around the wire.  To find which way the field wraps, point right thumb with current and curl fingers.

·         The magnitude of the field produced by a straight wire is (μ₀/2π)(I/d).

·         Here I represents the current creating the field, and d is the distance from the wire.

III.           A changing magnetic flux can produce a voltage

Magnetic flux through a loop of wire, F, is defined as BA.

·         B is the amount of magnetic field

·         A is the area of the wire loop through which the magnetic field directly penetrates

·         The units of magnetic flux are T∙m².

·         If the magnetic field is not straight through the wire loop, only use the component of the field that is straight through the loop.

Changing flux produces a voltage

·         This voltage is referred to as “induced emf,” e.

·         The equation for the amount of voltage induced is n ΔΦ/Δt .

·         n represents the number of wire loops.  Dt represents the time it took to change the flux.

·         The induced current in a wire of known resistance can be found using V = IR.

The direction of induced current is given by Lenz’s Law

·         Current must flow through a wire – thus, only two directions are even possible for an induced current.

·         Point right thumb in the direction of the magnetic field.

o   (If flux is increasing rather than decreasing, flip your thumb the other way.)

·         Curl your fingers; this is the direction of the induced current.

Special case: moving rectangular wire entering or leaving a magnetic field

·         The induced voltage in this special case is e = BLv

·         Here, B is the magnetic field, v is the speed of the wire

·         L represents the side of the rectangle that always stays completely in the field, not the side of the rectangle that is entering or leaving the field

·         If the wire isn’t entering or leaving the field, the induced voltage is zero because flux doesn’t change.

Induced EMF caused by a falling magnet, and a qualitative demonstration

Helmholtz coils of the type useful for demonstrating
the answer to today's question.

I've seen the multiple-choice question from at least two different sources:  a bar magnet is dropped through a circular wire coil from above.  The north end of the magnet points down.  What will a graph of induced EMF (or current) in the coil vs. time look like?

Answer:

This question requires use of Lenz's Law.  I teach an approach to Lenz's law described here.  Let's consider first the time while the magnet approaches the wire loop, and then separately the time when the magnet recedes toward the ground.

As the magnet approaches, the magnetic field is down toward the ground because the B field due to a bar magnet points out of the north end.  The magnetic flux is increasing because the bar magnet approaches the coil, increasing the magnetic field at the location of the coil.  Becuase flux increases, I flip my right thumb opposite the magnetic field, and curl my fingers -- the current in the coil is counter-clockwise when viewed from above.

After the magnet passes through the coil and is receding, the magnetic field still points down toward the ground!  This is because the B field due to a bar magnet points into the south end.  The magnetic flux is now decreasing because the bar magnet recedes from the coil, producing an ever-smaller magnetic field at the location of the coil.  Becuase the flux decreases, I keep my right thumb pointing in the direction of the magnetic field, curl my fingers, and find the current in the coil is the other way -- clockwise as viewed from above.

I asked my AP classes this question as a "check your neighbor" exercise today in class.  At first, when working alone, most of the class (~2/3) didn't recognize that the current switches direction when the magnet passes through the coil.  Upon discussion, I found that the key misconception was the direction of the magnetic field when the magnet receded.  Either they thought that the magnetic field must have a different direction near the other pole, or they thought that since the field was decreasing, the field itself must switch directions.  Good arguments among the class changed some views, such that after discussion about 2/3 of the class had the correct answer.

Setting up the demonstration:

Long time blog readers know that I don't think it good enough to explain something like this conceptually and/or mathematically without doing the experiment.  Nature is the ultimate arbiter of arguments.  So I set up the problem.

I used a 10-cm diameter wire coil with a gazillion loops.  I've on other occasions used a bigger diameter set of coils scavenged from a 20 year old broken q/m device.  You'll get the best results if you use a coil with enormous numbers of loops, like one designed to produce strong, uniform magnetic fields -- a Helmholtz coil, like those shown at the top of the post.

I hooked a Vernier voltage probe to my labpro, and set the labpro to take 1000 data points per second for 2 seconds.  A student clicked "collect;"  When data collection began, I dropped a small (~5 cm length) bar magnet through the coil.

A bit of zooming produced the graph shown above.  As expected, the voltage switched polarity (i.e. the current switched directions) as the magnet passed through the coil. 

Though it's not brutally obvious, the right-hand hump is not as wide on the graph as the left-hand hump.  Why not?  Because the magnet speeds up in free fall, and so recedes from the coil faster than it approached the coil.  I dropped the magnet from about shoulder height.  The ~0.10 s duration of the voltage spikes are substantially less than the ~0.5 s time I calculate for the magnet to fall to the ground, as it should be. 

Any more follow up thoughts?  This demonstration can produce a wealth of interesting calculations and experiments suitable for an end-of-year independent laboratory investigation.