Magnetic inductance & Solenoids. P.Ravindran, PHY041: Electricity & Magnetism 22 February 2013: Magnetic inductance, and Solenoid

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Lynn Peters
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1 Magnetic inductance & Solenoids

2 Changing Magnetic Flux A changing magnetic flux in a wire loop induces an electric current. The induced current is always in a direction that opposes the change in flux. These facts were discovered by Michael Faraday and represent a key connection between electricity and magnetism. One simple example of this is a magnet moving in and out of a wire loop. As a bar magnet approaches a wire loop along a line perpendicular to the loop, more and more field lines poke through the loop and the flux increases. To oppose this change in flux a current is induced in the direction shown. I N v S Note that the induced current produces its own magnetic field pointing to the right. Alsonotethatthere that there is no battery in the loop! This current will only exist when the flux inside the loop changes. When the magnet is withdrawn the flux decreases and current is induced in the other direction. There is no current when the magnet is still.

3 Induced emf s and Currents The current induced in a loop come not from a battery but from a changing magnetic flux. We can think of the loop containing an imaginarybattery that gets turned on whenever flux in the loop changes. The strength of this battery is called the emf (electromotive force); it s symbol is a script, and it is measured in volts. The induced current is given by: I = / R where R is the internal resistance it in the loop. itself depends on the rate at which the flux inside the loop is changing. If the flux is changing at a constant rate, This is Faraday s law of induction. The = - Ф B / t negative sign here indicates the emf opposes the change in flux. The greaterthethe change in flux the greater, the greaterthe the induced emf, and greater the induced current.

4 Faraday s Law (Maxwell s III equation) Changing magnetic field gives rise to electric current. Induced emf in the loop, due to changing magnetic flux. i.e. rate of change of magnetic flux is the e.m.f. induced in the circuit. If q 0 is the charge taken around the loop. Then Force F q E 0 Now work done in taking the charge around the loop will be dw F dl P E dl W q 0 E2r E Φ B B t q q E 2 r 0 0 B (Magnetic field inward) B E dl t t B P d B E l B ds t t. P S or Integral lform Differential form E B t

5 Faraday's Law If one pull a loop of wire to the right ihtthroughh a magnetic field (Fig. a). A current flow in the loop. If one move the magnet to the left, holding the loop still (Fig. b). Again, a current flow in the loop. With both the loop and the magnet at rest (Fig. c), if one change the strength th of the field (by varying the current in the coil of an electromagnet), current flow in the loop. Changing B

6 The first experiment, is an example of motional emf, conveniently expressed by the flux rule. But in the second experiment loop is stationary so force can not be magnetic which is responsible for producing current. Faraday thought: A changing magnetic field induces an electric field. the emf is again equal to the rate of change of the flux, P E dl B. da B t so E is related to the change in B by this equation

7 Last expression is the Faraday's law, in integral form. By applying Stokes' theorem: This is Faraday's law, in differential form Note: in the static case (constant B) as Faraday's law reduces to the old rule or xe=0 Edl. 0 In Experiment 3 the magnetic field changes for entirely different reasons, but according to Faraday's law an electric field will be induced, giving rise to an emf -d/dt.

8 For all three cases universal flux rule: Whenever (due to any reason) the magnetic flux through a loop changes, an emf will appear in the loop. B t Summary: In Faraday's first experiment it's the Lorentz force law at work; the emf is magnetic. But in the other two it's an electric field (induced by the changing magnetic field) that does the job.

9 General form of Faraday s Law V V V E ds b Ub Ua ba b a q a So the electromotive ti force around a closed path is: And Faraday s Law becomes: Eds Eds d dt A changing magnetic flux produces an electric field. This electric field is necessarily non conservative. B

10 Induced emf and Electric Fields An electric field is created in the conductor as a result of the changing magnetic flux Even in the absence of a conducting loop, a changing magnetic field will generate an electric field in empty space This induced delectric field is nonconservative Unlike the electric field produced by stationary charges The emf for any closed path can be expressed as the line integral of E. ds over the path

11 E produced by changing B d Ed B dt db 2 E2 r r dt db r E 2 dt

Faraday s law: Changing magnetic field induces electrical current (a) When a magnet is

shown, indicating that a current is induced in the loop.

even when the magnet is inside the loop.

12 Faraday s law: Changing magnetic field induces electrical current (a) When a magnet is moved toward a loop of wire connected to a galvanometer, the galvanometer deflects as shown, indicating that a current is induced in the loop. (b) When the magnet is held stationary, tti there is no induced d current in the loop, even when the magnet is inside the loop. (c) When the magnet is moved away from the loop, the galvanometer deflects in the opposite direction, indicating that the induced current is opposite that shown in part (a).

13 Magnetic Force on a Current Carrying Conductor For closed circuit of contour C carrying I, total magnetic force F m is: F m I In a uniform magnetic field, F m is zero for a closed circuit. C dl B N

14 Magnetic Force on a Current Carrying Conductor On a line segment, F m is proportional to the vector between the end points. F I B m

15 Straight Wire Practice Draw some magnetic field lines (loops in this case) along the wire. I Using x s s and dots to represent vectors into and out of the page, show the magnetic field for the same wire. Note B diminishes with distance from the wire. B out of page B into page I

16 Current Loops and Magnetic Fields The magnetic field inside a current loop tends to be strong; outside, it tends to be weak. Here s why: Using the right hand rule we see that each length of wire contributes to a B field into the page (all lengths reinforcing one another). Outside the loop, say at P, the field is weak since the left side of the wire produces a field out of the page, but the right side produces a field into the page. Explain why the field is weak above the top wire. The situation is the same with a circular loop. The effect is magnified with multiple turns of wire. Yet another right hand rule helps with current loops: Wrap your right hand in the direction of the loop and your thumb points in the direction of B inside. This is reminiscent of angular momentum for a spinning body. P weak field outside I I strong field inside loop, directed into page I I I strong field into page weak field

Current Loops and Bar Magnets Notice how similar the magnetic

Wrap your right hand along the loop in the direction of the

17 Current Loops and Bar Magnets Notice how similar the magnetic field of a current loop is to that of a simple bar magnet. Wrap your right hand along the loop in the direction of the current and your thumb points in the direction of the north pole of your electro magnet. Note also how the field lines are very close together inside the loop, just as they are when they thread through a bar magnet. I

18 Distributed coiled conductor Key parameter: n loops/metre Solenoid I Iffinitefinite length, sum individual loopsvia B S Law B If infinite length, apply Ampere s Law B constant and axial inside, zero outside Rectangular path, axial length L B vac nli B ni.d oiencl BvacL o vac o I L (use label B vac to distinguish from core filled solenoids) Solenoid is to magnetostatics what capacitor is to electrostatics

19 Solenoids Solenoids are one of the most common electromagnets. Solenoids consist of a tightly wrapped coil of wire, sometimes around an iron core. The multiple loops and the iron magnify the effect of the single loop electromagnet. A solenoid behaves as just like a simple bar magnet but only when current is flowing. The greater the current and the more turns per unit length, the greater the field inside. An ideal solenoid has a perfectly uniform magnetic field inside and zero field outside.

20 How Solenoids Work The cross section of a solenoid is shown. At point P inside the solenoid, the B field is a vector sum of the fields due to each section of wire. In the ideal case the magnetic field would be uniform inside and zero outside. B = 0 B P x x x x x x x x I out of the page I into the page

21 Solenoids and Bar Magnets A solenoid produces a magnetic field just like a simple bar magnet. Since it consists of many current loops, p, the resemblance to a bar magnet s field is much better than that of a single current loop.

22 Magnetic Fields: Overview Although the magnetic properties of electrons must ultimately be explained with quantum mechanics, we can think of magnetism arising ii whenever we have charge in motion. This motion can be that of an electron (either spinning or orbiting) or it can be in the form of a current. Remember: moving charges produce magnetic fields, and external magnetic fields exert a magnetic force on moving charges (at least if the charge has a component of its velocity perpendicular to the field).

Applications of Ampere s Law A Solenoid A solenoid is basically a bunch of

It is analogous to a capacitor which can produce a strong electric field.

Solenoids are important in engineering as they can convert electromagnetic

All automobiles use what is called a starter solenoid.

24 Applications of Ampere s Law A Solenoid A solenoid is basically a bunch of loops of wire that are tightly wound. It is analogous to a capacitor which can produce a strong electric field. In this case it can produce a strong MAGNETIC FIELD. Solenoids are important in engineering as they can convert electromagnetic energy into linear motion. All automobiles use what is called a starter solenoid. Inside this starter is a piston which is pushed out after receiving a small amount of current from the car s battery. This piston then completes a circuit between the car s battery and starter motor allowing the car to operate.

25 Applications of Ampere s Law A Solenoid The first thing you must understand is what is the enclosed current. It is basically the current, I, times the # of turns you enclose, N. It is important to understand that when you enclose a certain amount of turns that the magnetic field runs through the center of the solenoid. As a result the field lines and the length of the solenoid are parallel. This is a requirement for Ampere s Law. When you integrate all of the small current elements they ADD up to the length of the solenoid, L B dl I 0 enc B( L ) ( NI ) ( 0 N B 0 I, n # turnsper length L B ni o solenoid N L

26 Example A solenoid has a length L =1.23 m and an inner diameter d =3.55 cm, and it carries a current of 5.57 A. It consists of 5 close packed layers, each with 850 turns along length L. What is the magnetic field at the center? n # turnsper length B solenoid ni o N L B (1.26x10 )5 (5.57) 1.23 B T

27 APPLICATIONS OF ELECTROMAGNETS: LIFTING MAGNET

28 Applications Of Electromagnets: Electric Bell

29 Applications Of Electromagnets: Relay In this figure, you can see that a relay consists of two separate and completely independent circuits. The first is at the bottom and drives the electromagnet. In this circuit, a switch is controlling power to the electromagnet. When the switch is on, the electromagnet is on, and it attracts the armature (blue). The armature is acting as a switch in the second circuit. When the electromagnet is energized, the armature completes the second circuit and the light is on. When the electromagnet is not energized, the spring pulls the armature away and the circuit is not complete. In that case, the light is dark.

Chapter 27, 28 & 29: Magnetism & Electromagnetic Induction. Magnetic flux Faraday s and Lenz s law Electromagnetic Induction Ampere s law

Chapter 27, 28 & 29: Magnetism & Electromagnetic Induction Magnetic flux Faraday s and Lenz s law Electromagnetic Induction Ampere s law 1 Magnetic Flux and Faraday s Law of Electromagnetic Induction We