Magnetic Fields Part 3: Electromagnetic Induction

Transcription

1 Magnetic Fields Part 3: Electromagnetic Induction Last modified: 15/12/2017

2 Contents Links Electromagnetic Induction Induced EMF Induced Current Induction & Magnetic Flux Magnetic Flux Change in Flux Faraday s Law Lenz Law Lenz & Secondary Flux Lenz & Magnetic Force Electrical Generator Summary Electromagnetic Induction Recipe Maxwell s Equations & Electromagnetic Waes

3 Induced EMF Contents From preious lectures, we know that there will be a force F = q B on a charge q moing with elocity in a magnetic field B. Consider a conducting rod of length L moing in a uniform magnetic field as shown at right. Positie and negatie charges will experience forces in opposite directions. L + + F + F B This separation of charges creates a potential difference V between the ends of the rod, and thus an electric field E = V /L. There will then be a force due to E which opposes the magnetic force. + + F = qb + F = qe E B

4 Contents The motion of charges will stop when these opposing forces balance: qe = qb q V L = qb V = BL L E + + B This potential difference represents a gain in energy, thus it is an EMF E. The EMF E = BL is induced (i.e. caused) by the motion of the rod in the magnetic field. Kinetic energy of the moing rod has been conerted to potential energy of the separated charges.

5 Induced Current Contents As with any EMF, if a circuit is connected between the ends of the rod, then a current will flow in this circuit. Let s look at the simple case of a rectangular loop moing perpendicularly to a uniform magnetic field. When the loop is completely inside the magnetic field, then the same EMF E = BL is induced in each of the left and right sides. These EMFs are acting to create currents in opposite directions, but because the EMFs are equal in size, the total current flowing is zero. L w + + E E B More interesting are the cases when the loop is only partially within the field.

6 Contents When the loop is entering a region of uniform magnetic field, then the only EMF induced is on the right-hand side of the loop. Thus, there is a anti-clockwise current: i = E/R = BL R where R is the resistance of the loop. i B = 0 + E B A similar situation occurs when the loop is leaing a region of field. An EMF is induced only in the left side of the loop, and so the current flows in a clockwise direction. The magnitude of this current is the same as aboe. + E i B B = 0

7 Magnetic Flux Contents Looking at these three situations, we should notice that the key to whether a current flows or not is related to the number of magnetic field lines passing through the loop - i.e. the magnetic flux through the loop. When the loop is wholly within the field, there is no change in flux, and no induced current. As the loop enters the field, the number of field lines passing through is increasing, and there is a current flowing. When the loop leaes the field, the flux is decreasing and the induced current flows in the opposite direction. We can conclude therefore that the current is induced by changing the magnetic flux through the loop.

8 Contents B da The definition of the magnetic flux through a surface should look familiar. For a small flat surface of area da, the magnetic flux will be: dφ = B da To calculate the total flux through a cured surface, we diide it into infinitesmal flat surfaces and then add up (i.e. integrate oer) all of the fluxes of these surfaces: Φ = dφ = B da Unlike electric flux, a new named deried SI unit is used for magnetic flux. The SI unit of magnetic flux is the weber (Wb). 1 Wb = 1 T m 2

9 Contents For our example of the rectangular loop entering the region of magnetic field, the flux is simple to calculate as B is constant and the ectors B and A are parallel. Assuming that the loop begins entering the field at t = 0, we can calculate: L t φ = BA = BLt B = 0 B and so the time deriatie is: dφ dt = BL = E (same as our preious result!) Note that the area used to find the flux is the area of field inside the loop (shaded in blue) NOT the area of the whole loop.

10 Contents The case where the loop is leaing the region of field can be analyzed similarly. w t t Assuming that the loop begins leaing the field at t = 0, we hae: φ = BA = BL(w t) and the time deriatie is: L B B = 0 dφ dt = BL = E The minus sign is an indication that the current flows in the opposite direction to the preious case.

11 Faraday s Law Contents This result is general, applying to any shaped loop and to non-uniform magnetic fields, and is known as Faraday s Law. Faraday s Law A change in magnetic flux induces an EMF: E = dφ dt The induced current i can then be determined by using Ohm s Law: i = E R where R is the resistance of the loop.

12 Lenz Law Contents Notice that in this definition we hae used the absolute alue.. to gie a positie alue for E. It is possible, but tricky, to use ectors and this formula to gie the direcion of E (and hence i), but in practice it is easier to determine the magnitude (using Faraday) and direction of the induced current separately. The direction of an induced current is gien by Lenz Law. Lenz Law The direction of an induced current will OPPOSE the change in flux. Applying Lenz Law inoles using logic rather than calculation. Let s see how it works with our example of the rectangular loop in a uniform field.

13 Lenz & Secondary Flux Contents As we know, a current will produce a magnetic field. The flux of the field generated by an induced current is known as the secondary flux. (The original magnetic field proides the primary flux.) Lenz Law allows us to work out the direction of this secondary flux and hence the direction of the induced currrent. Consider the rectangular loop entering the region of magnetic field. Q: What is the change in flux? A: The flux is increasing. B So Lenz Law requires the induced current to decrease the flux. B = 0 B This can be achieed if the field of the secondary flux B is in the opposite direction to that of the primary flux, as shown.

14 Contents This secondary field is directed out of the page, and using the right hand rule for solenoids, the current must therefore be anti-clockwise. This is of course the same result found preiously. i B B = 0 B When the loop is leaing the field, the flux will be decreasing, so the secondary field must be in the same direction as the primary. The right hand rule gies a clockwise current. i B Again, this is the same result found preiously. B B = 0

15 Lenz & Magnetic Force Contents There is usually more than one way to apply Lenz Law. Depending on the problem, one way or the other might be easier. In our examples, instead of considering the secondary flux, we could use the magnetic drag force on the loop. Why is there a force acting? A current in a magnetic field will feel a force: F = il B The induced current is no exception. From Lenz Law, we know that the force on the induced current must oppose the change in flux.

16 Contents For our example, the change is that the loop is moing to the right. To oppose this, the magnetic force must act to the left. F i F i In each case the Right Hand Rule gies the direction of the induced current, which is consistent with the preious results. Both methods - the first using secondary flux and the second using force - are equally alid and you can use whicheer you prefer. Some problems howeer, may be easier with one method oer the other.

17 Contents This magnetic force will be, using the expression found earlier for the current i: F = ilb = L2 B 2 In the absence of other forces, this force will reduce the loop s speed. If we wish the loop to continue with constant speed, then an external force equal and opposite to this drag force must be applied. The work done by this external force is conerted, first to electrical energy in the induced current, then to heat/light/... in the resistance of the circuit. R

18 Electrical Generator Contents An important application of electromagnetic induction is the electrical generator. A conducting loop is placed inside a uniform magnetic field, and rotates at a constant angular speed ω. ω θ A B The magnetic flux through the loop is: φ = B A = BA cos θ = BA cos(ωt) The field B and loop area A are constant, so Faraday s Law gies the induced EMF: E = d (BA cos(ωt)) = ωba sin(ωt) dt

19 Contents A generator will often hae multiple turns of wire instead of a single loop. If such a coil has N turns, each turn will form a loop with EMF E connected in series, so the total EMF will be: NωBA sin(ωt) This situation is so common that Faraday s Law is often stated as: E = N dφ dt The EMF produced is arying in time and alternates in direction. It proides an AC current, which is what we obtain from a wall socket. A sinusoidal alternating current is ery easy to produce, and also has the bonus that it can be transmitted ery efficiently oer long distances. In Australia, this AC oltage has a peak of 340 V and a frequency of 50 Hz. These alues ary a little around the world.

20 Summary Contents A conducting loop will hae a an induced current flowing when there is a change in the magnetic flux, φ, through the loop. The magnitude of this current is calculated by first using Faraday s Law: E = dφ dt to find the EMF, then Ohms Law to find the current: i = E R where R is the resistance of the loop.

21 Contents The direction of the induced current is determined using Lenz Law: The direction of the induced current is such that it opposes the change in magnetic flux. Using Lenz Law is about applying logic rather than calculations.

22 Electromagnetic Induction Recipe Contents To find the magnitude of the induced current: Write down an expression for the magnetic flux φ = BA in terms of constants and time t. Differentiate this expression with respect to time. Take absolute alue of this result to find magnitude of E. Use Ohm s Law to find current i = E/R To find the direction of the induced current: Ask yourself: What is the change in flux? Increase or decrease? The induced current is opposing this. What does this mean in your problem in terms of the direction of (a) the secondary flux or (b) the magnetic force? Use the appropriate Right Hand rule to find current direction.

23 Maxwell s Equations & EM Waes Contents In the early 1860 s, James Clerk Maxwell was able to describe all electric and magnetic pheonomena using four basic (and mostly familiar) equations, known as Maxwell s equations: S S P P E da = 1 Q ε 0 B da = 0 E dl = d dt S B dl = µ 0 P P B da I + µ 0 ε 0 d dt P E da (Gauss Law) (Gauss Law for Magnetism) (Faraday s Law) (Ampère s Law+Maxwell term) Note: Ampére (and us!) only considered constant currents. Maxwell s extra term aboe includes currents that ary in time.

24 Contents These equations tell us that a arying magnetic field will produce an electric field, while a arying electric field will produce a magnetic field. In the correct circumstances, these ariations can continue indefinitely and propagate in space as an electromagnetic wae, which we hae preiously mentioned. y E c = E B = 1 µ0ε 0 z B c x