Magnetism
The strange thing about magnetism is that the magnetic field causes a force on a moving charge (unless it is moving in the field direction), and the direction of the force is perpendicular to both the field and the velocity. The other fields that you know about don't give a diddly whether the thing is moving, and the force is always in the direction of the field (or against the field). If gravity acted like magnetism on masses (in addition to its usual force), you would have trouble going in a straight line- it would push you sideways.
Another oddity is that although a ring magnet on a pencil will hover above another magnet, it cannot do it without the pencil. You cannot find a stationary combination of permanent magnets that will enable one to hover alone in equilibrium. There is a proof of this, Earnshaw’s theorem. It can be done with variable currents and feedback (maglev trains, for example).
Suppose charge q has velocity component vp perpendicular to
magnetic field B. Then the force F is proportional to qvpB, and the
field unit (Tesla) is defined to make F = qvpB. (This is like what
we do with
A simple application is a particle of mass m and charge q moving ^ to B with no other forces acting. Recall that mass times acceleration is mv2/r when force is ^ to velocity, so qvB = mv2/r. Divide both sides by v and solve for whatever.
Electric current I is moving charge, so it is easy to show that F = ILBp is the force on a straight wire of length L in a field which has component Bp perpendicular to the wire. If the wire is curved and/or it is in a non-uniform field, do it in little parts.
One way to explain the force on a moving charge is that it produces its own magnetic field that is proportional to qv. You know about magnets- like poles repel, unlike poles attract. The sketch below shows two magnets, and to keep it simple, one field line is drawn due to each magnet. (Note that magnetic field lines do not have a beginning and an end. Also note that the circular shape of the field was drawn for convenience. B is not really circular.) These magnets attract each other.
Positive charges moving toward you have magnetic field counterclockwise around them just like the magnets shown above. Hence they attract each other magnetically. The picture below shows one field line around + charges moving toward you or current toward you.
The force of attraction involves the field due to one at the location of the other. It turns out that the field due to a long straight wire is m oI/2p r, where the constant m o, known as the permeability of space, is 4p x 10-7 T m/A. This is a special case of Ampere's law, which in its simplest form is that if you follow the field around a closed path of distance L, then the average field times L = m o times the current intersecting the plane of your path on the inside of the closed path. For a long straight wire, the field has a circular pattern, so L is the circumference, 2p r. Actually, Ampere's law permits you to follow any closed path and use the component of B in the direction of motion. This will be illustrated in # 2 below.
A few other applications of Ampere's law:
1) The field inside a straight wire of circular cross section carrying constant
current density j (j= current/cross-sectional area): Imagine traveling around a
circular path of radius r inside the wire concentric to the wire's
cross-section. Be sure that you see that Ampere's law says that
B(2p r) = m ojp r2.
This is valid for any r less than or equal to the wire's radius. Show that
plugging in the wire's radius yields the same result as the formula in the
previous paragraph, so B is continuous.
2) The field inside a long uniformly wound coil with n turns per length:
Travel a distance L inside the coil parallel to the axis (the field direction).
Then complete a rectangle by going ^ to
the field, then outside the coil where the field is very small. When you have
completed the rectangle, you will have traveled around current nIL, and the
only time there was a significant field component in the direction of motion
was for the distance L inside the coil.
So BL = m onIL, or B = m onI. This is exact only for an
infinitely long coil. These are in rather short supply. For a long coil, the
equation is a good approximation for B at the center. You might find this more
useful: let N = total number of turns and L = length. Then n=N/L so B=m oNI/L. In truth,
B < m oNI/L, but just a
little less. (We need a symbol for that. << is much less, why not a
symbol for a little less?)
3) The field in a uniformly wound toroid (donut shape): Go in a circular path inside and show that B = m oNI/2p r. This is as exact as they get.
4) The field very close to a long current-carrying sheet of conductor such as
aluminum foil: travel around a small rectangle whose plane is ^ to the current direction so that you
enclose part of the current. If the current per width is l , it is easy to show that B is
m
o l /2.
These (above) are the basics- learn the stuff well.
Here is a question for you: The picture below shows a long conductor of rectangular cross-section with current toward you. The field around it would have some kind of oval shape. Would it be at all meaningful to choose a circular path around it and apply Ampere's law? Do it and get B = m oI/2p r. Is this valid?
Perhaps not useful, but it is the average component of B in the path direction.
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Comments, questions: fredrick.gram @tri-c.edu