Magnetic Field: Difference between revisions

Revision as of 17:41, 22 November 2016

Claimed by Seongshik Kim Spring 2016

This page discusses the general properties and characteristics of magnetic fields

Magnetic Field

Unlike electric fields, magnetic fields are made by moving charges. Stationary charges do not exert magnetic fields. In equilibrium, there is no net motion of charges inside a metal. Therefore, it should be noted that electrons inside a metal must move continuously in order to create a magnetic field. This is the very basic definition of the current, a non-equilibrium system in which there is a constant flow of electrons. Measuring electron current can be understood as counting the amount of electrons that pass through a particular cross section of a conductor. However, since it is extremely hard to count these electrons, we measure the current using other indirect methods, one being measuring the magnetic field.

To measure the direction of the magnetic field in comparison to the electric current, the RHR (right-hand rule) is very useful as shown in the image below.

Notice that the direction of the magnetic field is always perpendicular to the direction of the current. This is a very easy rule of thumn to keep in minf when solving complicated problems.

Before proceeding further into the discussion, recall that we defined equilibrium above as there being no net motion of charges inside a metal. Keep in mind that equilibrium and being continuous are two totally different things.

A Mathematical Model

The magnetic field created by a single charged particle is given by the equation [math]\displaystyle{ \vec{B} =\frac{\mu_0}{4\pi} \frac{q(\vec{v} \times \hat{r})}{|\vec{r}|^2} }[/math], where [math]\displaystyle{ \frac{\mu_0}{4\pi} }[/math], where [math]\displaystyle{ \frac{\mu_0}{4\pi} }[/math] is a fundamental constant equal to [math]\displaystyle{ 1 \times 10^{-7} T }[/math], [math]\displaystyle{ q }[/math] is the charge of the particle, [math]\displaystyle{ \vec{v} }[/math] is the velocity of the particle, and [math]\displaystyle{ \vec{r} }[/math] is the vector that points from source to observation location. This equation is called the Biot-Savart Law for a single moving charge.

There is, however, another version of this Biot-Savart Law. This definition focuses more on the electron currents that was explained in the introductory paragraph. It is defined by [math]\displaystyle{ \vec{B} =\frac{\mu_0}{4\pi} \frac{I(\vec{l} \times \hat{r})}{|\vec{r}|^2} }[/math]. You may notice these equations are very similar in format. The constants and the vectors are still the same, but in this equation, </math>, [math]\displaystyle{ I }[/math] is the conventional electron current in an observed wire and [math]\displaystyle{ \vec{l} }[/math] is the length of the segment of the wire.

A Computational Model

The Biot-Savart Law defined above can be visualized if you click on the link below. The link sends you to a GlowScript page where you will be exposed to VPython code that animates both this law and its RHR applications. Simply click the "Run" botton at the top left of the frame to view.

Characteristics of the Biot-Savart Law

Single Charged Moving Particle Version

Because the equation involves the cross product of velocity and the position vector, one can find out that there is no magnetic field in the direction of the movement of the charged particle, because the cross product of two vectors in the same direction is zero.

However, even in the absence of a magnetic field, an electric field may still be present.

By using a compass, one can calculate the magnitude of a current. The Earth exerts a magnitude that always points to the North. When a compass is near a current with a magnetic field, the needle would be deflected by the net magnetic field. Notice that although the magnetic field of the current is perpendicular to the direction of the movement of charges, the needle is not deflected 90 degrees because of the magnetic field of the Earth, which is usually larger than that of the current. Also, because the magnetic field is exerted in a circular pattern, the direction of the magnetic field above the source is exactly the opposite of the magnetic field under the source. As a result, depending on the location of the compass, the needle may deflect in the opposite direction but with the same magnitude.

Lastly, Because of the direction of the magnetic field is influenced by the charge of the source, charge, in this case, one must pay attention to the presence of Q in the equation above.

Electric Current Version

Connectedness

This is the foundation of Physics II, for many of the materials one learns is based on the electric field and the magnetic field. Once one understand how magnetic fields are created and affected by charges, one will be able to apply the knowledge to find out what is going on in currents. For instance, in a metal positively charged particles are not likely to move compared to electrons due to their heavy weight. However, electrons, being negatively charged, hinder one's ability to calculate relevant applications. Thus, the concept of the conventional current is introduced to explain not only the general flow of current in a circuit, but also describe macroscopic behaviors, including resistance and capacitance, of circuits. Therefore, one must be able to distinguish key differences between the magnetic field and the electric field and how they are applied. In addition, the cross product is used, so one must be familiar with the right hand rule as well.

Page initiated by --Spennell3 (talk) 14:20, 19 October 2015 (EDT)

    edited by Seongshik Kim 21:20, 17 April 2016

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 The magnetic field created by a single charged particle is given by the equation <math> \vec{B}  =\frac{\mu_0}{4\pi} \frac{q(\vec{v} \times \hat{r})}{|\vec{r}|^2}  </math>, where <math> \frac{\mu_0}{4\pi}</math>, where <math> \frac{\mu_0}{4\pi}</math> is a fundamental constant equal to <math> 1 \times 10^{-7} T </math>, <math>q</math> is the charge of the particle, <math> \vec{v}</math> is the velocity of the particle, and <math> \vec{r}</math> is the  vector that points from source to observation location.  This equation is called the Biot-Savart Law for a single moving charge.
-There is, however, another version of this Biot-Savart Law. This definition focuses more on the electron currents that was explained in the introductory paragraph. It is defined by <math> \vec{B}  =\frac{\mu_0}{4\pi} \frac{I(\vec{l} \times \hat{r})}{|\vec{r}|^2}  </math>. You may notice these equations are very similar in format. The constants and the vectors are still the same, but in this equation, </math>, <math>I</math> is the conventional electron current in an observed wire and <math> \vec{l}</math> is the length of the segment of the wire.
+There is, however, another version of this Biot-Savart Law. This definition focuses more on the electron currents that was explained in the introductory paragraph. It is defined by <math> \vec{B}  =\frac{\mu_0}{4\pi} \frac{I(\vec{l} \times \hat{r})}{|\vec{r}|^2}  </math>. You may notice these equations are very similar in format. The constants and the vectors are still the same, but in this equation, </math>, <math>I</math> is the conventional electron [[current]] in an observed wire and <math> \vec{l}</math> is the length of the segment of the wire.
 ===A Computational Model===