Physics 4 Magnetic Induction Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Physics 4

Magnetic Induction

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

Before we can talk about induction we need to understand magnetic flux.

You can think of flux as the number of field lines passing through an area.Here is the formula:

)cos(dABAdBflux B



Magnetic Flux

If the field is uniform and the area is flat, we have:

)cos( ABABflux B

Note that there are 3 parts to this formula: field strength, Area, and angle.We will learn that magnetic induction occurs if any of these change.In fact, the rate of change of flux is the induced voltage:

Here is Faraday’s Law of Induction:

The negative sign helps to remind us of what induction does – induction always opposes the change in the flux. We call this Lenz’s Law.

dtd B

induced

Next we will look at a few specific examples of a changing magnetic flux:



Faraday’s Law of Induction

)cos( ABABflux B

1) A loop of wire entering a uniform magnetic field.

X X X X X X

X X X X X X

X X X X X X

X X X X X X

v

B

Question: which direction is the induced current in the loop?




X X X X X X

X X X X X X

X X X X X X

X X X X X X

v

As the loop enters the B-field, the flux increases into the page, so there will be a current induced in the loop to oppose this increasing flux. The direction of the induced current is Counter-Clockwise, because that direction will create an induced B-field through the loop, out of the page (opposing the increase into the page).

I induced

B induced

B




X X X X X X

X X X X X X

X X X X X X

X X X X X X

v

You might also notice that there will be a net force on the loop to the right, opposing its motion (try the right hand rule on each side of the loop). Induction opposes the change by trying to push the loop back out of the B-field.

B I induced

B induced



X X X X X X

X X X X X X

X X X X X X

X X X X X X

2) A loop of wire rotating in a uniform magnetic field.


B



X X X X X X

X X X X X X

X X X X X X

X X X X X X

2) A loop of wire rotating in a uniform magnetic field.

As shown, the loop is intercepting the maximum flux. As it rotates, the flux will be decreasing, so there will be an induced current to oppose this change. The direction of the current will be clockwise, which creates an induced magnetic field through the loop, into the page.

I induced

X B induced

B



X X X X X X

X X X X X X

X X X X X X

X X X X X X

3) A stationary loop of wire in a magnetic field that is increasing in strength.


)sin( gincreaB



X X X X X X

X X X X X X

X X X X X X

X X X X X X

3) A stationary loop of wire in a magnetic field that is increasing in strength.

The increasing B-field causes the flux to increase, inducing current in the loop. The direction of the current is Counter-Clockwise, creating an induced B-field out of the page through the loop, opposing the change in the flux.

B induced

I induced)gsinincrea(B



4) A bar magnet approaching a stationary loop of wire.

N S

Question: which direction is the induced current in the loop, as viewed from the right?



4) A bar magnet approaching a stationary loop of wire.

N S

When viewed from the right, this looks like a circular loop with an increasing (because the magnet is approaching) B-field directed through the loop, to the left. The induced current will oppose this increasing field by creating some flux in the opposite direction (to the right, or out of the page in the view from the right). This means the induced current must be Counter-Clockwise.

magnetB X X

X X

X X

magnetB

View from the right

inducedB

inducedB

inducedI

inducedI



5) A rotating loop of wire near a stationary bar magnet.

N S

Question: which direction is the induced current in the loop, as viewed from the right?



5) A rotating loop of wire near a stationary bar magnet.

N S

As shown, the loop is intercepting maximum flux, so when it rotates, the flux decreases. Induction will create some flux through the loop to the right, to oppose the change. This means the induced current is Clockwise.

magnetB

inducedB

inducedI

X X

X X X

X X

magnetB

View from the right

inducedB

inducedI



6) A loop of wire approaching a straight wire with a steady current.

Current = I (steady)




6) A loop of wire approaching a straight wire with a steady current.

Current = I (steady)

X X X X X X X

X X X X X X X

X X X X X X X

wireB

In the vicinity of the loop, the wire creates a magnetic field directed into the page. As the loop approaches, the field gets stronger, so the flux is increasing. To oppose this, the induced current must be Counter-Clockwise to create some induced flux out of the page.

inducedB

inducedI



7) A stationary loop of wire near a straight wire with decreasing current.


Current = I (decreasing)



7) A stationary loop of wire near a straight wire with decreasing current.

Current = I (decreasing)

X X X X X X X

X

X X X X X X X

X X X X X X X

In the vicinity of the loop, the wire creates a magnetic field directed into the page. As the current decreases, the field gets weaker, so the flux is decreasing. To oppose this, the induced current must be Clockwise to create some induced flux into the page.

wireB

inducedI

inducedB



Here’s one where we can calculate the effect of the induction:

Suppose there is a uniform B-Field pointing into the page. A circuit is placed in the field so that it intercepts the maximum magnetic flux. One side of the rectangular circuit is free to slide back and forth along the rails. Let’s first try to figure out what happens qualitatively:

X B X X X X X X X X

X X X X X X X X X

X X X X X X X X X

X X X X X X X X X

RV





RV

First we can find the direction of the current in the given loop. Current will flow clockwise. In the diagram it is labeled Iinitial.

Iinitial

X B X X X X X X X X

X X X X X X X X X

X X X X X X X X X

X X X X X X X X X





RV

First we can find the direction of the current in the given loop. Current will flow clockwise. In the diagram it is labeled Iinitial.

Iinitial

The force on the sliding wire can also be found from the right hand rule for magnetic forces. This force points to the right. So the wire will slide to the right.

Fmag

X B X X X X X X X X

X X X X X X X X X

X X X X X X X X X

X X X X X X X X X





RV

Now that the wire is moving, we can talk about the induction effect (remember, induction only happens when the flux is changing). The magnetic flux through the loop is increasing into the page, and there will be an induced current in the wire that opposes this change.

Iinitial

Fmag

X B X X X X X X X X

X X X X X X X X X

X X X X X X X X X

X X X X X X X X X





RV


This induced current will flow counter-clockwise, creating some induced flux to oppose the increasing flux.

Iinitial

Fmag

Iinduced

X B X X X X X X X X

X X X X X X X X X

X X X X X X X X X

X X X X X X X X X





RV


This induced current will flow counter-clockwise, creating some induced flux to oppose the increasing flux.

Notice – this also creates an induced force on the wire to the left.

Iinitial

Fmag

IinducedFinduced

X B X X X X X X X X

X X X X X X X X X

X X X X X X X X X

X X X X X X X X X





RV

Iinitial

Fmag

IinducedFinduced

So the wire initially accelerates to the right, but the faster it moves, the stronger the induction, and the closer it gets to some maximum speed (where the induced force will match up with the initial Fmag. We can try to calculate this maximum speed.

X B X X X X X X X X

X X X X X X X X X

X X X X X X X X X

X X X X X X X X X



RV

Iinitial

Fmag

IinducedFinduced

To get the max speed, we will have to use our formula for the induced voltage. As soon as the induced voltage matches up with the battery voltage we will be at equilibrium, and the wire will be moving at maximum speed.

t

)cosAB(

tV B

induced

x

L

In this formula, the only thing changing is the horizontal distance which is labeled x in the diagram above. If we replace the Area with L*x and realize that we can ignore the angle (90 degrees), we get a simpler formula, involving the velocity of the wire.vLB

t

xLBVinduced

Setting this induced voltage equal to the battery voltage, we will get the maximum speed. Another option would be to set F induced = Fmag.

Should get the same result, after using Ohm’s Law.LB

Vv

VvLB

batterymax

batterymax

X B X X X X X X X X

X X X X X X X X X

X X X X X X X X X

X X X X X X X X X





Example: In the figure the loop is being pulled the right at constant speed v, and there is a constant current I in the straight wire.

Find the emf induced in the loop, and the direction of the induced current.





We will use Faraday’s law of induction for this one. The flux through the loop is due to the field produced by the wire, which has a formula:

𝐵𝑤𝑖𝑟𝑒=𝜇0 𝐼2𝜋𝑟







This field is not constant, so we will have to integrate to get the total flux through the loop:

Φ𝐵=𝐵 ∙𝑑𝐴








Φ𝐵=𝐵 ∙𝑑𝐴 Area of this slice is dA=b·dr

Φ𝐵=𝑟

𝑟+𝑎 𝜇0 𝐼2𝜋𝑟

𝑏𝑑𝑟=𝜇0 𝐼 𝑏2𝜋

𝑟

𝑟+𝑎𝑑𝑟𝑟𝑑𝑟=

𝜇0 𝐼 𝑏2𝜋 [𝑙𝑛(𝑟 +𝑎

𝑟 )]

The B-field points into the page, which is straight through the loop, so the dot product is just the magnitudes multiplied together.

Limits on our integral are from r to r+a









Φ𝐵=𝑟



𝑟



𝑟 )]



For Faraday’s law we take the derivative to find the induced emf:

𝜀𝑖𝑛𝑑𝑢𝑐𝑒𝑑=−𝑑Φ𝐵

𝑑𝑡

The distance r is changing at speed v as the loop is pulled away, so dr/dt = v









Φ𝐵=𝑟



𝑟



𝑟 )]



For Faraday’s law we take the derivative to find the induced emf:

𝜀𝑖𝑛𝑑𝑢𝑐𝑒𝑑=−𝑑Φ𝐵

𝑑𝑡

𝜀𝑖𝑛𝑑𝑢𝑐𝑒𝑑=𝜇0 𝐼 𝑎𝑏𝑣2𝜋 𝑟 (𝑟+𝑎)

The distance r is changing at speed v as the loop is pulled away, so dr/dt = v

The current direction will oppose the change in the flux. Since the flux decreases as the loop is pulled away, the induced current is clockwise – creating an induced B-field into the page, as desired.

Displacement Current



In a region where there is a changing electric flux, we find that there is a magnetic field produced. The field generated is the same as if a current flowed through the region, even though no wires are present there. We have a formula for this displacement current:

𝑖𝑑=𝜖𝑑Φ𝐸

𝑑𝑡

Maxwell’s Equations



Maxwell’s equations summarize the relationships between electric and magnetic fields:

∮𝐸 ∙𝑑𝐴=𝑄𝑒𝑛𝑐𝑙

𝜖0𝑮𝒂𝒖𝒔 𝒔′ 𝒔𝑳𝒂𝒘 𝒇𝒐𝒓 𝑬

∮𝐵 ∙𝑑𝐴=0𝑮𝒂𝒖𝒔 𝒔′ 𝒔 𝑳𝒂𝒘 𝒇𝒐𝒓 𝑩

∮𝐸 ∙𝑑 𝑙=−𝑑Φ𝐵

𝑑𝑡𝑭𝒂𝒓𝒂𝒅𝒂𝒚 ′ 𝒔𝑳𝒂𝒘

∮𝐵 ∙𝑑 𝑙=𝜇0(𝑖𝑐+𝜖0 𝑑Φ𝐸

𝑑𝑡 )𝑒𝑛𝑐𝑙

𝑨𝒎𝒑𝒆𝒓 𝒆 ′ 𝒔𝑳𝒂𝒘

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Physics 4 Magnetic Induction Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB