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--- 184_notes:i_thru [2017/10/26 21:50] – dmcpadden
+++ 184_notes:i_thru [2020/08/24 13:29] (current) – dmcpadden
@@ Line 1: / Line 1: @@
-===== Current through a loop =====
+Section 21.6 in Matter and Interactions (4th edition)
-As we discussed earlier, our canonical example with be the long straight wire. In this case, the wire is a bit thick, so that we can investigate what happens inside the wire as well. That's a tough job with the Biot-Savart law. We will use Ampere's law to find the magnetic field both inside and outside the thick wire. On this page, we will focus on the right-hand side of Ampere's law, namely, the current through our Amperian loop.
+/*[[184_notes:amp_law|Next Page: Putting together Ampere's Law]]
-In this case, we will consider that the wire is a bit thicker than the average wire, so that it has a current density $J=I/A$. We will also //__assume that the current density is uniform__// in this class (upper-division courses may address non-uniform current densities). That is, at every point in the wire the same amount of charge per unit time per unit area exists. This will help us understand the power of Ampere's Law. The figure below shows a cross-section of the wire.
+[[184_notes:loop|Previous Page: Magnetic Field along a Closed Loop]]*/
-{{184_notes:week10_2.png?500}}
+===== Current through a loop =====
+Now that we have the left side of the equation, the next step is to talk about the right side of Ampere's law - namely the $\mu_0 I_{enc}$ bit. For an Amperian loop that is outside of a single wire, this part is actually rather simple, but it starts to become more complicated when we consider loops inside the wire. As we said before, if the wire is a bit thick, we can investigate what happens to the magnetic field inside the wire as well outside. That's a tough job with the Biot-Savart law and is one of the places where Ampere's law can be super useful. On this page, we will focus on the right-hand side of Ampere's law, namely, the current through our Amperian loop.
 {{youtube>394V31p-mrI?large}}
 ==== What is the current enclosed? ====
-Ampere's Law relates how curly the magnetic field is to how much current produces it.
+The point of Ampere's Law is that it relates how curly the magnetic field is to how much current produces it.
+ $\oint \vec{B} \bullet d \vec{l} = \mu_0 I_{enc}$
+The right hand side describes the amount of current enclosed by the Amperian loop - that is, how much current runs through the inside of the loop. The figure below describes relationship between the loop and the enclosed current.
+[{{  184_notes:week10_4.png?450|Thick wire with current,  $I$ , enclosed by an Amperian loop}}]
+For the purposes of these notes, let's assume we have a thick wire with a total uniform current,  $I_{tot}$ .
-$$\oint \vec{B}\cdot d \vec{l} = \mu_0 I_{enc}$$
+=== Enclosing all the current ===
-The right hand side describes the amount of current enclosed by the Amperian loop. That is, how much current runs through the area of the loop face. The figure below describes relationship between the loop and the enclosed current.
+If we are looking for the magnetic field outside of the wire, we will choose to draw the loop so that it's radius is bigger than the radius of wire. In this case, the total current that runs through the wire will pass through the Amperian loop, meaning the loop will enclose all the current. This is the simplest of cases where  $I_{enc} = I_{tot}$ .
-{{  184_notes:week10_4.png?450}}
+=== Enclosing some of the current ===
-For the purposes of these notes, let's assume with have a thick wire with total uniform current,  $I_{tot}$ .
+When we want to find the magnetic field inside the wire, then we need pick the radius of the loop to be smaller than the radius of the wire. In this case, **some of the current will pass through the loop but not all of the current**. We need to be able to determine what fraction of the total current will pass through our loop.
-== Enclosing all the current ==
+[{{  184_notes:week10_2.png?500|Current density ( $J$ )}}]
-In many cases the radius of the loop will be larger than the radius outside of a wire, that is, it will enclose all the current. This is the simplest of cases where $I_{enc} = I_{tot}$. This will be the case for finding the magnetic field outside the wire.
+Just like how we used [[184_notes:dq|charge density when talking about the charge from a line]], we will use the current density to help us find the  $I_{encl}$  (or the fraction of the current that passes through the Amperian loop). If we //__assume a constant, uniform current__//, then the current density (represented by a "J") is given by:
+$$J=\frac{I_{tot}}{A_{tot}}$$
+where $I_{tot}$ is the total current going through the wire and  $A_{tot}$  is the total cross-sectional area of the wire. The units of current density would then be  $A/m^2$ . Technically, current density is a vector ( $\vec{J}$ ) that points in the direction that the charges are moving, but for our purposes, we will only be using the magnitude J.
-== Enclosing some of the current ==
+Once we know the current density, we can use that to find the total enclosed current ( $I_{enc}$ ). If we multiply the current density by the enclosed area (the area of the Amperian loop you chose), then that will give use the fraction of the current that passes through the loop. Mathematically, this would be:
-In some cases, the radius of the loop will be smaller than the radius of the wire. In that case, you will enclose some but not all of the current $I_{tot}$. To find $I_{encl} $, you will need to know the current density$ J=I/A_{wire}$. For our purposes, we will //__assume a constant current density__// because we will deal with uniform currents. Hence the enclosed current is a fraction of the total,
+$$I_{enc} = J A_{enc} = \dfrac{I_{tot}}{A_{tot}} A_{enc}$$
-$$I_{enc} = J A_{enc} = I_{tot} \dfrac{A_{enc}}{A_{tot}}$$
+[{{  184_notes:week10_5.png?450|Current density when the amperian loop encloses an area smaller than the area of the wire}}]
-{{  184_notes:week10_5.png?450}}
+where again $A_{tot} $is the total cross-sectional area of the wire,$ A_{enc} $is the cross-sectional area enclosed by the loop, and$ I_{tot}$ is the total current in the wire. This is very similar to how you will find [[184_notes:q_enc|$Q_{encl}$ with Gauss's Law]] for Electric Fields next week.
-where both of these areas are cross-sectional areas.  $A_{tot}$  is the total cross-sectional area of the wire and  $A_{enc}$  is the cross-sectional area enclosed by the loop. This is very similar to how you found [[184_notes:q_enc| $Q_{encl}$  with Gauss's Law]] for Electric Fields.