8/24/2006 Faraday ( F.Robilliard) 1

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1 P 8/24/2006 Faraday ( F.obilliard) 1

2 Introduction: In previous topics, we have seen that electric fields can cause electric currents, that cause magnetic fields. Thus magnetic and electric fields are fundamentally linked together. We identify them together as producing one of the four fundamental forces of nature - the electromagnetic force. We will now consider the reverse link - how magnetic fields are linked back to electric currents, and consequently to electric fields. We will generally be looking at things from the point of view of a conducting loop in the field we will take a loop perspective!. We will consider what happens to a conducting loop if magnetic flux cuts through it, and, in particular, if changes in that flux take place To start, we will need to develop the idea of magnetic flux φ, analogously to that of electric flux. 8/24/2006 Faraday ( F.obilliard) 2

3 Magnetic Flux Lines As we have seen, magnetic fields can be represented graphically by flux lines. A magnetic flux line is a line drawn in a magnetic field, with shape such that, a tangent to the line at any point corresponds to the direction of the magnetic field vector, B, at that point. N stronger uniform field B flux line The strength of the field is represented by the closeness of lines drawn in a given region of the field. weaker uniform field A uniform field will have uniformly spaced, parallel flux lines. 8/24/2006 Faraday ( F.obilliard) 3

4 Closeness of Flux lines: Consider uniformly spaced flux lines cutting, perpendicularly, through a flat surface of unit area. We draw the closeness of the flux lines, such that the number of them, that cut a unit area of surface, is numerically equal to the magnitude of the field vector, B. A =1 m B=5 T B=5 T (side elevation) No of flux lines cutting through unit area, here = φ = B A = 5 x 1 = 5 We think of B as the number of perpendicular flux lines cutting per unit surface area. 8/24/2006 Faraday ( F.obilliard) 4

5 Drawing convention for flux direction: There is a convention that allows us to represent the direction of flux lines, when they cut through the plane of a diagram. We think of a flux line as an arrow. eye or eye or If the arrow is viewed from the tail end, it will appear thus: If the arrow is viewed from the head end, it will appear thus: 8/24/2006 Faraday ( F.obilliard) 5

6 Magnetic Flux dφ We can now develop a more formal quantitative definition of flux. We use an element of area, da, to define flux. Because da is a tiny element of a surface, it is flat, and consequently has a definable vector direction. Because it is small, the field will be locally uniform, over the element, even though it may be non-uniform on a larger scale. Flux, dφ, should be thought of as the total number of flux lines, that cut through an element of surface area da. Case 1. The vector da is aligned in the direction of the flux lines: flux da B Flux and da parallel To quantify flux, dφ, we draw the closeness of flux lines such that, the number of them cutting, per unit surface area, numerically equals the magnitude of B Thus ( Nr flux lines cutting unit area) ( nr flux lines cutting area da) Hence we define dφ = dφ B = B. da B.dA 8/24/2006 Faraday ( F.obilliard) 6

7 da not Aligned with Flux: Case 2: da is not parallel to the flux ( vector B direction) We resolve the B vector in the direction of da = B cosθ, and perpendicular to da = B sinθ. The flux associated with the (B sin θ) component is along the surface of the da area, and therefore does not cut the surface. It contributes zero flux to the surface. The flux associated with the (B cos θ) component is perpendicular to the surface, and works as for the previous Case 1, namely: dφ = (B cosθ ) da or, in vector notation, using the dot product - dφ = B.dA dot product B sinθ θ da B B cosθ 8/24/2006 Faraday ( F.obilliard) 7

8 General Definition of φ: Case 3: Define the total flux φ cutting through a surface of any shape, in a non-uniform magnetic field. Consider some surface in a magnetic field da B da is an element of the surface B is the local magnetic field vector, at the surface element da (Flux, dφ, cutting single surface element da) = dφ = B.dA (Total flux, φ, cutting the entire surface) = sum of all the fluxes cutting all the surface elements, like da, from which the entire surface is composed = = φ dφ over surface over surface B.dA General Definition of flux φ dot product (Case 2) 8/24/2006 Faraday ( F.obilliard) 8

9 Thought Experiment with a Conducting Loop: What happens to a conducting loop, when it moves through a region of uniform magnetic field? P v=const A flat rectangular conducting loop P is moved through the region of uniform magnetic field, at a constant velocity, v. (Magnetic flux is perpendicular to the plane of the loop.) Because the loop is conducting, it contains free electrons, that are carried along as passenger charges, by the conductor. We will analyse the magnetic forces that act on such passenger charges, as the loop traverses the magnetic field region. 8/24/2006 Faraday ( F.obilliard) 9

10 Magnetic forces on passenger Charges Although the passenger charges are actually negative electrons, it is equivalent to imagine positive passenger charges to be carried along with the loop. Because they move through a magnetic field, these passenger charges will experience a magnetic force, F = q v x B (cross product). The mauve arrows represent the magnetic forces at different stages of the motion P v No field, no forces, no current. 8/24/2006 Faraday ( F.obilliard) 10

11 Magnetic forces on passenger Charges F = q v x B P i v Unbalanced force from to - anticlockwise current. 8/24/2006 Faraday ( F.obilliard) 11

12 Magnetic forces on passenger Charges F = q v x B P v Force from to P balances force from to - no current. 8/24/2006 Faraday ( F.obilliard) 12

13 Magnetic forces on passenger Charges F = q v x B P i v Unbalanced force from to P clockwise current. 8/24/2006 Faraday ( F.obilliard) 13

14 Magnetic forces on passenger Charges F = q v x B P v No field, no forces, no current 8/24/2006 Faraday ( F.obilliard) 14

15 Loop Current, i: Consider how the loop current, i, varies as the leading edge of the loop moves through the field region (anticlockwise positive). P i i=0 8/24/2006 Faraday ( F.obilliard) 15

16 Loop Current, i: Consider how the loop current, i, varies as the leading edge of the loop moves through the field region (anticlockwise positive). P i i 8/24/2006 Faraday ( F.obilliard) 16

17 Loop Current, i: Consider how the loop current, i, varies as the leading edge of the loop moves through the field region (anticlockwise positive). P i i=0 8/24/2006 Faraday ( F.obilliard) 17

18 Loop Current, i: Consider how the loop current, i, varies as the leading edge of the loop moves through the field region (anticlockwise positive). P i i 8/24/2006 Faraday ( F.obilliard) 18

19 Loop Current, i: Consider how the loop current, i, varies as the leading edge of the loop moves through the field region (anticlockwise positive). P i i=0 8/24/2006 Faraday ( F.obilliard) 19

20 Loop Current, i: Consider how the loop current, i, varies as the leading edge of the loop moves through the field region (anticlockwise positive). P i i=0 8/24/2006 Faraday ( F.obilliard) 20

21 Generalization of Experiment: From the previous thought experiment, we can draw the following generalization: If the number of flux lines, φ,cutting the loop, changes with time, there is a net magnetic force around the loop ( an electromotive force = emf ), which causes a loop current. (For a uniform field, this occurs when the loop is either entering, or leaving, the field region, but not when it is totally outside of, or totally within, the region.) This loop current is called the induced current, and the emf producing it, the induced emf. Many detailed experimental investigations were carried out into this phenomenon, which led to a quantitative formulation of the relationship between the change of flux and the loop current produced. This relationship is called Faraday s Law. The direction in which the current flows is described by the Lenz Law. 8/24/2006 Faraday ( F.obilliard) 21

22 Faraday s Law: This law was originally formulated for conducting coils, which are equivalent to a number of conducting loops in series. N = the number of turns in the coil. When the number of magnetic flux lines, φ, cutting the loops of a coil, changes, an emf is induced in the coil. The magnitude of the induced emf, ε, in each loop is proportional to the time-rate at which φ changes. Because the coil has N loops in series - d ε = N φ dt Faraday s Law where: dφ = the change in the number of flux lines cutting the loops of the coil that occurred in time interval dt. Note: the constant of proportionality = 1 in the I system. Note: the negative sign is an expression of the Lenz Law. Note: The current, i, in the coil will be given by the Kirchoff 2 rule, namely ε = ir, where r = the resistance of the coil. 8/24/2006 Faraday ( F.obilliard) 22

23 Lenz Law: Faraday gives the magnitude of an induced current, whereas Lenz gives its direction. The initial flux that changed to produce the induced current is called the primary flux. The induced current, like all currents, will produce its own magnetic field (flux). This flux is called the induced, or secondary flux. The reasoning of Lenz sees the secondary flux as a reaction to the change in the primary flux that produced it. emember the Field Produced By a Loop Current: i B ight Hand Grip ule (H): Clasp the loop in the right hand, with the fingers in the direction of the loop current. i The thumb then points in the direction of the magnetic field produced through the centre of the loop. 8/24/2006 Faraday ( F.obilliard) 23

24 tatement of Lenz Law: If a conducting loop is threaded by a primary magnetic flux, and the number of primary flux lines cutting the loop changes, then a current will be induced in the loop. This current will produce its own secondary flux through the loop, in a direction as given by the H. The direction of the induced current will be consistent with the following: the direction of the secondary flux, will oppose the change in the primary flux that created it. Note: This statement is from the point of view of the loop. The secondary current is unaware of its own field. Note: the secondary flux opposes the change in the primary flux, not the primary flux itself. Note: this opposition is in terms of the number of flux lines. If the number of primary flux lines through the loop is increasing, then the secondary flux will be in the opposite direction to the primary flux, thereby opposing the increase; if the number of primary flux lines is decreasing, then the secondary flux will be in the same direction, thereby opposing the decrease. 8/24/2006 Faraday ( F.obilliard) 24

25 Faraday & the ectangular Loop: ectangular loop P enters a region of uniform magnetic field with field vector B, at constant velocity v. B is perpendicular to the plane of the loop. P = b, P = a. esistance of loop = r. P v B a t=0 b 8/24/2006 Faraday ( F.obilliard) 25

26 Faraday & the ectangular Loop: ectangular loop P enters a region of uniform magnetic field with field vector B, at constant velocity v. B is perpendicular to the plane of the loop. P = b, P = a. esistance of loop = r. Loop Area in field = A=a.vt P v B Faraday: a t i vt dφ loop emf = = ir = N = dt Bav loop current = i = - r (Bavt) = Bav 8/24/2006 Faraday ( F.obilliard) 26 d dt Flux cutting loop at time t = φ = B.A = BA = B.a.vt Faraday (N=1) The loop current is proportional to the velocity of the loop.

27 Lenz & the ectangular Loop: The primary flux through the loop is downward and increasing. To oppose this downward increase, the secondary flux will need to be upward. Thus, using the H, the loop current, i, will need to be anticlockwise. v P a i t The direction of the secondary current can also be got from Faraday using a sign convention. The direction of the primary flux is taken as positive The negative in i = - Bav/r indicates that the induced current must produce a secondary field in the opposite direction to that of the primary field must be anticlockwise! (using the H). 8/24/2006 Faraday ( F.obilliard) 27

28 z Find the Emf of a otating Loop in a Uniform Field: V View V y ω ω θ θ A A B ectangular loop. B is a uniform magnetic field, pointing in the (+x)-direction x B A rectangular loop is shown, with vector area, A. θ is the angle from the B vector, to the A vector. Looking edge-on at the loop (in the (-z)-direction) View V. The rectangular loop is rotated, at a constant angular velocity, ω, with the z-axis as its spin axis. The ω vector points in the (+z)-direction. 8/24/2006 Faraday ( F.obilliard) 28

29 View V ω θ Find an expression for the emf, ε, induced in the loop. A Total flux through the loop, at time t: B Find emf in loop: Assume that A is parallel to B, ( θ = 0 ) at time t = 0. d = const. dt Transposing : dot product = t = t φ B.dA = B.A = BA cos = BA cos t Thus, from Faraday, the emf induced in the loop, at time t = dφ d = N = (BA cost) = dt dt BA sin t N=1 B, A & ω const. 8/24/2006 Faraday ( F.obilliard) 29

30 = BA sin t Graphically: Thus loop current, i, is given by (r = loop resistance) i = = r BA r sin t The emf (and loop current) are sinusoidal functions of time. Firstly, the current goes anticlockwise round the loop, then reverses and goes clockwise. uch a current is called an alternating current. Amplitude = BAω ε Period =2π/ω t A rotating loop in a uniform magnetic field is the basis for almost all electrical power generation. In Australia, mains power is alternating, with an amplitude of 339 V (240 V M), and a period of 1/50 s. 8/24/2006 Faraday ( F.obilliard) 30

31 ω B 8/24/2006 Faraday ( F.obilliard) 31