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

Physics 4 Magnetic Induction

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: flux B da B da cos( ) B Magnetic Flux If the field is uniform and the area is flat, we have: flux B B A B A cos()

Faraday s Law of Induction flux B B A B A cos() 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: induced d dt B 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. Next we will look at a few specific examples of a changing magnetic flux:

1) A loop of wire entering a uniform magnetic field. B v Question: which direction is the induced current in the loop?

1) A loop of wire entering a uniform magnetic field. B I induced v B induced 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).

1) A loop of wire entering a uniform magnetic field. B I induced v B induced 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.

2) A loop of wire rotating in a uniform magnetic field. B Question: which direction is the induced current in the loop?

2) A loop of wire rotating in a uniform magnetic field. B X B induced I induced 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.

3) A stationary loop of wire in a magnetic field that is increasing in strength. B ( increa sing) Question: which direction is the induced current in the loop?

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

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. I induced View from the right N S B X X magnet B induced B magnet X X X B induced X I induced 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.

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. View from the right N S B magnet B induced X B magnet X I induced X X X B induced X X I induced 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.

6) A loop of wire approaching a straight wire with a steady current. Current = I (steady) Question: which direction is the induced current in the loop?

6) A loop of wire approaching a straight wire with a steady current. I induced X B wire B induced X X Current = I (steady) 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.

7) A stationary loop of wire near a straight wire with decreasing current. Current = I (decreasing) Question: which direction is the induced current in the loop?

7) A stationary loop of wire near a straight wire with decreasing current. I induced X B wire XB induced X X Current = I (decreasing) 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.

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: I initial F mag R V First we can find the direction of the current in the given loop. Current will flow clockwise. In the diagram it is labeled I initial. 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.

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: I initial F mag R V 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.

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: I initial F mag R V I induced 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. This induced current will flow counter-clockwise, creating some induced flux to oppose the increasing flux.

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: F mag I initial R V F induced I induced 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. 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.

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: F mag I initial R V F induced I induced 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 F mag. We can try to calculate this maximum speed.

x L I initial F mag R V F induced I induced 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. V induced V induced vmax t B (B A cos) t x B L B L v t B L vmax Vbattery Vbattery B 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. Setting this induced voltage equal to the battery voltage, we will get the maximum speed. Another option would be to set F induced = F mag. Should get the same result, after using Ohm s Law.

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: B wire = μ 0I 2πr This field is not constant, so we will have to integrate to get the total flux through the loop: Φ B = B d A Φ B = r+a μ0 I r 2πr bdr = μ 0Ib 2π r+a dr 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 r dr = μ 0Ib 2π r + a ln r Area of this slice is da=b dr For Faraday s law we take the derivative to find the induced emf: ε induced = dφ B dt The distance r is changing at speed v as the loop is pulled away, so dr/dt = v

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: B wire = μ 0I 2πr This field is not constant, so we will have to integrate to get the total flux through the loop: Φ B = B d A Φ B = r+a μ0 I r 2πr bdr = μ 0Ib 2π r+a dr 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 r dr = μ 0Ib 2π r + a ln r Area of this slice is da=b dr For Faraday s law we take the derivative to find the induced emf: ε induced = dφ B dt ε induced = μ 0Iabv 2πr(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: i d = ε dφ E dt

Maxwell s Equations Maxwell s equations summarize the relationships between electric and magnetic fields: E d A = Q encl ε 0 B d A = 0 Gauss s Law for E Gauss s Law for B B d l = μ 0 i c + ε 0 dφ E dt encl Ampere s Law E d l = dφ B dt Faraday s Law