Magnets and Magnetic Fields Magnetism Magnets and Magnetic Fields We studied electric forces and electric fields yesterday… Now we study magnetism and magnetic fields. Rather than defining magnetism, we begin by discussing properties of magnetic materials. Recall how there are two kinds of charge (+ and -), and likes repel, opposites attract.

Similarly, there are two kinds of magnetic poles (North and South), and like poles repel, opposites attract.* S N S N Thanks to Dr. Waddill for the nice pictures! Attract S N S N Repel S N Repel Attract *Recall also that I have a mental defect which often causes me to say “likes attract and unlikes repel” when I mean the opposite. I am not to be penalized for a mental defect!

There is one difference between magnetism and electricity: it is possible to have isolated + or – electric charges, but isolated N and S poles have never been observed. - + S N I.E., every magnet has BOTH a N and a S pole, no how many times you “chop it up.” S N = S N + S N

The earth has associated with it a magnetic field, with poles near the geographic poles. The pole of a magnet attracted to the earth’s north geographic pole is the magnet’s North pole. N The pole of a magnet attracted to the earth’s south geographic pole is the magnet’s South pole. S http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magearth.html Just as we used the electric field to help us “explain” and visualize electric forces in space, we use the magnetic field to help us “explain” and visualize magnetic forces in space.

Magnetic field lines are tangent to the magnetic field. Magnetic field lines point in the same direction that the north pole of a compass would point. Magnetic field lines are tangent to the magnetic field. The more magnetic field lines in a region in space, the stronger the magnetic field. Outside the magnet, magnetic field lines point away from N poles (*why?). Huh? Nooooooo…. *The N pole of a compass would “want to get to” the S pole of the magnet.

Is the earth’s north pole a magnetic N or a magnetic S? It has to be a S, otherwise, the compass N would not point to it. Unless the N of a compass needle is really S. Dang! This is too much for me! Yup, it’s confusing. Here’s a “picture” of the magnetic field of a bar magnet, using iron filings to map out the field. The magnetic field ought to “remind” you of the earth’s field. Later I’ll give a better definition for magnetic field direction.

Here’s what the magnetic field looks like when you put unlike or like poles next to each other. The magnetic field B is a vector which points in the direction of magnetic field lines. We will quantify the magnitude of B later.

Electric Current Produces Magnetism An electric current produces a magnetic field.* The direction of the current is given by the right-hand rule. Grasp the current-carrying wire in your right hand, with your thumb pointing in the direction of the current. Curl your fingers around the wire. Your fingers indicate the direction of the magnetic field. *Experimentally observed, then demanded by theory as a logical consequence of Maxwell’s equations.

Field comes out of page here. “Turns around” and goes into page here. Picture on previous page is from http://physics.mtsu.edu/~phys232/Lectures/ L12-L16/L17/Current_Loops/current_loops.html This picture also illustrates the magnetic field due to a current-carrying loop of wire. These symbols mean “out of page” and “into page.” See next section. Field comes out of page here. “Turns around” and goes into page here.

Here’s a simpler case: the magnetic field due to a straight wire. Field comes out of page here. Thanks again to Dr. Waddill for the nice pictures! Field turns around and goes into page here. If the wire is grasped in the right hand with the thumb pointing in the direction of the current, the fingers will curl in the direction of B.

Force on an Electric Current in a Magnetic Field; Definition of B As seen above, an electric current gives rise to a magnetic field, which must exert a force on a magnet. Does a magnet exert a force on a current-carrying wire? (Newton’s 3rd Law says it should.) Yes—a current-carrying wire in a magnetic field “feels” a force. The direction is given by the right-hand rule: Point your outstretched fingers in the direction of the current. Bend your fingers 90º and orient your hand to point the bent fingers in the direction of the magnetic field. Your thumb points in the direction of the force.

I’ll demonstrate another right-hand rule in class. You may need to re-orient your hand as you go through this procedure. During exams, I see all sorts of gyrations as students try to figure out directions. I’ll demonstrate another right-hand rule in class. Here is a web “physics toy” to help you visualize the force on a current-carrying conductor. (Select Lorentz Force.) Below is another picture to help you visualize. It came from http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/forwir2.html. The web page even has a built-in calculator that gives you numerical answers to your force problems!

The force is perpendicular to both the current and the magnetic field. You would expect the magnitude of the force to depend on the magnitudes of the magnetic field and the current. In fact, it does. The force also depends on how much of the wire is in the magnetic field. If the direction of the current is perpendicular to the magnetic field, then F = I ℓ B. I’ll write the lowercase l in italics (ℓ) to help you distinguish it from the number 1.

If the current and magnetic field are not perpendicular, the force is given by OSE: F = I ℓ B sin , where  is the angle between the current vector and the magnetic field vector. (Smallest angle from the current vector to the magnetic field vector.) I ℓ  B

The right-hand rule and the equation above actually serve as the definition of the magnetic field B. The SI unit for magnetic field is the tesla: 1 T = 1 N / (1 A · 1 m). A 1 tesla magnetic field is extremely strong. The earth’s magnetic field is a few hundredths of a tesla.

I have to go to a lot of effort to explain magnetic field and force direction when I teach the non-calculus course. It’s so much easier with calculus and vectors. The force on a charge q moving with a velocity v in a magnetic field B is found to obey The magnitude of the cross product is qvB sin . But it’s so much easier learning the right-hand rule for the vector cross product, and applying it to torques, charged particles, etc., instead of learning a seemingly new right hand rule for each new topic. The elegance of math! If you take a number of charged particles in a volume of wire that has a length ℓ in a magnetic field, it is easy to derive the vector form of our OSE:

Example In the figure two slides back, B=0 Example In the figure two slides back, B=0.9 T, I=30 A, ℓ=12 cm, and =60°. What is the force on the wire? F = I ℓ B sin  = (30 A) (0.12 m) (0.9 T) (sin 60°) F = 2.8 N I ℓ  B

Hold it! Force is a vector quantity! What is the direction. The force is perpendicular to both current direction and magnetic field direction. Apply either version of the right-hand rule and you find it is into the paper. You could use the right-hand screw rule, which is the way I best visualize the direction. Or just let the math tell you! We need to have a way to draw 3-d vectors on 2-d paper. We will use the symbol for a vector pointing directly out of the page, towards us (that is supposed to look like the sharp point of an arrowhead coming right towards your eye). We will use the symbol for a vector pointing directly into the page, away from us (that is supposed to look like the feathered end of an arrow going away from your eye).

Example A rectangular loop of wire hangs vertically in a magnetic field B as shown. B is uniform along the 10 cm horizontal length of wire, and the top portion of the wire is outside the field. The loop hangs from a balance which measures a downward force F=3.48x10-2 N in excess of the wire weight when the current is 0.245 A. What is the magnitude of B? I I 10 cm                         B         Huh? F Sorry, that’s not an acceptable answer. You do know how to work this problem!

The forces on these vertical segments of the wire are equal and opposite in direction. We need not worry about them further. I I 10 cm I can figure out the directions of those two forces. Can you?         The force on the lower horizontal segment is downward, as shown in the drawing. Could you verify that?                 B         The angle between current and magnetic field is 90°. sin(90°)=1. F OSE: F = I ℓ B sin  solve for B!

Force on an Electric Charge Moving in a Magnetic Field B = F  ( I ℓ sin(90) ) B = (3.48x10-2 N)  (0.245 A) (0.1 m) B = 1.42 T Hey, that wasn’t so bad! Only the direction bit is hard work at this point. Force on an Electric Charge Moving in a Magnetic Field We’ve kind of done this already—what’s the difference between a moving charge and a current in a wire? The current was confined to a wire, but we don’t expect that to alter the forces involved.

The algebra-based textbook does thing backwards from normal—force on wire first, then force on charge. I already gave the equation for the force on a charge: You can see the “derivation” in lec27_long.ppt, supplementary material.

Here’s the equation we use if we “can’t” use cross products: If the charged particle is moving perpendicular to B,  = 90° and the force is greatest: F = q v B. The above OSE gives the magnitude of the force. The right hand rule gives the direction for positive charges. For negative charges, just reverse the direction (determine the direction as if it were for a positive charge and the force on the negative charge is in the opposite direction).

          Bout           v v         + - FB FB                               Thanks again to Dr. Waddill for the nice picture. Don’t you wish you were taking Physics 24 too?

Magnetic Field due to a Straight Wire We already saw how the magnetic field due to a current “curls around” a wire. This tells us the direction of the magnetic field. What about the magnitude?

Experimentally it is found (and verified by theory) that the larger the current, the larger the magnetic field, and the further away from the wire, the weaker the magnetic field. Mathematically, where I is the current in the wire, r is the distance away from the wire at which B is being measured, and 0 is a constant: This “funny” definition of 0 allows us to more elegantly define current (later).

Example A vertical electric wire in the wall of a building carries a current of 25 A upward. What is the magnetic field at a point 10 cm due north of this wire? Let’s make north be to the left in this picture, and up be up. up I=25 A According to the right hand rule, the magnetic field is to the west, coming out of the plane of the “paper.” N d=0.1 m B To calculate the magnitude, B:

Definition of the Ampere and the Coulomb We defined the ampere of current this morning as being 1 C of charge flowing past a point in 1 s: 1 A = 1 C / 1 s. That’s the way I learned it many years ago. Now the ampere is actually defined as the current flowing in two parallel wires 1 m apart which produces a force per unit length of 2x10-7 N/m. A coulomb is then defined as 1 A · 1 s. Physics is constantly being “tweaked” as new knowledge and experimental techniques become available.

Ampere’s Law What is a solenoid? A solenoid is a coil of wire with many loops. Each loop produces a magnetic field that looks like this.

“When the coils of the solenoid are closely spaced, each turn can be regarded as a circular loop, and the net magnetic field is the vector sum of the magnetic field for each loop. This produces a magnetic field that is approximately constant inside the solenoid, and nearly zero outside the solenoid.” Thanks again to Dr. Waddill for the pictures and text.

“The ideal solenoid is approached when the coils are very close and the length of the solenoid is much greater than its radius. Then we can approximate the magnetic field as constant inside and zero outside the solenoid.” B The vectors in and out of the page represent the current (and therefore the wires), so imagine this picture as a slice through the center of the solenoid, perpendicular to the wires.   I The slice is made perpendicular to the wires and parallel to the solenoid axis. Textbooks show that the magnetic field inside the solenoid is

B is the magnitude of the magnetic field inside the solenoid (the direction is given by the right-hand rule), n is the number of loops per unit length (loops per meter), and I is the current in the wire. I’ll write the “official” version like this: N is the total number of loops (sometimes called “turns”) and L is the total length of the solenoid. More about solenoids on-line here.

The magnetic field of a solenoid looks like the magnetic field of a bar magnet. (http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html#c1)

The BIG IDEAS There are two BIG IDEA equations buried in this lecture. It is not obvious where they are, because we are so focused on details when we learn this material for the first time. One of the big ideas arises from the observation that magnetic poles always come in pairs, unlike + and – charged particles. In the next lecture, I’ll introduce the idea of magnetic flux, which is “like” the idea of electric flux.

You can calculate the magnetic flux through a given area: If you integrate the magnetic flux over a closed area (e.g., a sphere, or a cylinder closed at both ends), the result is zero:

The integral is zero because wherever you find a N pole, you also find a S pole, and the net flux going out of the surface must equal the net flux going into the surface (kind of like the N “cancels” the S). The equation is called Gauss’ law for magnetism, and is one of Maxwell’s four equations. It also says there is no such thing as a magnetic monopole. Some quantum theories suggest that magnetic monopoles might exist. We have not found them. If we do, then the right hand side of the equation above will need modified.

You also saw Ampere’s law, which appeared in the context of a solenoid You also saw Ampere’s law, which appeared in the context of a solenoid. The law is far more general than that. It also appeared in the equation for a magnetic field due to a current in a wire, except then we didn’t call it Ampere’s law. If you integrate this expression over a closed path, you get a result proportional to the current I and the total path length.

In the next lecture, we will find that electric fields which change with time also give rise to magnetic fields, so the full version of the Ampere’s law Maxwell equation is You’ve seen three out of Maxwell’s four equations. One more lecture on E&M, one more equation!