Physics Book - User contributions [en]

Magnetic Field of a Long Thick Wire Using Ampere's Law

2017-04-10T04:32:49Z

Otoyobo3:

'''Claimed by Oluwasanmi Toyobo (Spring 2017)'''

==The Main Idea==

This page will explain and show how to use Ampere's Law to find the magnetic field of a long thick wire at a specified distance. It is much easier to calculate the magnetic field using Ampere's Law as opposed to Biot-Savart Law.

===Mathematical Model for Ampere's Law===

Ampere's Law is a quantitative relationship between the pattern of the magnetic field along an Amperian loop and the amount of constant current passing through that path. An Amperian loop is essentially a closed loop that can be any size. On this page, the Amperian loop will be circular. Also, It's important to remember that the equation for Ampere's Law only applies to an Amperian loop that has a constant current flowing through it. Ampere's Law is shown below:

[[File:AmperesLaw11112.jpg]]

The following pictures display how to use Ampere's Law when trying to calculate the magnetic field of a current flowing out of the page at distance ''r'' away from the wire.

[[File:AmperesLaw2.jpg]]

The diagram above shows a current-carrying wire with its current flowing out of the page. Using the right-hand rule, we can determine the direction of the magnetic field to be as shown. The dotted circle represents the Amperian loop of radius ''r''.

The workings below, shows how Ampere's Law is used to calculate the magnetic field of the current-carrying wire at a distance ''r''. Note that on the third line of working, the dot product disappears. This occurs because the direction of the magnetic field is the same as the direction of ''dl''. More specifically, the angle between ''dl'' and the magnetic field ''B'' is 0. Remember the equation of the dot product is '''|a| x |b| x cos (theta)'''. So in this case, ''a = B'', ''b = dl'' and ''theta = 0''. So if ''theta = 0'' we know that ''cos (0) = 1''. In the next line of working ''B'' is pulled out of the closed integral, this occurs because all along the Amperian loop the magnetic field is the same, it does not change in magnitude. In the next line of working, once we added up all the ''dl'' 's on the Amperian loop, that will give us the circumference of the Amperian loop. From there we make ''B'' the subject of the equation.

[[File:AmperesLaw3.jpg]]

We will see how one can use Ampere's Law to calculate the magnetic field of a long thick wire in examples below.

==Examples==

===Simple===

A long thick wire has radius ''r = 0.1 m'' and a constant current of ''I = 0.15 A''. The Amperian loop within the wire has radius ''r = 0.03 m''. Calculate the magnetic field through the Amperian loop.

[[File:Question1.jpg]]

Explanation of the solution:

[[File:Question12.jpg]]

* 1) Write out Ampere's Law. The ''enc'' written as a subscript of ''I'' is short for enclosed.

* 2) For the current, you only want the current that is enclosed within the loop of radius ''r'', not all of the current enclosed within the loop of radius ''R''. To calculate the desired current, multiply the given current ''I'' by a ratio of the Amperian loop's area over the area of long thick wire.

* 3) Substitute the desired current into the equation.

[[File:Question22.jpg]]

* 4) Remove the dot product on the right hand side of the equation because the direction of the magnetic field on the Amperian loop is the same as the direction of ''dl''

* 5) Pull ''B'' out of the closed integral because it is the same all along the Amperian loop.

* 6) Complete the integral to give you the circumference of the Amperian loop and make ''B'' the subject of the equation.

* 7) Substitute in the values to give you '''9e-8 T'''

===Middling===

A long thick wire of radius ''R = 0.1 m'' has constant current ''I = 0.15 A'' flowing out of the page. An Amperian loop outside of the wire has radius ''r = 0.17 m''. Calculate the magnetic field through the Amperian loop.

[[File:Question3.jpg]]

Explanation of the solution:

[[File:Question31.jpg]]

* 1) Write out Ampere's Law. The ''AL'' written as a subscript is short for Amperian Loop.

* 2) For the current, you can not calculate a ratio similar to the one calculated in the ''Simple'' question. Because the Amperian Loop is outside of the wire, the current you need is the current ''I'' flowing in the wire.

* 3) Remove the dot product on the right hand side of the equation because the direction of the magnetic field on the Amperian loop is the same as the direction of ''dl''

* 4) Pull ''B'' out of the closed integral because it is the same all along the Amperian loop.

* 5) Complete the integral to give you the circumference of the Amperian loop. Keep in mind that the radius of the Amperian loop is not ''r'' (the radius of the wire), but it is ''R'' (the radius of the Amperian loop)

* 7) Substitute in the given values to give you '''1.76e-7 T'''

===Hard===

A long thick wire has radius ''r = 0.1 m'' and a constant current of ''I = 0.15 A''. The Amperian loop within the wire has radius ''r = 0.05 m''. Calculate the magnetic force due to the Amperian loop, exerted on an electron moving at ''4e7 m/s''.

[[File:Question1.jpg]]

Explanation:

[[File:ppp.jpg]]

* 1) Refer to '''Simple''' question to calculate the magnetic field.

* 2) Once you've calculated the magnetic field, substitute this value into the equation for magnetic force.

* 3) Substitute in the given values to give you '''4.8e-20 N'''.

==Connectedness==

Ampere's Law is particularly interesting because of how it is used as an alternative to the Biot-Savart Law. When studying this, I realized how that the relationships and laws in physics can be used to explain a multitude of things no matter how specific they may seem. Calculating the magnetic field of a long wire is connected to Mechanical Engineering because mechanical engineers need to know what the interacting forces are doing within a motor for example. Using parts of Ampere's Law, the magnetic field of wires within motors can be calculated. Magnetic fields and calculating their magnitudes is important in the motor industry, especially in the case of alternating current generators. For large-scale power alternating current generators, bigger magnetic fields are needed. These are produced from permanent magnets or coils of long wires.

==History==

In 1826, it was Hans Christian Oersted (in Denmark) who had discovered the connection between magnetism and electricity when he found that the needle of a compass he was holding was deflected when he moved it close to a current carrying wire during one of his lectures. It was this discovery that prompted the French physicist, Andre-Marie Ampere, to start his own research. Ampere was conducting experiments to learn more about the magnetic fields created by currents. Ampere found that when he placed two wires next to each other with current running in the same direction, the wires were attracted to each other and when the current was running in opposite directions, the wires repelled each other! Experiments like this and more gave way to the creation of Ampere's Law that helps people calculate the strength of magnetic fields and currents in particular situations.

== See also ==

See the page on Ampere's Law for a more in depth look at the law itself: [[Ampere's Law]]

For more applications of Ampere's Law see: [[Magnetic Field of a Toroid Using Ampere's Law]] and [[Magnetic Field of Coaxial Cable Using Ampere's Law]]

===Further reading===

Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley.

Buschauer, Robert. (December 2013). Derivation of the Biot-Savart Law from Ampere's Law Using the Displacement Current. Vol. 51 Issue 9. ''Physics Teacher''. pp. 542-543. ''Academic Search Complete''.

===External links===

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magcur.html

https://www.youtube.com/watch?v=ryGzpGpTtIM

https://www.youtube.com/watch?v=JHNloU9Rfow&t=9s

==References==

Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley. page 887.

[[Category:Maxwell's Equations]]

File:Ppp.jpg

2017-04-10T04:10:13Z

Otoyobo3:

Magnetic Field of a Long Thick Wire Using Ampere's Law

2017-04-10T03:59:04Z

Otoyobo3:

'''Claimed by Oluwasanmi Toyobo (Spring 2017)'''

==The Main Idea==

This page will explain and show how to use Ampere's Law to find the magnetic field of a long thick wire at a specified distance. It is much easier to calculate the magnetic field using Ampere's Law as opposed to Biot-Savart Law.

===Mathematical Model for Ampere's Law===

Ampere's Law is a quantitative relationship between the pattern of the magnetic field along an Amperian loop and the amount of constant current passing through that path. An Amperian loop is essentially a closed loop that can be any size. On this page, the Amperian loop will be circular. Also, It's important to remember that the equation for Ampere's Law only applies to an Amperian loop that has a constant current flowing through it. Ampere's Law is shown below:

[[File:AmperesLaw11112.jpg]]

The following pictures display how to use Ampere's Law when trying to calculate the magnetic field of a current flowing out of the page at distance ''r'' away from the wire.

[[File:AmperesLaw2.jpg]]

The diagram above shows a current-carrying wire with its current flowing out of the page. Using the right-hand rule, we can determine the direction of the magnetic field to be as shown. The dotted circle represents the Amperian loop of radius ''r''.

The workings below, shows how Ampere's Law is used to calculate the magnetic field of the current-carrying wire at a distance ''r''. Note that on the third line of working, the dot product disappears. This occurs because the direction of the magnetic field is the same as the direction of ''dl''. More specifically, the angle between ''dl'' and the magnetic field ''B'' is 0. Remember the equation of the dot product is '''|a| x |b| x cos (theta)'''. So in this case, ''a = B'', ''b = dl'' and ''theta = 0''. So if ''theta = 0'' we know that ''cos (0) = 1''. In the next line of working ''B'' is pulled out of the closed integral, this occurs because all along the Amperian loop the magnetic field is the same, it does not change in magnitude. In the next line of working, once we added up all the ''dl'' 's on the Amperian loop, that will give us the circumference of the Amperian loop. From there we make ''B'' the subject of the equation.

[[File:AmperesLaw3.jpg]]

We will see how one can use Ampere's Law to calculate the magnetic field of a long thick wire in examples below.

==Examples==

===Simple===

A long thick wire has radius ''r = 0.1 m'' and a constant current of ''I = 0.15 A''. The Amperian loop within the wire has radius ''r = 0.03 m''. Calculate the magnetic field through the Amperian loop.

[[File:Question1.jpg]]

Explanation of the solution:

[[File:Question12.jpg]]

* 1) Write out Ampere's Law. The ''enc'' written as a subscript of ''I'' is short for enclosed.

* 2) For the current, you only want the current that is enclosed within the loop of radius ''r'', not all of the current enclosed within the loop of radius ''R''. To calculate the desired current, multiply the given current ''I'' by a ratio of the Amperian loop's area over the area of long thick wire.

* 3) Substitute the desired current into the equation.

[[File:Question22.jpg]]

* 4) Remove the dot product on the right hand side of the equation because the direction of the magnetic field on the Amperian loop is the same as the direction of ''dl''

* 5) Pull ''B'' out of the closed integral because it is the same all along the Amperian loop.

* 6) Complete the integral to give you the circumference of the Amperian loop and make ''B'' the subject of the equation.

* 7) Substitute in the values to give you '''9e-8 T'''

===Middling===

A long thick wire of radius ''R = 0.1 m'' has constant current ''I = 0.15 A'' flowing out of the page. An Amperian loop outside of the wire has radius ''r = 0.17 m''. Calculate the magnetic field through the Amperian loop.

[[File:Question3.jpg]]

Explanation of the solution:

[[File:Question31.jpg]]

* 1) Write out Ampere's Law. The ''AL'' written as a subscript is short for Amperian Loop.

* 2) For the current, you can not calculate a ratio similar to the one calculated in the ''Simple'' question. Because the Amperian Loop is outside of the wire, the current you need is the current ''I'' flowing in the wire.

* 3) Remove the dot product on the right hand side of the equation because the direction of the magnetic field on the Amperian loop is the same as the direction of ''dl''

* 4) Pull ''B'' out of the closed integral because it is the same all along the Amperian loop.

* 5) Complete the integral to give you the circumference of the Amperian loop. Keep in mind that the radius of the Amperian loop is not ''r'' (the radius of the wire), but it is ''R'' (the radius of the Amperian loop)

* 7) Substitute in the given values to give you '''1.76e-7 T'''

===Hard===

Using the same values as in the middling problem calculate the magnetic force on the electron due to the wire.

Solution: The formula for magnetic force is <math>qv×B</math>. So by using the value for B computed above and calculating the cross product between B and qv we find that this equals < 2.49e-17, -2.07e-17, 0 > N. Don't forget that because this is the force on an electron q is negative.

==History==

In 1826, it was Hans Christian Oersted who had discovered the connection between magnetism and electricity when he found that the needle of a compass he was holding was deflected when he moved it close to a current carrying wire during one of his lectures. It was this discovery that prompted Andre-Marie Ampere to start his own research. Ampere was conducting experiments to learn more about the magnetic fields created by currents. Ampere found that when he placed two wires next to each other with current running in the same direction, the wires were attracted to each other and when the current was running in opposite directions, the wires repelled each other! Experiments like this and more gave way to the creation of Ampere's Law that helps people calculate the strength of magnetic fields and currents in particular situations.

== See also ==

See the page on Ampere's Law for a more in depth look at the law itself: [[Ampere's Law]]

For more applications of Ampere's Law see: [[Magnetic Field of a Toroid Using Ampere's Law]] and [[Magnetic Field of Coaxial Cable Using Ampere's Law]]

===Further reading===

Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley.

===External links===

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magcur.html

https://www.khanacademy.org/science/physics/magnetic-forces-and-magnetic-fields/magnetic-field-current-carrying-wire/v/magnetism-6-magnetic-field-due-to-current

https://en.wikibooks.org/wiki/Electrodynamics/Ampere%27s_Law

==References==

Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley. page 887.

[[Category:Maxwell's Equations]]

Magnetic Field of a Long Thick Wire Using Ampere's Law

2017-04-10T03:58:29Z

Otoyobo3:

'''Claimed by Oluwasanmi Toyobo (Spring 2017)'''

==The Main Idea==

This page will explain and show how to use Ampere's Law to find the magnetic field of a long thick wire at a specified distance. It is much easier to calculate the magnetic field using Ampere's Law as opposed to Biot-Savart Law.

===Mathematical Model for Ampere's Law===

Ampere's Law is a quantitative relationship between the pattern of the magnetic field along an Amperian loop and the amount of constant current passing through that path. An Amperian loop is essentially a closed loop that can be any size. On this page, the Amperian loop will be circular. Also, It's important to remember that the equation for Ampere's Law only applies to an Amperian loop that has a constant current flowing through it. Ampere's Law is shown below:

[[File:AmperesLaw11112.jpg]]

The following pictures display how to use Ampere's Law when trying to calculate the magnetic field of a current flowing out of the page at distance ''r'' away from the wire.

[[File:AmperesLaw2.jpg]]

The diagram above shows a current-carrying wire with its current flowing out of the page. Using the right-hand rule, we can determine the direction of the magnetic field to be as shown. The dotted circle represents the Amperian loop of radius ''r''.

The workings below, shows how Ampere's Law is used to calculate the magnetic field of the current-carrying wire at a distance ''r''. Note that on the third line of working, the dot product disappears. This occurs because the direction of the magnetic field is the same as the direction of ''dl''. More specifically, the angle between ''dl'' and the magnetic field ''B'' is 0. Remember the equation of the dot product is '''|a| x |b| x cos (theta)'''. So in this case, ''a = B'', ''b = dl'' and ''theta = 0''. So if ''theta = 0'' we know that ''cos (0) = 1''. In the next line of working ''B'' is pulled out of the closed integral, this occurs because all along the Amperian loop the magnetic field is the same, it does not change in magnitude. In the next line of working, once we added up all the ''dl'' 's on the Amperian loop, that will give us the circumference of the Amperian loop. From there we make ''B'' the subject of the equation.

[[File:AmperesLaw3.jpg]]

We will see how one can use Ampere's Law to calculate the magnetic field of a long thick wire in examples below.

==Examples==

===Simple===

A long thick wire has radius ''R = 0.1 m'' and a constant current of ''I = 0.15 A'' flowing out of the page. The Amperian loop within the wire has radius ''r = 0.03 m''. Calculate the magnetic field through the Amperian loop.

[[File:Question1.jpg]]

Explanation of the solution:

[[File:Question12.jpg]]

* 1) Write out Ampere's Law. The ''AL'' written as a subscript of ''I'' is short for Amperian Loop.

* 2) For the current, you only want the current that is enclosed within the loop of radius ''r'', not all of the current enclosed within the loop of radius ''R''. To calculate the desired current, multiply the given current ''I'' by a ratio of the Amperian loop's area over the area of long thick wire.

* 3) Substitute the desired current into the equation.

* 4) Remove the dot product on the right hand side of the equation because the direction of the magnetic field on the Amperian loop is the same as the direction of ''dl''

* 5) Pull ''B'' out of the closed integral because it is the same all along the Amperian loop.

* 6) Complete the integral to give you the circumference of the Amperian loop and make ''B'' the subject of the equation.

* 7) Substitute in the given values to give you '''9e-8 T'''

===Middling===

A long thick wire of radius ''R = 0.1 m'' has constant current ''I = 0.15 A'' flowing out of the page. An Amperian loop outside of the wire has radius ''r = 0.17 m''. Calculate the magnetic field through the Amperian loop.

[[File:Question3.jpg]]

Explanation of the solution:

[[File:Question31.jpg]]

* 1) Write out Ampere's Law. The ''enc'' written as a subscript of ''I'' is short for enclosed.

* 2) For the current, you only want the current that is enclosed within the loop of radius ''r'', not all of the current enclosed within the loop of radius ''R''. To calculate the desired current, multiply the given current ''I'' by a ratio of the Amperian loop's area over the area of long thick wire.

* 3) Substitute the desired current into the equation.

[[File:Question32.jpg]]

* 4) Remove the dot product on the right hand side of the equation because the direction of the magnetic field on the Amperian loop is the same as the direction of ''dl''

* 5) Pull ''B'' out of the closed integral because it is the same all along the Amperian loop.

* 6) Complete the integral to give you the circumference of the Amperian loop and make ''B'' the subject of the equation.

* 7) Substitute in the given values to give you '''9e-8 T'''

===Hard===

Using the same values as in the middling problem calculate the magnetic force on the electron due to the wire.

Solution: The formula for magnetic force is <math>qv×B</math>. So by using the value for B computed above and calculating the cross product between B and qv we find that this equals < 2.49e-17, -2.07e-17, 0 > N. Don't forget that because this is the force on an electron q is negative.

==History==

In 1826, it was Hans Christian Oersted who had discovered the connection between magnetism and electricity when he found that the needle of a compass he was holding was deflected when he moved it close to a current carrying wire during one of his lectures. It was this discovery that prompted Andre-Marie Ampere to start his own research. Ampere was conducting experiments to learn more about the magnetic fields created by currents. Ampere found that when he placed two wires next to each other with current running in the same direction, the wires were attracted to each other and when the current was running in opposite directions, the wires repelled each other! Experiments like this and more gave way to the creation of Ampere's Law that helps people calculate the strength of magnetic fields and currents in particular situations.

== See also ==

See the page on Ampere's Law for a more in depth look at the law itself: [[Ampere's Law]]

For more applications of Ampere's Law see: [[Magnetic Field of a Toroid Using Ampere's Law]] and [[Magnetic Field of Coaxial Cable Using Ampere's Law]]

===Further reading===

Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley.

===External links===

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magcur.html

https://www.khanacademy.org/science/physics/magnetic-forces-and-magnetic-fields/magnetic-field-current-carrying-wire/v/magnetism-6-magnetic-field-due-to-current

https://en.wikibooks.org/wiki/Electrodynamics/Ampere%27s_Law

==References==

Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley. page 887.

[[Category:Maxwell's Equations]]

File:Question31.jpg

2017-04-10T03:47:52Z

Otoyobo3:

File:Question3.jpg

2017-04-10T03:47:46Z

Otoyobo3:

Magnetic Field of a Long Thick Wire Using Ampere's Law

2017-04-10T03:42:37Z

Otoyobo3:

'''Claimed by Oluwasanmi Toyobo (Spring 2017)'''

==The Main Idea==

This page will explain and show how to use Ampere's Law to find the magnetic field of a long thick wire at a specified distance. It is much easier to calculate the magnetic field using Ampere's Law as opposed to Biot-Savart Law.

===Mathematical Model for Ampere's Law===

Ampere's Law is a quantitative relationship between the pattern of the magnetic field along an Amperian loop and the amount of constant current passing through that path. An Amperian loop is essentially a closed loop that can be any size. On this page, the Amperian loop will be circular. Also, It's important to remember that the equation for Ampere's Law only applies to an Amperian loop that has a constant current flowing through it. Ampere's Law is shown below:

[[File:AmperesLaw11112.jpg]]

The following pictures display how to use Ampere's Law when trying to calculate the magnetic field of a current flowing out of the page at distance ''r'' away from the wire.

[[File:AmperesLaw2.jpg]]

The diagram above shows a current-carrying wire with its current flowing out of the page. Using the right-hand rule, we can determine the direction of the magnetic field to be as shown. The dotted circle represents the Amperian loop of radius ''r''.

The workings below, shows how Ampere's Law is used to calculate the magnetic field of the current-carrying wire at a distance ''r''. Note that on the third line of working, the dot product disappears. This occurs because the direction of the magnetic field is the same as the direction of ''dl''. More specifically, the angle between ''dl'' and the magnetic field ''B'' is 0. Remember the equation of the dot product is '''|a| x |b| x cos (theta)'''. So in this case, ''a = B'', ''b = dl'' and ''theta = 0''. So if ''theta = 0'' we know that ''cos (0) = 1''. In the next line of working ''B'' is pulled out of the closed integral, this occurs because all along the Amperian loop the magnetic field is the same, it does not change in magnitude. In the next line of working, once we added up all the ''dl'' 's on the Amperian loop, that will give us the circumference of the Amperian loop. From there we make ''B'' the subject of the equation.

[[File:AmperesLaw3.jpg]]

We will see how one can use Ampere's Law to calculate the magnetic field of a long thick wire in examples below.

==Examples==

===Simple===

A long thick wire has radius ''r = 0.1 m'' and a constant current of ''I = 0.15 A''. The Amperian loop within the wire has radius ''r = 0.03 m''. Calculate the magnetic field through the Amperian loop.

[[File:Question1.jpg]]

Explanation of the solution:

[[File:Question12.jpg]]

* 1) Write out Ampere's Law. The ''enc'' written as a subscript of ''I'' is short for enclosed.

* 2) For the current, you only want the current that is enclosed within the loop of radius ''r'', not all of the current enclosed within the loop of radius ''R''. To calculate the desired current, multiply the given current ''I'' by a ratio of the Amperian loop's area over the area of long thick wire.

* 3) Substitute the desired current into the equation.

[[File:Question22.jpg]]

* 4) Remove the dot product on the right hand side of the equation because the direction of the magnetic field on the Amperian loop is the same as the direction of ''dl''

* 5) Pull ''B'' out of the closed integral because it is the same all along the Amperian loop.

* 6) Complete the integral to give you the circumference of the Amperian loop and make ''B'' the subject of the equation.

* 7) Substitute in the values to give you '''9e-8 T'''

===Middling===

A long straight wire suspended in the air carries a conventional current of 7.4 amperes in the -x direction as shown (the wire runs along the x-axis). At a particular instant an electron at location < 0, -0.004, 0 > m has velocity < -3.5 e5, -4.2 e5, 0 > m/s. What is the magnetic field due to the wire at the location of the electron?

Solution: <math>{B = \frac{μ_0(7.4)}{2π(0.004)}}</math> and use right hand rule. The answer is < 0, 0, 3.7e-4 > T.

===Hard===

Using the same values as in the middling problem calculate the magnetic force on the electron due to the wire.

Solution: The formula for magnetic force is <math>qv×B</math>. So by using the value for B computed above and calculating the cross product between B and qv we find that this equals < 2.49e-17, -2.07e-17, 0 > N. Don't forget that because this is the force on an electron q is negative.

==History==

In 1826, it was Hans Christian Oersted who had discovered the connection between magnetism and electricity when he found that the needle of a compass he was holding was deflected when he moved it close to a current carrying wire during one of his lectures. It was this discovery that prompted Andre-Marie Ampere to start his own research. Ampere was conducting experiments to learn more about the magnetic fields created by currents. Ampere found that when he placed two wires next to each other with current running in the same direction, the wires were attracted to each other and when the current was running in opposite directions, the wires repelled each other! Experiments like this and more gave way to the creation of Ampere's Law that helps people calculate the strength of magnetic fields and currents in particular situations.

== See also ==

See the page on Ampere's Law for a more in depth look at the law itself: [[Ampere's Law]]

For more applications of Ampere's Law see: [[Magnetic Field of a Toroid Using Ampere's Law]] and [[Magnetic Field of Coaxial Cable Using Ampere's Law]]

===Further reading===

Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley.

===External links===

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magcur.html

https://www.khanacademy.org/science/physics/magnetic-forces-and-magnetic-fields/magnetic-field-current-carrying-wire/v/magnetism-6-magnetic-field-due-to-current

https://en.wikibooks.org/wiki/Electrodynamics/Ampere%27s_Law

==References==

Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley. page 887.

[[Category:Maxwell's Equations]]

File:Question22.jpg

2017-04-10T03:42:17Z

Otoyobo3:

File:Question12.jpg

2017-04-10T03:42:04Z

Otoyobo3:

File:Question1.jpg

2017-04-10T03:39:59Z

Otoyobo3:

File:AmperesLaw11112.jpg

2017-04-10T03:23:12Z

Otoyobo3:

File:AmperesLaw11111.jpg

2017-04-10T03:22:15Z

Otoyobo3:

Magnetic Field of a Long Thick Wire Using Ampere's Law

2017-04-10T03:04:52Z

Otoyobo3:

File:AmperesLaw22.png

2017-04-10T03:03:52Z

Otoyobo3:

File:AmperesLaw3.jpg

2017-04-10T02:57:21Z

Otoyobo3:

File:AmperesLaw2.jpg

2017-04-10T02:54:15Z

Otoyobo3:

File:AmperesLaw1111.jpg

2017-04-10T02:52:51Z

Otoyobo3:

File:AmperesLaw111.jpg

2017-04-10T02:52:01Z

Otoyobo3:

Magnetic Field of a Long Thick Wire Using Ampere's Law

2017-04-10T02:50:29Z

Otoyobo3: /* Mathematical Model for Ampere's Law */

File:AmperesLaw11.jpg

2017-04-10T02:50:23Z

Otoyobo3:

Magnetic Field of a Long Thick Wire Using Ampere's Law

2017-04-10T02:48:02Z

Otoyobo3:

File:AmperesLaw1.jpg

2017-04-10T02:47:42Z

Otoyobo3:

File:Mple.jpg

2017-04-10T01:30:09Z

Otoyobo3:

Magnetic Field of a Long Thick Wire Using Ampere's Law

2017-04-10T01:28:44Z

Otoyobo3: /* Mathematical Model for Ampere's Law */

'''Claimed by Oluwasanmi Toyobo (Spring 2017)'''

==The Main Idea==

This page will explain and show how to use Ampere's Law to find the magnetic field of a long thick wire. It is much easier to calculate the magnetic field using Ampere's Law as opposed to Biot-Savart Law.

===Mathematical Model for Ampere's Law===

Ampere's Law is a quantitative relationship between the pattern of the magnetic field along a closed path and the amount of constant current passing through that path. It's important to remember that the equation for Ampere's Law applies to any kind of loop that has a constant current flowing through it. Ampere's Law is shown below:

[[File:Example.jpg]]

We will see how one can use Ampere's Law to calculate the magnetic field of a long thick wire in examples below.

==Examples==

===Simple===

What is the magnetic field at a point 0.03 m away from a wire that has a current of 7 amperes?

Solution: <math>I=7</math> and <math>r=0.03</math>. So inserting this into the formula gives <math>{B = \frac{μ_0(7)}{2π(0.03)}}</math>. This results in 4.67e-5 T

===Middling===

A long straight wire suspended in the air carries a conventional current of 7.4 amperes in the -x direction as shown (the wire runs along the x-axis). At a particular instant an electron at location < 0, -0.004, 0 > m has velocity < -3.5 e5, -4.2 e5, 0 > m/s. What is the magnetic field due to the wire at the location of the electron?

Solution: <math>{B = \frac{μ_0(7.4)}{2π(0.004)}}</math> and use right hand rule. The answer is < 0, 0, 3.7e-4 > T.

===Hard===

Using the same values as in the middling problem calculate the magnetic force on the electron due to the wire.

Solution: The formula for magnetic force is <math>qv×B</math>. So by using the value for B computed above and calculating the cross product between B and qv we find that this equals < 2.49e-17, -2.07e-17, 0 > N. Don't forget that because this is the force on an electron q is negative.

==History==

In 1826, it was Hans Christian Oersted who had discovered the connection between magnetism and electricity when he found that the needle of a compass he was holding was deflected when he moved it close to a current carrying wire during one of his lectures. It was this discovery that prompted Andre-Marie Ampere to start his own research. Ampere was conducting experiments to learn more about the magnetic fields created by currents. Ampere found that when he placed two wires next to each other with current running in the same direction, the wires were attracted to each other and when the current was running in opposite directions, the wires repelled each other! Experiments like this and more gave way to the creation of Ampere's Law that helps people calculate the strength of magnetic fields and currents in particular situations.

== See also ==

See the page on Ampere's Law for a more in depth look at the law itself: [[Ampere's Law]]

For more applications of Ampere's Law see: [[Magnetic Field of a Toroid Using Ampere's Law]] and [[Magnetic Field of Coaxial Cable Using Ampere's Law]]

===Further reading===

Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley.

===External links===

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magcur.html

https://www.khanacademy.org/science/physics/magnetic-forces-and-magnetic-fields/magnetic-field-current-carrying-wire/v/magnetism-6-magnetic-field-due-to-current

https://en.wikibooks.org/wiki/Electrodynamics/Ampere%27s_Law

==References==

Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley. page 887.

[[Category:Maxwell's Equations]]

Magnetic Field of a Long Thick Wire Using Ampere's Law

2017-04-10T01:27:28Z

Otoyobo3:

'''Claimed by Oluwasanmi Toyobo (Spring 2017)'''

==The Main Idea==

This page will explain and show how to use Ampere's Law to find the magnetic field of a long thick wire. It is much easier to calculate the magnetic field using Ampere's Law as opposed to Biot-Savart Law.

===Mathematical Model for Ampere's Law===

Ampere's Law is a quantitative relationship between the pattern of the magnetic field along a closed path and the amount of constant current passing through that path. It's important to remember that the equation for Ampere's Law applies to any kind of loop that has a constant current flowing through it. Ampere's Law is shown below:

*insert picture of Ampere's Law* [[File:ampereslaw.jpg]]

We will see how one can use Ampere's Law to calculate the magnetic field of a long thick wire in examples below.

==Examples==

===Simple===

What is the magnetic field at a point 0.03 m away from a wire that has a current of 7 amperes?

Solution: <math>I=7</math> and <math>r=0.03</math>. So inserting this into the formula gives <math>{B = \frac{μ_0(7)}{2π(0.03)}}</math>. This results in 4.67e-5 T

===Middling===

A long straight wire suspended in the air carries a conventional current of 7.4 amperes in the -x direction as shown (the wire runs along the x-axis). At a particular instant an electron at location < 0, -0.004, 0 > m has velocity < -3.5 e5, -4.2 e5, 0 > m/s. What is the magnetic field due to the wire at the location of the electron?

Solution: <math>{B = \frac{μ_0(7.4)}{2π(0.004)}}</math> and use right hand rule. The answer is < 0, 0, 3.7e-4 > T.

===Hard===

Using the same values as in the middling problem calculate the magnetic force on the electron due to the wire.

Solution: The formula for magnetic force is <math>qv×B</math>. So by using the value for B computed above and calculating the cross product between B and qv we find that this equals < 2.49e-17, -2.07e-17, 0 > N. Don't forget that because this is the force on an electron q is negative.

==History==

In 1826, it was Hans Christian Oersted who had discovered the connection between magnetism and electricity when he found that the needle of a compass he was holding was deflected when he moved it close to a current carrying wire during one of his lectures. It was this discovery that prompted Andre-Marie Ampere to start his own research. Ampere was conducting experiments to learn more about the magnetic fields created by currents. Ampere found that when he placed two wires next to each other with current running in the same direction, the wires were attracted to each other and when the current was running in opposite directions, the wires repelled each other! Experiments like this and more gave way to the creation of Ampere's Law that helps people calculate the strength of magnetic fields and currents in particular situations.

== See also ==

See the page on Ampere's Law for a more in depth look at the law itself: [[Ampere's Law]]

For more applications of Ampere's Law see: [[Magnetic Field of a Toroid Using Ampere's Law]] and [[Magnetic Field of Coaxial Cable Using Ampere's Law]]

===Further reading===

Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley.

===External links===

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magcur.html

https://www.khanacademy.org/science/physics/magnetic-forces-and-magnetic-fields/magnetic-field-current-carrying-wire/v/magnetism-6-magnetic-field-due-to-current

https://en.wikibooks.org/wiki/Electrodynamics/Ampere%27s_Law

==References==

Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley. page 887.

[[Category:Maxwell's Equations]]

File:Ampereslaw.jpg

2017-04-10T01:26:29Z

Otoyobo3:

Magnetic Field of a Long Thick Wire Using Ampere's Law

2017-04-10T00:24:41Z

Otoyobo3:

'''Claimed by Oluwasanmi Toyobo (Spring 2017)'''

==The Main Idea==

This page will explain and show how to use Ampere's Law to find the magnetic field of a long thick wire. It is much easier to calculate the magnetic field using Ampere's Law as opposed to Biot-Savart Law.

===Formula for Ampere's Law===

Here is the proof:

[[File:Ampere's_Law_Proof.png]]

[[File:Ampere's_Law.png]]

==How to Find Magnetic Field of A Long Thick Wire==

[[File:Wire.png]]

To find the magnetic field <math>B</math> at a distance <math>r</math> from the center of the long wire apply Ampere's Law. By the symmetry of the wire <math>B</math> will always be constant and tangential to the circular path at every point around the wire.

The magnetic field is everywhere parallel to the path for a circular path centered on wire. The direction of magnetic field can be determined by the right hand rule. The magnetic field's direction is perpendicular to the wire and is in the direction the fingers curl if you wrap the wire around. The direction of current is where your thumb points to.

[[File:Ampere's_Picture.png]]

The path integral <math>{{\oint}d\vec{l}}</math> in this situation is equal to the circumference of the circular path around the wire. This is equal to <math>2πr</math>.

Using the formula above and plugging in <math>{d\vec{l}}</math> we have: <math>{B(2πr) = μ_0I}</math>. To solve for <math>B</math> divide both sides by <math>2πr</math>.

This results in the equation: <math>{B = \frac{μ_0I}{2πr}}</math> which is equal to <math>{\frac{μ_02I}{4πr}}</math>. This is the equation for the magnetic field of a long thick wire that is found using the Biot-Savart law.

==Examples==

===Simple===

What is the magnetic field at a point 0.03 m away from a wire that has a current of 7 amperes?

Solution: <math>I=7</math> and <math>r=0.03</math>. So inserting this into the formula gives <math>{B = \frac{μ_0(7)}{2π(0.03)}}</math>. This results in 4.67e-5 T

===Middling===

A long straight wire suspended in the air carries a conventional current of 7.4 amperes in the -x direction as shown (the wire runs along the x-axis). At a particular instant an electron at location < 0, -0.004, 0 > m has velocity < -3.5 e5, -4.2 e5, 0 > m/s. What is the magnetic field due to the wire at the location of the electron?

Solution: <math>{B = \frac{μ_0(7.4)}{2π(0.004)}}</math> and use right hand rule. The answer is < 0, 0, 3.7e-4 > T.

===Hard===

Using the same values as in the middling problem calculate the magnetic force on the electron due to the wire.

Solution: The formula for magnetic force is <math>qv×B</math>. So by using the value for B computed above and calculating the cross product between B and qv we find that this equals < 2.49e-17, -2.07e-17, 0 > N. Don't forget that because this is the force on an electron q is negative.

== See also ==

See the page on Ampere's Law for a more in depth look at the law itself: [[Ampere's Law]]

For more applications of Ampere's Law see: [[Magnetic Field of a Toroid Using Ampere's Law]] and [[Magnetic Field of Coaxial Cable Using Ampere's Law]]

===Further reading===

Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley.

===External links===

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magcur.html

https://www.khanacademy.org/science/physics/magnetic-forces-and-magnetic-fields/magnetic-field-current-carrying-wire/v/magnetism-6-magnetic-field-due-to-current

https://en.wikibooks.org/wiki/Electrodynamics/Ampere%27s_Law

==References==

Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley. page 887.

[[Category:Maxwell's Equations]]