Biot-Savart Law for Currents: Difference between revisions

Latest revision as of 14:13, 15 November 2020

Claimed by Abigail Ochal Fall 2020

Biot-Savart Law

The Biot-Savart Law can be used for more than just single moving charges; a notable application of this law is its ability to calculate the magnetic field for an extremely large number of charges - an example of thousands of charges moving together is within a current carrying wire (current being the amount of charges moving over a specific amount of time).

When using Biot-Savart Law to find the magnetic field of a short wire, we can extend this concept to a variety of different shapes - long current carrying wires, current carrying loops, etc.

One main point to note is that the application of the Biot-Savart law is specifically for steady state current (current that is not changing over time ----- [math]\displaystyle{ dI/dt = 0 }[/math] )!

A Mathematical Model

First We start off with the original version of the Biot-Savart Law. [math]\displaystyle{ \vec B=\frac{\mu_0}{4 \pi } \frac{q\vec v\times\hat r}{r^2} }[/math], where [math]\displaystyle{ \frac{\mu_0}{4 \pi } = 1 \times 10^{-7}\frac{Tm^2}{Cm/s}, }[/math]

Because we are dealing with a portion of wire [math]\displaystyle{ \mathrm{d}\boldsymbol{\ell} }[/math] long with an Area A containing n moving particles with charge q, we find that the total number of moving charges is equal to |q|(nAv) which is also equal to I, the current in the wire.

[math]\displaystyle{ B = \frac{\mu_0I}{4\pi}\frac{\mathrm{d}\boldsymbol{\ell} \times \mathbf{\hat r}}{r^2}, }[/math]

Because the shape of the current carrying wire can vary from a straight wire to a loop, we must integrate over the region of the wire.

[math]\displaystyle{ \Delta B = \frac{\mu_0I}{4\pi}\int_{\mathrm{wire}}\frac{\mathrm{d}\boldsymbol{\ell} \times \mathbf{\hat r}}{r^2}, }[/math]

The key point is that there are [math]\displaystyle{ nA\Delta l }[/math] electrons in a short length of wire, each moving with average speed [math]\displaystyle{ \vec v }[/math], so that the sum of all the [math]\displaystyle{ q\vec v }[/math] contributions is [math]\displaystyle{ nA \Delta l|q|\vec v = I\Delta l. }[/math]

When applying the Biot-Savart Law to a Long Straight Wire, we follow a set of steps,

Step 1: Cut Up the Distribution into Pieces ([math]\displaystyle{ \Delta l }[/math]) and Draw [math]\displaystyle{ \Delta B }[/math].

Step 2: Write an Expression for the Magnetic Field Due to One Piece.

Step 3: Add Up the Contributions of All the Pieces (each [math]\displaystyle{ \Delta l }[/math] adds together to make the entirety of the length).

Step 4: Check the Result.

Where the Magnetic Field of a Straight Wire is shown by,

[math]\displaystyle{ B = \frac{\mu_0}{4\pi}\frac{LI}{r(r^2 + (L/2)^2)^{1/2}} }[/math] for length [math]\displaystyle{ L }[/math], conventional current [math]\displaystyle{ I }[/math], a perpendicular distance [math]\displaystyle{ r }[/math] from the center of the wire, or, [math]\displaystyle{ B = \frac{\mu_0}{4\pi}\frac{2I}{r} }[/math] if [math]\displaystyle{ L\;\gt \;\gt r. }[/math]

Right Hand Rule

When using the Biot-Savart Law for Currents, it is crucial to understand the direction of the magnetic field created by a current.

In order to get the direction of the Magnetic Field, as shown in the image on the right, we:

1. Point our thumb in the direction of the current, and

2. automatically curl our fingers around in one direction.

This curling direction is the direction of the magnetic field itself.

If we are looking at an observation location directly to the left of the wire, then we know the magnetic field is pointing out of the page. If we are looking at an observation location directly to the right of the wire, then the magnetic field is pointing into the page. If the observation location is directly in front of the wire (closest to us), then the magnetic field is pointing to the right. If the observation location is directly behind the wire, then the magnetic field is to the left.

Picking a few different observation locations can help to visualize the circular nature of the magnetic field due to the current in the wire.

Another way to determine the direction of the magnetic field due to the current would be to use cross product of the vector dl and rhat. When dl and rhat are parallel, the magnetic field will be 0.

Long Wire Integration Example

For a long wire of length L positioned along the x axis with current flowing in the positive x direction (as shown by image on right)

First, we start off with our adjusted Biot-Savart Formula for a slice of wire.

[math]\displaystyle{ \Delta B = \frac{\mu_0I}{4\pi}\int_{\mathrm{wire}}\frac{\mathrm{d}\boldsymbol{\ell} \times \mathbf{\hat r}}{r^2}, }[/math]

Second, we must find [math]\displaystyle{ r }[/math], the vector pointing from the source to the observation location. In this case, we will choose an observation location y above the rod.

[math]\displaystyle{ r = obs - source = \lt 0,y,0\gt - \lt x,0,0\gt = \lt -x,y,0\gt }[/math]. which has a magnitude of [math]\displaystyle{ \sqrt(x^2+y^2) }[/math]

We see that [math]\displaystyle{ \hat r = \frac{r}{|r|} }[/math] .

[math]\displaystyle{ \hat r = \frac{\lt -x,y,0\gt }{\sqrt(x^2+y^2))} }[/math]

We then have to express [math]\displaystyle{ \Delta \boldsymbol{\ell} }[/math] in terms of our variable of integration, x. [math]\displaystyle{ \Delta \boldsymbol{\ell} }[/math] = [math]\displaystyle{ \Delta x\lt 1,0,0,\gt }[/math]

Our new equation after substituting our new variables is [math]\displaystyle{ \Delta B = \frac{\mu_0I\Delta x\lt 1,0,0,\gt }{4\pi(x^2+y^2)} \times \frac{\lt -x,y,0\gt }{\sqrt(x^2+y^2))} }[/math]

Finding the cross product of the above vectors gives us a product in the +z direction. [math]\displaystyle{ \Delta B = \frac{\mu_0I\Delta x}{4\pi(x^2+y^2)^(\frac{3}{2})} }[/math]

We are finally ready to integrate. Because we are integrating the entire rod our limits are

[math]\displaystyle{ \int\limits_{-L/2}^{L/2}\ }[/math] [math]\displaystyle{ \int\limits_{-L/2}^{L/2}\ = \frac{\mu_0I\Delta x}{4\pi(x^2+y^2)^(\frac{3}{2})} }[/math]

We find that our final answer is [math]\displaystyle{ B= \frac{\mu_0}{4\pi}\frac{LI}{y\sqrt(y^2+(L/2)^2)}\hat z }[/math]

Current Loop Integration Example

Saved for Elizabeth Clayton 10/7/2020

A Computational Model

The following link shows the magnetic field produced by small segments of wire in a loop individually. For a long straight wire, we see that there is a circular magnetic field surrounding the wire with current. The following link does a stepwise visual of the contributions of each part of the wire at an observation location a distance r from the wire.

http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/17-Bwire-with-r

We see that along the axis of the wire, each contribution not on the axis is negated due to symmetry and the resulting magnetic field is all along the wire.

http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/17-B-loop-with-r-dB.

Connectedness

In biochemistry, there are lots of instruments that rely on magnets and magnetic fields to gather information about different molecules.

One example is proton nuclear magnetic resonance (H+ NMR), which creates peaks corresponding to hydrogen atoms (protons) on adjacent carbon atoms in organic molecules.

Another example, which is commonly used in medicine, is magnetic resonance imaging (MRI), which takes images using magnets to interact with protons in the body. For more information, go to the "External Links" Section.

History

Jean-Baptiste Biot and Félix Savart worked together to understand current and magnetic fields during the 1820's. Find more information about the history in "External Links"

External Links

More information on MRI's

History of Biot-Savart

@@ Line 1: / Line 1: @@
-==Thermodynamics==
-This topics focuses on energy work of a system but it can only deal with a large scale response to heat in a system.  '''Thermodynamics''' is the study of the work, heat and energy of a system.  The smaller scale gas interactions can explained using the kinetic theory of gases.  There are three fundamental laws that go along with the topic of thermodynamics.  They are the zeroth law, the first law, and the second law.  These laws help us understand predict the the operation of the physical system.  In order to understand the laws, you must first understand thermal equilibrium.  [[Thermal equilibrium]] is reached when a object that is at a higher temperature is in contact with an object that is at a lower temperature and the first object transfers heat to the latter object until they approach the same temperature and maintain that temperature constantly.  It is also important to note that any thermodynamic system in thermal equilibrium possesses internal energy.
+  '''Claimed by Abigail Ochal Fall 2020'''
-===Zeroth Law===
+==Biot-Savart Law==
+The Biot-Savart Law can be used for more than just single moving charges; a notable application of this law is its ability to calculate the magnetic field for an extremely large number of charges - an example of thousands of charges moving together is within a current carrying wire (current being the amount of charges moving over a specific amount of time).
+When using Biot-Savart Law to find the magnetic field of a short wire, we can extend this concept to a variety of different shapes - long current carrying wires, current carrying loops, etc.
+*One main point to note is that the application of the Biot-Savart law is specifically for steady state current (current that is not changing over time ----- <math> dI/dt = 0 </math> )!
-The zeroth law states that if two systems are at thermal equilibrium at the same time as a third system, then all of the systems are at equilibrium with each other.  If systems A and C are in thermal equilibrium with B, then system A and C are also in thermal equilibrium with each other.  There are underlying ideas of heat that are also important.  The most prominent one is that all heat is of the same kind.  As long as the systems are at thermal equilibrium, every unit of internal energy that passes from one system to the other is balanced by the same amount of energy passing back.  This also applies when the two systems or objects have different atomic masses or material.
 ====A Mathematical Model====
-If A = B and A = C, then B = C
+First We start off with the original version of the Biot-Savart Law.
-A = B = C
+<math>\vec B=\frac{\mu_0}{4 \pi } \frac{q\vec v\times\hat r}{r^2}</math>, where <math> \frac{\mu_0}{4 \pi } = 1 \times 10^{-7}\frac{Tm^2}{Cm/s}, </math>
+Because we are dealing with a portion of wire <math>\mathrm{d}\boldsymbol{\ell}</math> long with an Area A containing n moving particles with charge q, we find that the total number of moving charges is equal to |q|(nAv) which is also equal to I, the current in the wire.
+<math>B = \frac{\mu_0I}{4\pi}\frac{\mathrm{d}\boldsymbol{\ell} \times \mathbf{\hat r}}{r^2},</math>
+[[File:biotsavartlawforcurrentsaochal.png|400 px|thumb|alt text]]
+Because the shape of the current carrying wire can vary from a straight wire to a loop, we must integrate over the region of the wire.
+<math>\Delta B = \frac{\mu_0I}{4\pi}\int_{\mathrm{wire}}\frac{\mathrm{d}\boldsymbol{\ell} \times \mathbf{\hat r}}{r^2},</math>
+The key point is that there are <math> nA\Delta l </math> electrons in a short length of wire, each moving with average speed <math>\vec v </math>, so that the sum of all the <math> q\vec v </math> contributions is <math> nA \Delta l|q|\vec v = I\Delta l. </math>
-====A Computational Model====
-How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]
+When applying the Biot-Savart Law to a Long Straight Wire, we follow a set of steps,
-===First Law===
+Step 1: Cut Up the Distribution into Pieces (<math> \Delta l </math>) and Draw <math> \Delta B </math>.
-The first law of thermodynamics defines the internal energy (E) as equal to the difference between heat transfer (Q) ''into'' a system and work (W) ''done by'' the system.  Heat removed from a system would be given a negative sign and heat applied to the system would be given a positive sign.  Internal energy can be converted into other types of energy because it acts like potential energy.  Heat and work, however, cannot be stored or conserved independently because they depend on the process.  This allows for many different possible states of a system to exist.  There can be a process known as the adiabatic process in which there is no heat transfer.  This occurs when a system is full insulated from the outside environment.  The implementation of this law also brings about another useful state variable, '''enthalpy'''.
+Step 2: Write an Expression for the Magnetic Field Due to One Piece.
-====A Mathematical Model====
+Step 3: Add Up the Contributions of All the Pieces (each <math> \Delta l </math> adds together to make the entirety of the length).
+Step 4: Check the Result.
+Where the Magnetic Field of a Straight Wire is shown by,
+<math> B = \frac{\mu_0}{4\pi}\frac{LI}{r(r^2 + (L/2)^2)^{1/2}} </math> for length <math> L </math>, conventional current <math> I </math>, a perpendicular distance <math> r </math> from  the center of the wire, or,
+<math> B = \frac{\mu_0}{4\pi}\frac{2I}{r} </math> if <math> L\>>\>> r. </math>
+==Right Hand Rule==
+[[File:Wire diagram 1 aochal.png|300px|thumb|alt text]]
+When using the Biot-Savart Law for Currents, it is crucial to understand the direction of the magnetic field created by a current.
+In order to get the direction of the Magnetic Field, as shown in the image on the right, we:
+. Point our thumb in the direction of the current, and
+. automatically curl our fingers around in one direction.
+This curling direction is the direction of the magnetic field itself. [[File:RHR aochal.png|300px|thumb|alt text]]
+If we are looking at an observation location directly to the left of the wire, then we know the magnetic field is pointing out of the page.
+If we are looking at an observation location directly to the right of the wire, then the magnetic field is pointing into the page.
+If the observation location is directly in front of the wire (closest to us), then the magnetic field is pointing to the right.
+If the observation location is directly behind the wire, then the magnetic field is to the left.
+Picking a few different observation locations can help to visualize the circular nature of the magnetic field due to the current in the wire.
+Another way to determine the direction of the magnetic field due to the current would be to use cross product of the vector dl and rhat. When dl and rhat are parallel, the magnetic field will be 0.
+==Long Wire Integration Example==
+[[File:IMG_2469.jpg|300px|thumb|alt text]]
+====For a long wire of length L positioned along the x axis with current flowing in the positive x direction (as shown by image on right)====
+First, we start off with our adjusted Biot-Savart Formula for a slice of wire.
+<math>\Delta B = \frac{\mu_0I}{4\pi}\int_{\mathrm{wire}}\frac{\mathrm{d}\boldsymbol{\ell} \times \mathbf{\hat r}}{r^2},</math>
-E2 - E1 = Q - W
+Second, we must find <math>r</math>, the vector pointing from the source to the observation location.  In this case, we will choose an observation location y above the rod.
-==Second Law==
+<math> r = obs - source = <0,y,0> - < x,0,0> = <-x,y,0> </math>. which has a magnitude of <math>\sqrt(x^2+y^2)</math>
-The second law states that there is another useful variable of heat, entropy (S).  Entropy can be described as the disorder or chaos of a system, but in physics, we will just refer to it as another variable like enthalpy or temperature.  For any given physical process, the combined entropy of a system and the environment remains a constant if the process can be reversed.  The second law also states that if the physical process is irreversible, the combined entropy of the system and the environment must increase.  Therefore, the final entropy must be greater than the initial entropy.
+We see that <math>\hat r = \frac{r}{|r|} </math> .
-===Mathematical Models===
+<math>\hat r = \frac{<-x,y,0>}{\sqrt(x^2+y^2))}</math>
-delta S = delta Q/T
+We then have to express <math> \Delta \boldsymbol{\ell}</math> in terms of our variable of integration, x.
-Sf = Si (reversible process)
+<math> \Delta \boldsymbol{\ell}</math> = <math> \Delta x<1,0,0,></math>
-Sf > Si (irreversible process)
-===Examples===
+Our new equation after substituting our new variables is
+<math> \Delta B = \frac{\mu_0I\Delta x<1,0,0,>}{4\pi(x^2+y^2)} \times \frac{<-x,y,0>}{\sqrt(x^2+y^2))} </math>
-'''Reversible process''': Ideally forcing a flow through a constricted pipe, where there are no boundary layers. As the flow moves through the constriction, the pressure, volume and temperature change, but they return to their normal values once they hit the downstream.  This return to the variables' original values allows there to be no change in entropy.  It is often known as an isentropic process.
+Finding the cross product of the above vectors gives us a product in the +z direction.
+<math> \Delta B = \frac{\mu_0I\Delta x}{4\pi(x^2+y^2)^(\frac{3}{2})}  </math>
-'''Irreversible process''': When a hot object and cold object are put in contact with each other, eventually the heat from the hot object will transfer to the cold object and the two will reach the same temperature and stay constant at that temperature, reaching equilibrium.  However, once those objects are separated, they will remain at that equilibrium temperature until something else acts upon it.  The objects do not go back to their original temperatures so there is a change in entropy.
+We are finally ready to integrate. Because we are integrating the entire rod our limits are
-==Connectedness==
-#How is this topic connected to something that you are interested in?
-#How is it connected to your major?
-#Is there an interesting industrial application?
-==History==
+<math> \int\limits_{-L/2}^{L/2}\    </math>   <math> \int\limits_{-L/2}^{L/2}\    = \frac{\mu_0I\Delta x}{4\pi(x^2+y^2)^(\frac{3}{2})} </math>
-Thermodynamics was brought up as a science in the 18th and 19th centuries.  However, it was first brought up by Galilei, who introduced the concept of temperature and invented the first thermometer.  G. Black first introduced the word 'thermodynamics'.  Later, G. Wilke introduced another unit of measurement known as the calorie that measures heat.   The idea of thermodynamics was brought up by Nicolas Leonard Sadi Carnot.  He is often known as "the father of thermodynamics".  It all began with the development of the steam engine during the Industrial Revolution.  He devised an ideal cycle of operation.  During his observations and experimentations, he had the incorrect notion that heat is conserved, however he was able to lay down theorems that led to the development of thermodynamics.  In the 20th century, the science of thermodynamics became a conventional term and a basic division of physics.  Thermodynamics dealt with the study of general properties of physical systems under equilibrium and the conditions necessary to obtain equilibrium.
+We find that our final answer is <math> B= \frac{\mu_0}{4\pi}\frac{LI}{y\sqrt(y^2+(L/2)^2)}\hat z </math>
-== See also ==
+==Current Loop Integration Example==
+'''Saved for Elizabeth Clayton 10/7/2020'''
+==A Computational Model==
+The following link shows the magnetic field produced by small segments of wire in a loop individually.
+For a long straight wire, we see that there is a circular magnetic field surrounding the wire with current. The following link does a stepwise visual of the contributions of each part of the wire at an observation location a distance r from the wire.
-Are there related topics or categories in this wiki resource for the curious reader to explore?  How does this topic fit into that context?
+http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/17-Bwire-with-r
-===Further reading===
+We see that along the axis of the wire, each contribution not on the axis is negated due to symmetry and the resulting magnetic field is all along the wire.
-Books, Articles or other print media on this topic
+http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/17-B-loop-with-r-dB.
+==Connectedness==
+In biochemistry, there are lots of instruments that rely on magnets and magnetic fields to gather information about different molecules.
-===External links===
+One example is proton nuclear magnetic resonance (H+ NMR), which creates peaks corresponding to hydrogen atoms (protons) on adjacent carbon atoms in organic molecules.
-Internet resources on this topic
+Another example, which is commonly used in medicine, is magnetic resonance imaging (MRI), which takes images using magnets to interact with protons in the body. For more information, go to the "External Links" Section.
-==References==
+==History==
+Jean-Baptiste Biot and Félix Savart worked together to understand current and magnetic fields during the 1820's. Find more information about the history in "External Links"
+== See also ==
+[Right Hand Rule[http://www.physicsbook.gatech.edu/Right-Hand_Rule]]
-https://www.grc.nasa.gov/www/k-12/airplane/thermo0.html
+[Direction of magnetic fields[http://www.physicsbook.gatech.edu/Direction_of_Magnetic_Field]]
-http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/thereq.html
+Are there related topics or categories in this wiki resource for the curious reader to explore?  How does this topic fit into that context?
-https://www.grc.nasa.gov/www/k-12/airplane/thermo2.html
-http://www.phys.nthu.edu.tw/~thschang/notes/GP21.pdf
-http://www.eoearth.org/view/article/153532/
+==External Links==
 [[Category:Which Category did you place this in?]]
+[https://www.nibib.nih.gov/science-education/science-topics/magnetic-resonance-imaging-mri More information on MRI's]
+[https://www.britannica.com/biography/Jean-Baptiste-Biot History of Biot-Savart]