Motional Emf using Faraday's Law - Revision history

Patelanuj: /* The Main Idea */

2026-04-27T04:24:12Z

The Main Idea

← Older revision		Revision as of 00:24, 27 April 2026
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	Claimed by Anuj Patel, Spring 2026		Claimed by Anuj Patel, Spring 2026
	==The Main Idea==		==The Main Idea==
			Motional EMF is an electromotive force (EMF) induced in a conductor due to relative motion between the conductor and a magnetic field. The magnetic field itself does not need to change in time. What matters is that the magnetic flux through a loop changes, and motion is one way to cause that. This is captured by Faraday's Law:
	Motional EMF is an electromotive force (EMF) induced in a conductor due to relative motion between the conductor and a magnetic field. The magnetic field itself does ~~'''~~not~~'''~~ need to change in time. What matters is that the magnetic flux through a loop changes, and motion is one way to cause that. This is captured by Faraday's Law:

	<div align="center"><math>emf = -\frac{d\Phi_B}{dt}</math></div>		<div align="center"><math>emf = -\frac{d\Phi_B}{dt}</math></div>
			The negative sign is an important aspect to Lenz's Law: the induced current always flows in a direction such that its own magnetic field opposes the change in flux that caused it. This comes from conservation of energy: if the induced current helped the flux change instead of opposing it, the bar would accelerate without any external force, creating energy from nothing.

	The negative sign encodes '''Lenz's Law''': the induced current always flows in a direction such that its own magnetic field opposes the change in flux that caused it. This comes from conservation of energy. If the induced current helped the flux change instead of opposing it, it would create energy from nothing.		To understand why a moving conductor develops an EMF, think about what happens at the microscopic level. When a conducting bar moves with velocity <math>{\vec{v}}</math> through a magnetic field <math>{\vec{B}}</math>, the free electrons inside are carried along with it. These moving electrons experience a magnetic force <math>{\vec{F} = q\vec{v} \times \vec{B}}</math>. For electrons (charge <math>{-e}</math>), this pushes them toward one end of the bar, causing charge separation. One end becomes negative, the other positive. This creates an internal electric field that opposes further charge buildup. In an isolated bar, this equilibrium is reached quickly and no current flows. But if the bar is part of a closed loop, the separated charges can flow continuously around the circuit, driving a continuous induced current through the circuit.

	To understand ''why'' a moving conductor develops an EMF, think about what happens at the microscopic level. When a conducting bar moves with velocity <math>{\vec{v}}</math> through a magnetic field <math>{\vec{B}}</math>, the free electrons inside are carried along with it. These moving electrons experience a magnetic force <math>{\vec{F} = q\vec{v} \times \vec{B}}</math>. For electrons (charge <math>{-e}</math>), this pushes them toward one end of the bar, causing charge separation. One end becomes negative, the other positive. This creates an internal electric field that opposes further charge buildup. In an isolated bar, this equilibrium is reached quickly and no current flows. But if the bar is part of a ~~'''~~closed loop~~'''~~, the separated charges can flow continuously around the circuit, ~~giving us an~~ induced current ~~driven by the motional EMF.~~

	~~Now consider a bar magnet approaching a horizontal wire loop. As the magnet gets closer, the flux~~ through the loop increases. This induces an EMF which drives a current. By Lenz's Law, that current flows in the direction that opposes the increasing flux, pushing back against the approaching magnet. The faster the magnet moves, the stronger the effect.

	To ~~help visualize~~ this, ~~I built an interactive simulation in the Computational Model section where you can move~~ a sliding bar ~~along a U-shaped rail in a uniform~~ magnetic field and ~~see~~ the loop area ~~change~~, the induced ~~EMF update~~ in ~~real time~~, and the current ~~direction flip depending on which way~~ the bar moves~~. You can also flip~~ the ~~direction of~~ the ~~magnetic field~~. ~~Here it is:~~		To see this in action, consider a rectangular loop formed by two rails and a sliding bar. As the bar moves, the area of the loop changes, which changes the magnetic flux <math>\Phi_B = BA</math> through it even though the field <math>{\vec{B}}</math> stays constant. When the bar moves outward and increases the loop area, the flux increases, and by Lenz's Law the induced current flows in the direction that opposes that increase. When the bar moves inward and decreases the loop area, the flux decreases, and the induced current reverses to oppose the decrease. The faster the bar moves, the greater <math>\frac{d\Phi_B}{dt} = Bwv</math> and the stronger the induced EMF.

	https://glowscript.org/#/user/anujpatel/folder/MyPrograms/program/wiki		An interactive simulation in the Computational Model section allows you to move the sliding bar and see the induced EMF update in real time and the current direction arrows flip depending on which way the bar moves. The direction of the magnetic field can also be flipped. [https://glowscript.org/#/user/anujpatel/folder/MyPrograms/program/wiki View the simulation here.]

	~~Originally~~ motional EMF ~~was calculated~~ directly as <math>{emf = vBL}</math>, where <math>{\vec{v}}</math> is the velocity of the conductor and <math>{L}</math> is its length. Writing EMF in terms of magnetic flux using Faraday's Law is more general though, since it applies whether flux changes due to motion, a changing field, or a changing loop orientation. For more detail see [[Motional Emf]] and [[Faraday's Law]].		The motional EMF can also be expressed directly as <math>{emf = vBL}</math>, where <math>{\vec{v}}</math> is the velocity of the conductor and <math>{L}</math> is its length. This follows directly from the magnetic force on the charges in the bar. Writing EMF in terms of magnetic flux using Faraday's Law is more general though, since it applies whether flux changes due to motion, a changing field, or a changing loop orientation. For more detail see [[Motional Emf]] and [[Faraday's Law]].

	==A Mathematical Model==		==A Mathematical Model==

Patelanuj: added the glowscript simulator

2026-04-27T04:15:10Z

added the glowscript simulator

← Older revision		Revision as of 00:15, 27 April 2026
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	Now consider a bar magnet approaching a horizontal wire loop. As the magnet gets closer, the flux through the loop increases. This induces an EMF which drives a current. By Lenz's Law, that current flows in the direction that opposes the increasing flux, pushing back against the approaching magnet. The faster the magnet moves, the stronger the effect.		Now consider a bar magnet approaching a horizontal wire loop. As the magnet gets closer, the flux through the loop increases. This induces an EMF which drives a current. By Lenz's Law, that current flows in the direction that opposes the increasing flux, pushing back against the approaching magnet. The faster the magnet moves, the stronger the effect.

	To help visualize this, I built an interactive simulation in the Computational Model section where you can move a sliding bar along a U-shaped rail in a uniform magnetic field and see the loop area change, the induced EMF update in real time, and the current direction flip depending on which way the bar moves.		To help visualize this, I built an interactive simulation in the Computational Model section where you can move a sliding bar along a U-shaped rail in a uniform magnetic field and see the loop area change, the induced EMF update in real time, and the current direction flip depending on which way the bar moves. You can also flip the direction of the magnetic field. Here it is:

			https://glowscript.org/#/user/anujpatel/folder/MyPrograms/program/wiki

	Originally motional EMF was calculated directly as <math>{emf = vBL}</math>, where <math>{\vec{v}}</math> is the velocity of the conductor and <math>{L}</math> is its length. Writing EMF in terms of magnetic flux using Faraday's Law is more general though, since it applies whether flux changes due to motion, a changing field, or a changing loop orientation. For more detail see [[Motional Emf]] and [[Faraday's Law]].		Originally motional EMF was calculated directly as <math>{emf = vBL}</math>, where <math>{\vec{v}}</math> is the velocity of the conductor and <math>{L}</math> is its length. Writing EMF in terms of magnetic flux using Faraday's Law is more general though, since it applies whether flux changes due to motion, a changing field, or a changing loop orientation. For more detail see [[Motional Emf]] and [[Faraday's Law]].

Patelanuj: Changed main section. still have to add computational model

2026-04-27T03:51:41Z

Changed main section. still have to add computational model

← Older revision		Revision as of 23:51, 26 April 2026
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			Claimed by Anuj Patel, Spring 2026
	==The Main Idea==		==The Main Idea==

	~~We have previously shown that~~ an electromotive force (~~emf) can be generated by changing the magnetic flux (see week 10, Motional~~ EMF)~~. The emf generated is directly proportional to the negative rate of change of the magnetic flux. The negative sign is~~ a ~~representative of the conservation of energy or Lenz's Law. Basically the magnetic field created by the induced current (induced by the external magnetic field) needs~~ to ~~oppose~~ the external magnetic field otherwise new energy is created which breaks the law of conservation of energy. In the examples provided in the section "Motional Emf" the magnetic flux is changed by changing the areas of the loops. But as per the Magnetic Flux equations, the flux can also be changed by varying the magnitude and~~/or direction of the applied~~ magnetic field. ~~So what happens when the~~ magnetic field ~~changes and~~ not ~~the area? This is were Faraday comes~~ in. ~~He discovered through his experiments~~ that the ~~Magnetic flux equation is valid no matter how the~~ flux is ~~changing which allowed him~~ to ~~relate the electric and magnetic fields in a new law~~. ~~Which~~ is ~~named after him as~~ Faraday's Law ~~but Maxwell also wrote down the differential form of the same law before him.~~		Motional EMF is an electromotive force (EMF) induced in a conductor due to relative motion between the conductor and a magnetic field. The magnetic field itself does '''not''' need to change in time. What matters is that the magnetic flux through a loop changes, and motion is one way to cause that. This is captured by Faraday's Law:

	~~When a wire moves through an area with applied magnetic field with velocity~~ <~~math~~>~~{\vec{v}}~~</math>~~, a current begins to flow across it. The wire moving means that the electrons inside the wire are also moving with the same velocity <math>{~~\vec{v}}</math> as the wire. As you remember from previous sections that a charge moving in an external magnetic field experience a magnetic force. This force then pushes the electrons to a direction determined by the cross product of the velocity of the electrons <math>{\~~vec{v}~~}~~</math> and the magnetic field <math>{\vec~~{B}}</math> ~~times the charge of an electron which is <math>{-e}~~</~~math~~>. This makes the electrons accumulate on one side of the wire, thus polarizing the wire. This polarization gives rise to an electric field inside the wire which creates a force on the electrons opposite to the magnetic force on the electrons, hence, in steady the electrons in a wire experience zero net force. Now imagine the ends of the wire are connected to a load. This allows the accumulated electrons to flow around the loop. This means that the electric field decreases hence the net force is doesn't go to zero in steady-state which means a current is induced in the loop and hence and electromotive force (emf).		<div align="center"><math>emf = -\frac{d\Phi_B}{dt}</math></div>

	~~Originally, we calculated~~ the ~~motional emf~~ in a moving bar ~~by using~~ the ~~equation~~ <math>{\~~frac~~{q(\vec{v} \times \vec{B})L}{q}}</math> where <math>{\vec{v}}</math> is the velocity of the ~~bar~~ and <math>{L}</math> is ~~the bar~~ length. ~~However, writing an equation for emf~~ in terms of magnetic flux ~~can yield simpler calculations. Motional emf has~~ a ~~differential relationship to magnetic flux. If an enclosed magnetic~~ field ~~remains constant but the~~ loop ~~changes shape or~~ orientation~~, the resulting change in the area leads to a change in magnetic flux~~. For ~~an in-depth conceptual breakdown of motional emf~~ see [[Motional Emf]]~~, for more details on other applications of Faraday's law see~~ [[Faraday's Law]].		The negative sign encodes '''Lenz's Law''': the induced current always flows in a direction such that its own magnetic field opposes the change in flux that caused it. This comes from conservation of energy. If the induced current helped the flux change instead of opposing it, it would create energy from nothing.

			To understand ''why'' a moving conductor develops an EMF, think about what happens at the microscopic level. When a conducting bar moves with velocity <math>{\vec{v}}</math> through a magnetic field <math>{\vec{B}}</math>, the free electrons inside are carried along with it. These moving electrons experience a magnetic force <math>{\vec{F} = q\vec{v} \times \vec{B}}</math>. For electrons (charge <math>{-e}</math>), this pushes them toward one end of the bar, causing charge separation. One end becomes negative, the other positive. This creates an internal electric field that opposes further charge buildup. In an isolated bar, this equilibrium is reached quickly and no current flows. But if the bar is part of a '''closed loop''', the separated charges can flow continuously around the circuit, giving us an induced current driven by the motional EMF.

			Now consider a bar magnet approaching a horizontal wire loop. As the magnet gets closer, the flux through the loop increases. This induces an EMF which drives a current. By Lenz's Law, that current flows in the direction that opposes the increasing flux, pushing back against the approaching magnet. The faster the magnet moves, the stronger the effect.

			To help visualize this, I built an interactive simulation in the Computational Model section where you can move a sliding bar along a U-shaped rail in a uniform magnetic field and see the loop area change, the induced EMF update in real time, and the current direction flip depending on which way the bar moves.

			Originally motional EMF was calculated directly as <math>{emf = vBL}</math>, where <math>{\vec{v}}</math> is the velocity of the conductor and <math>{L}</math> is its length. Writing EMF in terms of magnetic flux using Faraday's Law is more general though, since it applies whether flux changes due to motion, a changing field, or a changing loop orientation. For more detail see [[Motional Emf]] and [[Faraday's Law]].

	==A Mathematical Model==		==A Mathematical Model==

Patelanuj at 03:31, 27 April 2026

2026-04-27T03:31:20Z

← Older revision		Revision as of 23:31, 26 April 2026
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	~~Claimed by Anuj Patel, Spring 2026~~
	==The Main Idea==		==The Main Idea==

Patelanuj at 03:19, 27 April 2026

2026-04-27T03:19:32Z

← Older revision		Revision as of 23:19, 26 April 2026
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	Claimed by ~~Aditya Bhairampally~~, ~~Fall 2025~~		Claimed by Anuj Patel, Spring 2026
	==The Main Idea==		==The Main Idea==

Abhairampally3: /* Python Visual */

2025-12-02T03:30:45Z

Python Visual

← Older revision		Revision as of 23:30, 1 December 2025
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	print("our conductor is now polerized")		print("our conductor is now polerized")

	~~[[File:Polarization of a metal bar.webm\|autoplay\|loop\|width=400\|alt=Video for class Wiki]]~~

	Here is the GlowScript simulation: https://www.glowscript.org/#/user/abhairampally/folder/Public/program/MotionalEMFbyFaraday'sLaw. Be sure to play around with the code and rates of the loops to properly see the movement of charge as the bar magnet moves towards the capacitor!		Here is the GlowScript simulation: https://www.glowscript.org/#/user/abhairampally/folder/Public/program/MotionalEMFbyFaraday'sLaw. Be sure to play around with the code and rates of the loops to properly see the movement of charge as the bar magnet moves towards the capacitor!

Abhairampally3: /* Python Visual */

2025-12-02T03:28:40Z

Python Visual

← Older revision		Revision as of 23:28, 1 December 2025
Line 237:		Line 237:
	print("our conductor is now polerized")		print("our conductor is now polerized")

	[[File:Polarization of a metal bar.webm\|~~thumb~~\|Video for class Wiki]]		[[File:Polarization of a metal bar.webm\|autoplay\|loop\|width=400\|alt=Video for class Wiki]]

	Here is the GlowScript simulation: https://www.glowscript.org/#/user/abhairampally/folder/Public/program/MotionalEMFbyFaraday'sLaw. Be sure to play around with the code and rates of the loops to properly see the movement of charge as the bar magnet moves towards the capacitor!		Here is the GlowScript simulation: https://www.glowscript.org/#/user/abhairampally/folder/Public/program/MotionalEMFbyFaraday'sLaw. Be sure to play around with the code and rates of the loops to properly see the movement of charge as the bar magnet moves towards the capacitor!

Abhairampally3: /* Python Visual */

2025-12-02T03:27:20Z

Python Visual

← Older revision		Revision as of 23:27, 1 December 2025
Line 236:		Line 236:

	print("our conductor is now polerized")		print("our conductor is now polerized")

			[[File:Polarization of a metal bar.webm\|thumb\|Video for class Wiki]]

	Here is the GlowScript simulation: https://www.glowscript.org/#/user/abhairampally/folder/Public/program/MotionalEMFbyFaraday'sLaw. Be sure to play around with the code and rates of the loops to properly see the movement of charge as the bar magnet moves towards the capacitor!		Here is the GlowScript simulation: https://www.glowscript.org/#/user/abhairampally/folder/Public/program/MotionalEMFbyFaraday'sLaw. Be sure to play around with the code and rates of the loops to properly see the movement of charge as the bar magnet moves towards the capacitor!

Abhairampally3: /* Accelerating rod simulation */

2025-12-02T02:43:52Z

Accelerating rod simulation

← Older revision		Revision as of 22:43, 1 December 2025
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	The following simulation shows the motional EMF in a conducting rod that accelerates along two parallel rails within a uniform magnetic field. As the rod moves, the charges inside experience a magnetic force given by <math>\vec{F} = q \, \vec{v} \times \vec{B}</math>, where <math>\mathbf{\vec{v}}</math> is the instantaneous velocity of the rod. This force separates positive and negative charges, creating an induced electric field along the rod, which produces an EMF.		The following simulation shows the motional EMF in a conducting rod that accelerates along two parallel rails within a uniform magnetic field. As the rod moves, the charges inside experience a magnetic force given by <math>\vec{F} = q \, \vec{v} \times \vec{B}</math>, where <math>\mathbf{\vec{v}}</math> is the instantaneous velocity of the rod. This force separates positive and negative charges, creating an induced electric field along the rod, which produces an EMF.

	Because the rod is accelerating, its velocity v = at increases over time causing the motional EMF which is calculated as <math>E = B L \, \mathbf{v}</math> to increase linearly as the rod moves.		Because the rod is accelerating, its velocity <math display="block">v = at</math> increases over time causing the motional EMF which is calculated as <math>E = B L \, \mathbf{v}</math> to increase linearly as the rod moves.

	This simulation shows how Faraday's law of induction applies when a conductor moves with changing velocity through a magnetic field.		This simulation shows how Faraday's law of induction applies when a conductor moves with changing velocity through a magnetic field.

Abhairampally3: /* References */

2025-12-02T02:20:47Z

References

← Older revision		Revision as of 22:20, 1 December 2025
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	https://www.physicsbootcamp.org/Motional-EMF.html		https://www.physicsbootcamp.org/Motional-EMF.html

			https://en.wikipedia.org/wiki/Faraday's_law_of_induction#Motional_emf

	[[Category: Faraday's Law]]		[[Category: Faraday's Law]]