chapter27

__**Chapter 27 Sources of the Magnetic Field**__ **by Justin Horn**
This chapter expands upon the previous one's discussion of magnetic fields by getting into how magnetic fields are actually produced and shows us how to determine the magnitude of magnetic fields from different sources. Most importantly, the chapter establishes that any moving charge creates a magnetic field, whether it is a point charge, current moving through a wire, or even an electron moving around the outside of an atom.

__﻿Important Formulas:__
Magnetic Field Due to a Point Charge:

The Biot-Savart Law:

Gauss's Law for Magnetism:

Ampere's Law:

**__﻿27- 1: The Magnetic Field of Moving Point Charges__**
A magnetic field is induced by not just a current, but any kind of moving charge. This includes any kind of of moving point charge. When you have a point charge, the magnetic field can be derived by the following expression: where q is the charge of the point charge, v is the velocity of the of the charge, r-hat is a unit vector pointing in the direction of the radius, and r is the radius.

The only really confusing thing about this formula is the unit vector, r-hat. Because r-hat is a unit vector, it has a magnitude of one. The unit vector, therefore only exists to provide the magnetic field with a direction. So when crossing v with r-hat, just remember that its magnitude is always 1 and its direction will be the same as the direction of the normal radius vector. Also, 4π is included in the formula to ensure simplicity when using Ampere's Law, which will be discussed later.

Sample Problem:



__**27-2: The Magnetic Field of Currents: The Biot-Savart Law**__
When deriving the magnetic field created as a result of a moving current, a formula known as the BIot-Savart Law must be used: The expression is similar to the expression for the magnetic field of a moving point charge in that you are multipling the velocity at which the charge is moving with the magnitude of the charge, then crossing it with the unit vector r-hat that lies in the direction of the radius. The difference is that in this case, both magnitude of charge and velocity of charge are represented by the expression Idl. Remember that current is equal to dq/dt. This means that current is essentially the velocity at which a certain quantity of charge is moving. In crossing r-hat with current, you are crossing it with both velocity and quantity of charge, just like you would with a point charge. A wire, with a current running through it, however, does not have all its charge concentrated at one point, so you have to integrate along the length of the wire in order to get the magnetic field created by i.

Sample Problem:



**__﻿27-3: Gauss's Law For Magnetism__**
Gauss's Law For Magnetism states that the net magnetic flux through any closed Gaussian surface is 0. In simple terms, this is saying that there are no magnetic monopoles. Because you can't have a single magnetic pole, no matter how you draw the Guasssian surface, the magnetic flux will have to be 0, as every field line that enters the field must also leave the field in order to go back to the other pole of the magnet.

In mathematical terms, Gausss's Law for Magnetism states that

__**27-4: Ampere's Law**__
Ampere's Law is an expression relating the magnetic field at the edge of a closed curve, C to the current that is enclosed by said curve.

where B is the component of the magnetic field tanget to C, mu nought is the permeability of space, a fundamental constant, and I is the current enclosed by C. By integrating around the curve, dL, we create a situation where radius by itself does not matter and will not change the value of the expression. Radius only matters in that it can change how much current is enclosed by the surface.

When you have certain symmetric situations in which B has the same value at all points on the curve C, Ampere's Law makes it much easier to determine the magnitude of the magnetic field. Take, for example, the magnetic field due to a long straight wire. When B is constant at all points, it can be pulled out of the integral and we are just integrating dL. The integral of dL is just L, and, in this case the length is equal to the circumference of curve C. The circumference is equal to 2 πR, where R is the radius of C. After substituting in and solving for B, we end up with the expression B = m 0 I / (2 πR). This is the same expression we would get if we used the Biot-Savart Law to calculate the magnetic field due to a long straight wire, but the derivation is much simpler when using Ampere's Law.

Sample Problem:

__** 27-5: Magnetism In Matter: **__
We've established that magnetism is the result of moving electric charge and this explains a certain type of magnetic (electromagnets), but we have not yet accounted for permanent magnets. Permanent magnets are certain materials that always create a magnetic field, regardless of whether or not a current is moving through them. Like electromagnets, permanent magnets are the result of moving charges. In this case, the moving charges are the electrons that move about the outside of the atom's nucleus and the intrinsic spin of the electron itself. In certain substances, the magnetic dipole moments created by these forces align and create a strong magnetic field that is independent of any kind of external moving charge.

Materials can be either paramagnetic, diamagnetic, or ferromagnetic depending upon how their magnetic moments respond to an external magnetic field. Paramagnetic materials do not tend to produce any significant magnetic field. Their magnetic moments are usually randomized and do not line up to produce a strong magnetic field, however, when exposed to an external magnetic field, the magnetic moments of a paramagnetic material will at least partially line up with the lines of the external field. There will probably be a very slight increase in the magnitude of an external field due to a paramagnetic object. Ferromagnetic materials have a much higher degree of alignment and, as such, they often carry a magnetic charge even in the absence of an external magnetic field. The magnetic moments of a diamagnetic object oppose those of the external magnetic field, thereby weakening it.

Magnetization and Magnetic Susceptibility
The magnetization of a material is the net magnetic dipole moment per unit of volume or M = dμ/dV. The magnetic moments created by the movements of electrons within a substance create a magnetic field which cancels out within the material itself, but leaves a net magnetic field on the surface of the material. This surface current is known as Amperian Current.

**__Lab: Ampere's Law__**

 * __Applet Used:__** http://phet.colorado.edu/en/simulation/magnets-and-electromagnets

Ampere's Law relates the amount of current enclosed by a closed curve with the strength of the magnetic field generated by that current. In order to test this, we are going to use a physics applet to see how magnetic field strength changes with the current. The particular applet we are using involves an electromagnet, a solenoid that is attached the a battery and has a current running through it. We are going to increase the current going through the wire by increasing the voltage going through the battery. We are going to start at 0 and increase the voltage to 10 volts in 1-volt increments, recording the magnetic field at each voltage. We will measure the magnetic field at the same point every time so that the radius isn't changing. We are also going to keep the number of loops the solenoid has consistent at 4. Nothing will be changing except the amount of current flowing through the solenoid and, if Ampere's Law holds, the strength of the magnetic field at this particular point.

Results:
 * Volts || Magnetic Field Strength (T) ||
 * 1 || 0.003 ||
 * 2 || 0.006 ||
 * 3 || 0.009 ||
 * 4 || 0.012 ||
 * 5 || 0.015 ||
 * 6 || 0.018 ||
 * 7 || 0.021 ||
 * 8 || 0.024 ||
 * 9 || 0.027 ||
 * 10 || 0.03 ||

As we would expect, as we increase the current flowing through the solenoid, the magnetic field strength increases. If we were able to calculate the increase in current (using something other than Ampere's Law obviously) instead of the increase in voltage, we would find that, for every ampere the current increases by, the magnetic field increases by a factor of 4π×10 −7 in accordance with Ampere's Law. Even without knowing the exact current, we can still see that this data reinforces the positive relationship that Ampere's Law implies between magnetic field strength and current.


 * CATEGORY ||  SCORE (1-4)  ||  POINTS (0-20)  ||
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