Motional Emf using Faraday's Law

The Main Idea

When a wire moves through an area of magnetic field, a current begins to flow along the wire as a result of magnetic forces. Originally, we calculated the motional emf in a moving bar by using the equation [math]\displaystyle{ {\frac{q(\vec{v} \times \vec{B})L}{q}} }[/math] where v is the velocity of the bar and L 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 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.

A Mathematical Model

Motional emf results when the area enclosing a constant magnetic field changes. Let's observe a specific scenario in which a bar of length L slides along two frictionless bars. We can observe the change in area over a short time as [math]\displaystyle{ \Delta{A} = L\Delta{x} = Lv\Delta{t} }[/math]. We already know that magnetic flux is defined by the formula: [math]\displaystyle{ \Phi_m = \int\! \vec{B} \cdot\vec{n}dA }[/math]. In the case that v is perpendicular to B, we combine these to get: [math]\displaystyle{ \frac{\Delta{\Phi_m}}{\Delta{t}} = BLv }[/math].

Emf is said to be the work done per unit charge: [math]\displaystyle{ emf = \frac{FL}{q} = \frac{qvBL}{q} = vBL }[/math] (again, we are assuming v is perpendicular to B).

Comparing the above two formulas, we can clearly see that [math]\displaystyle{ |{emf}| = |\frac{d\Phi_m}{dt}| }[/math]. This is exactly what Faraday's Law tells us!

Faraday's Law is defined as: [math]\displaystyle{ emf = \int\! \vec{E} \cdot d\vec{l} = -\frac{d}{dt} \int\! \vec{B} \cdot \vec{n}dA }[/math]

where [math]\displaystyle{ \vec{E} }[/math] is the Non-Coulomb electric field along the path, [math]\displaystyle{ l }[/math] is the length of the path you're integrating on, [math]\displaystyle{ \vec{B} }[/math] is the magnetic field inside the area enclosed, and [math]\displaystyle{ \vec{n} }[/math] is the unit vector perpendicular to area A.

A Computational Model

NOTE: The magnitude of the magnetic field is constant, the phrase B increasing refers to the area on the loop enclosing a larger area of magnetic field as time passes

In the image shown to the left, a bar of length [math]\displaystyle{ L }[/math] is moving along two other bars from right to left. The blue circles containing "x"s represent a magnetic field directed into the page. As the bar moves to the right, the system encloses a greater amount of magnetic field.

To explain this concept more clearly, take a look at the figures above. This image shows a bar moving in a magnetic field at two different times. In the first picture, at time [math]\displaystyle{ t_1 }[/math], the system encircles half of two individual magnetic field circles. However, in the second picture taken at time [math]\displaystyle{ t_2 }[/math], the system now encircles 6 full magnetic field circles. Of course, this explanation isn't using technical terms, but the point still stands: the enclosed magnetic field is increasing as time increases.

Returning to the scenario in the first image, because the magnetic field is not constant, we can use Faraday's Law to solve for the motional emf.

As stated above, the formula is as follows:

[math]\displaystyle{ emf = -\frac{d}{dt} \int\! \vec{B} \cdot \vec{n}dA }[/math]

First, integrate the integral with respect to the area of the rectangle enclosed.

[math]\displaystyle{ emf = -\frac{d}{dt} (\vec{B} \cdot \vec{n}A) }[/math]

We have the dimensions of the bar in variables: length [math]\displaystyle{ L }[/math] and width [math]\displaystyle{ x }[/math]. Substitute these values for the area, [math]\displaystyle{ A }[/math]

[math]\displaystyle{ emf = -\frac{d}{dt} (\vec{B} \cdot \vec{n}(L)(x)) }[/math]

Now that we have this formula, we have to figure out how to take its derivative with respect to [math]\displaystyle{ t }[/math]. Which of the magnitudes of these values is changing?

The magnitude of the magnetic field is constant. (More "circles" are added as time increases, but the magnitude of each "circle" does not change.

The magnitude of the normal vector is constant.

The length, [math]\displaystyle{ L }[/math], of the bar is constant.

The width of the surface enclosed, [math]\displaystyle{ x }[/math], changes.

As a result, the formula now becomes:

[math]\displaystyle{ emf = (\vec{B} \cdot \vec{n}(L))\left(-\frac{d}{dt}(x)\right) }[/math]

In this case, [math]\displaystyle{ \frac{dx}{dt} = \vec{v} }[/math] because [math]\displaystyle{ x }[/math] is a function of time, where [math]\displaystyle{ \vec{v} }[/math] is the velocity of the moving bar. Substituting that in, we get:

[math]\displaystyle{ emf = (\vec{B} \cdot \vec{n}(L))\vec{v} }[/math]

Plugging in these values, we can solve for the motional emf of the bar.

Because the magnetic field is changing with time, however, there is also an induced current flowing through the circuit. We can find the direction of the current using the right hand rule. To do this, we can use 2 different methods:

1. We can use the equation [math]\displaystyle{ \vec{F} = q\vec{v} \times \vec{B} }[/math], where [math]\displaystyle{ \vec{F} }[/math] is the force on the bar, and [math]\displaystyle{ \vec{v} }[/math] is the velocity of the bar. Using the right hand rule, we can point our fingers in the direction of the velocity of the bar and curl them in the direction of the magnetic field. The direction that our thumb points is the direction of the force on a positive charge. In this case, [math]\displaystyle{ \vec{F} }[/math] points upward, so the positive charges in the bar will move to the top, causing it to polarize with positive charges at the top and negative charges at the bottom. We can now visualize the bar as a battery that causes a current [math]\displaystyle{ I }[/math] to run out of the positive end. In this case, since the bar is polarized with the positive charges at the top, the current will flow out of the top of the bar and continue around the circuit.

2. We can use the negative direction of the change in magnetic field, [math]\displaystyle{ -\frac{dB}{dt} }[/math] to find the direction of the current. To do this, make a diagram comparing the magnitude of the magnetic field enclosed at time [math]\displaystyle{ t_1 }[/math] and at time [math]\displaystyle{ t_2 }[/math]. Then, draw an arrow representing the direction of change of the magnetic field. Now, flip the arrow to take the negative of that vector's direction. Using the right hand rule, point your thumb in the direction of [math]\displaystyle{ -\frac{dB}{dt} }[/math], and the curl of your fingers will give you the direction of the induced current, [math]\displaystyle{ I }[/math].

If the magnetic field is NOT constant, meaning it changes with time, the derivative [math]\displaystyle{ \frac{d}{dt} }[/math] will be distributed to both [math]\displaystyle{ \vec{B} }[/math] and [math]\displaystyle{ x }[/math] in the formula. In this case, we must use the product rule to be able to set up the equation and continue solving for [math]\displaystyle{ emf }[/math].

[math]\displaystyle{ emf = -(\frac{d}{dt} \vec{B})A \cdot B(\frac{d}{dt}A) }[/math]

The B (dA/dt) can be replaced by BLv.

The first term, [math]\displaystyle{ (\frac{d}{dt}\vec{B})A }[/math], represents Faraday's law and is nonzero of there is a varying magnetic field. The second term, [math]\displaystyle{ B(\frac{d}{dt}A) }[/math], represents motional emf and is nonzero if there is a change in the amount of enclosed area.

Examples

Using the figure below, identify the following.

a) Direction of magnetic field

b) Direction of change in magnetic field, [math]\displaystyle{ \frac{d\vec{B}}{dt} }[/math]

c) Direction of negative change in magnetic field, [math]\displaystyle{ -\frac{d\vec{B}}{dt} }[/math]

d) Direction of current, [math]\displaystyle{ I }[/math]

e) Polarization of moving bar

f) Direction of electric field inside bar due to polarization

g) Direction of force on bar

Click for Solution

SOLUTION:

a) Into the page

A circle with an 'x' inside of it represents a vector into the page. A circle with a dot inside represents a vector out of the page.

b) Into the page

Initially, at the time of the image, there are 4 circles representing magnetic field enclosed by the bars. However, as the bar moves, at some time t, the number of circles enclosed by the bar will increase; therefore, there is more magnetic field inside the loop. This means that the change in magnetic field is in the direction of the magnetic field.

c) Out of the page

The negative change in magnetic field is in the opposite direction as change in magnetic field.

d) Counterclockwise

Point your thumb in the direction of [math]\displaystyle{ -\frac{d\vec{B}}{dt} }[/math]. Your fingers will curl in the direction of current.

e) Positive charges at the top, negative charges at the bottom

The magnetic force on a particle is [math]\displaystyle{ \vec{F} = q\vec{v} \times \vec{B} }[/math], so point your fingers in the direction of the velocity of the bar and curl them in the direction of magnetic field. The direction of your thumb is the direction of force on a positive particle.

f) Down

Positive charges have an electric field that points away from them while negative particles have an electric field that point towards them. If the top of the bar is positively charged, the field will point downward toward the negative particles.

g) Left

When a current is involved, [math]\displaystyle{ \vec{F} = I\vec{l} \times \vec{B} }[/math], so point your fingers in the direction of the length of the bar (in the direction of current) and curl them in the direction of magnetic field. The direction of your thumb is the direction of force on the bar.

Medium

A bar of length [math]\displaystyle{ L = 2 }[/math] is moving across two other bars in a region of magnetic field, [math]\displaystyle{ B = 0.0013T }[/math] directed into the page. The bar is moving with a velocity of 10 m/s, and [math]\displaystyle{ x }[/math] is the width of the area enclosed. What is the magnitude of the [math]\displaystyle{ emf }[/math] produced?

Click for Solution

SOLUTION:

Because the amount of magnetic field enclosed by the system is changing with time, we must use Faraday's Law: [math]\displaystyle{ |emf| = \frac{d}{dt} \int\! \vec{B} \cdot \vec{n}dA }[/math]

First, integrate through the formula: [math]\displaystyle{ |emf| = \frac{d}{dt} \left(\vec{B} \cdot A\right) }[/math]

Change in area [math]\displaystyle{ \Delta{A} = L\Delta{x} }[/math]

In this case, the distance [math]\displaystyle{ x }[/math] is changing and resulting in a change in area, so the formula becomes: [math]\displaystyle{ |emf| = \vec{B} \cdot L\frac{d}{dt}x }[/math]

The derivative of distance is velocity. [math]\displaystyle{ \frac{dx}{dt} = v }[/math]

Therefore, |emf| in this problem is equal to [math]\displaystyle{ BLv = .026 V }[/math]

Difficult

A long straight wire carrying current I = .3 A is moving with speed v = 5 m/s toward a small circular coil of radius R = .005 and 10 turns. The long wire is in the plane of the coil. The coil is very small, so that, at any fixed moment in time, you can neglect the spatial variation of the wire's magnetic field over the area of the coil.

a) Is the induced current in the coil flowing clockwise or counterclockwise?

b) At the instant when the long wire is a distance x = 4 m from the center of the coil, determine the magnitude of the induced emf in the coil.

Now consider the case where the wire is stationary and the coil is moving down parallel to the wire with a constant speed, [math]\displaystyle{ v = 2 m/s }[/math].

c) At the instant when the long wire is a distance x = 4 m from the center of the coil, determine the magnitude of the induced emf in the coil.

Click for Solution

SOLUTION:

a) Counterclockwise

Using the right hand rule, if you point your thumb in the direction of current (+y), your fingers will curl in the direction of magnetic field. In this case, magnetic field is pointing into the page at the coil. At the location of the coil, the magnitude of the magnetic field due to the wire is increasing as the wire moves closer; therefore, [math]\displaystyle{ \frac{d\vec{B}}{dt} }[/math] is pointing into the page, and [math]\displaystyle{ -\frac{d\vec{B}}{dt} }[/math] is pointing out of the page. If you point your thumb in the direction of [math]\displaystyle{ -\frac{d\vec{B}}{dt} }[/math], your fingers curl in the direction of the induced current.

b) [math]\displaystyle{ |emf| = 1.47E-11 V }[/math]

After integrating Faraday's Law, we get [math]\displaystyle{ |emf| = \frac{d}{dt} (\vec{B} \cdot A) }[/math]

Notice that distance [math]\displaystyle{ x }[/math] is changing with time.

After doing this derivative, we get [math]\displaystyle{ |emf| = \frac{\mu_0IR^2v}{2x^2} }[/math]

This is the magnitude of emf for one loop in the coil, so we have to multiply it by the number of loops, [math]\displaystyle{ N }[/math].

[math]\displaystyle{ |emf| = \frac{N\mu_0IR^2v}{2x^2} }[/math]

c) |emf| = 0

Remember that the emf relies on a changing magnetic field, which was dependent on a changing [math]\displaystyle{ x }[/math] in the previous example. Now, however, the coil is moving parallel to the wire, meaning there is no change in [math]\displaystyle{ x }[/math], and no change in magnetic field.

Connectedness

Believe it or not, Faraday's law can be applied to musical instruments such as the electric guitar. In many electric instruments, 'pickup coils' sense the vibration of the strings, which causes variations in magnetic flux. These pickup coils often consist of magnet wrapped with a coil of copper wire, where the magnet creates a magnetic field and the vibrations of the string disturb the field, inducing a current in the coiled wire.

I am a biomedical engineering student, and one application of Faraday's law in the medical field is transcranial magnetic stimulation. During this procedure, magnetic coils are used to stimulate small regions of the brain through electromagnetic induction. Current is discharged from a capacitor into the coil to produce pulsed magnetic fields. This technique can be used to evaluate and diagnose various conditions affecting the connection between the brain and muscles, including strokes and motor neuron diseases. It has also been said to alleviate the symptoms of major depressive disorder.

I am currently majoring in mechanical engineering, and in this field, we are required to work with both mechanics and circuit-like scenarios. Personally, I am interested in going into the car manufacturing industry, where motional emf plays a very important role. When you move an object through a magnetic field, it resists movement and generates electricity in the loop. If this is done with enough force, it could be used to stop a small car or roller-coaster.

History

Prior to 1831, the only known way to make an electric current flow through a conducting wire was to connect the ends of the wire to the positive and negative terminals of a battery. We know from the loop rule that around a closed loop, [math]\displaystyle{ V = emf = \oint \vec{E} \cdot d\vec{l} = 0 }[/math]. However, Michael Faraday discovered through his experiments 2 ways in which current could be induced in a closed loop of wire in the absence of a battery: by changing the magnetic field around the loop, or by moving the loop through a constant magnetic field. In his first experiment, Faraday wrapped two wires around opposite sides of an iron ring and plugged one wire into a galvanometer and the other into a battery. He observed that when he held a bar magnet was held stationary with respect to the loop, the galvanometer did not read a current. However, when he moved the bar magnet towards or away from the loop, the galvanometer read a non-zero current. If a current is flowing, that means there must be some emf. Based off of the results of his experiments, Faraday eventually came up with a relationship telling us that the emf generated in a loop of wire in some magnetic field is proportional to the rate of change of the magnetic flux through the loop. This is what we know today as Faraday's law.

However, at the time, his theory was rejected until James Clerk Maxwell took it up again and incorporated it into his Maxwell's equations.

See also

You may want to explore the process of calculating motional emf before the use of Faraday's Law. Maxwell's equations and circuits with resistance are also relevant and may be worth looking into.

Motional emf problems can be pretty tricky depending on what the question is asking you to do. It's always a good idea to know how each formula came about, and how it can change bases on different scenarios. This includes the formula for resistance in a circuit, [math]\displaystyle{ V = IR }[/math]. A problem could go as far as to give you a resistance for a circuit, ask you to solve for the potential difference, [math]\displaystyle{ V }[/math], or [math]\displaystyle{ emf }[/math], and then ask you to solve for the current as well.

Lastly, I advise you to become familiar with Lenz's law because it gives the direction of the induced emf and current resulting from electromagnetic induction.

Further reading

SparkNotes: SAT Physics

Matter & Interactions, Vol. II: Electric and Magnetic Interactions, 4nd Edition by R. Chabay & B. Sherwood (John Wiley & Sons 2015)

External links

Video Explanation: https://www.youtube.com/watch?v=Wgtw5lPKFXI

Text Explanation: https://www.boundless.com/physics/textbooks/boundless-physics-textbook/induction-ac-circuits-and-electrical-technologies-22/magnetic-flux-induction-and-faraday-s-law-161/motional-emf-570-6257/

References

https://en.wikipedia.org/wiki/Faraday%27s_law_of_induction

http://farside.ph.utexas.edu/teaching/em/lectures/node43.html

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elevol.html#c4

https://en.wikipedia.org/wiki/Pickup_(music_technology)

http://www.physics.princeton.edu/~mcdonald/examples/guitar.pdf

https://en.wikipedia.org/wiki/Transcranial_magnetic_stimulation#Technical_information