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Physics 106

Outline - Induction

  1. Background

    1. Magnetic field lines

      1. Magnetic Field Lines for Current in Coil of Wire (Fig. 1a)

        1. Curl the fingers of your right hand in the direction of the current. Your thumb points in the direction of the field along the axis of the coil.

        2. The field at other points is tangent to the field lines which are dashed in Fig. 1a.

      2. Magnetic Field Lines for Bar Magnet (Fig. 1b)

        1. Magnetic field lines leave North pole and go to South pole. The field at any point is tangent to the field line.

        2. North pole of magnet acts like current carrying coil of wire with current in counterclockwise direction as you view that end of the coil.

        3. Before we knew the cause of magnetism was a net flow of charge, we said a circular coil with current in counterclockwise direction as you view that end of the coil acted like a North pole.

        4. By experiment we find that a North Pole repels another North pole or a coil with the current in a counterclockwise direction.

    2. Magnetic Flux

      1. ΦB = B . A = BA cos angle symbolB,A for B constant.
      2. For Fig. 2a,  angle symbolB,A = 90o.  cos 90o = 0  and  ΦB = 0.
      3. For Fig. 2b,  ΦB = BA cos angle symbolB,A.
      4. For Fig. 2c,  angle symbolB,A = 0o.  cos 0o = 1  and  ΦB = BA.
      5. For Fig. 2d,  notice that
      6. If B, A,  or  angle symbolB,A  vary,  ΦB = ∫ B . dA.

    3. The Experiment

      1. First we calibrate a galvanometer to detect the presence and the direction of a current. In Fig. 3 when current enters the terminal on the right hand side of the galvanometer, the needle deflects to the left.

      2. In Fig. 4a, the bar magnet is at rest. There is no current in coil of wire and the galvanometer does not deflect.

      3. In Fig. 4b, the magnet moves toward the coil and the galvanometer deflects toward the left indicating an induced current and associated induced magnetic field (shown by dashed line) opposing the field of the magnet.

      4. In Fig. 4c, the rate of change of the magnetic flux produced by the magnet has increased and there is a greater current in the galvanometer.

      5. In Fig. 4d, the magnet is inside the coil at rest. Current is zero.

      6. In Fig. 4e, the magnet moves to the right. Current reverses.

    4. Results of the Experiment

      The induced current creates an induced magnetic field that opposes the change produced by the external magnetic field. In this experiment the bar magnet provided the external field.

      1. When the magnet was at rest there was no induced current
        (Fig. 4a above).

      2. When the North pole of the magnet with its field toward the left moved toward the right face of the coil, the induced current in the coil was counterclockwise as viewed from its right end with the Coil's induced field to the right. Another way of looking at this is the right end of the coil acts like a North pole and repels the North pole of the magnet as you move the magnet toward it. You must exert a force through a distance doing work to induce a current in the coil (Fig. 4b above).

      3. The greater rate of change of magnetic flux, the greater induced current. As the magnet gets nearer to the coil, the flux changes more rapidly. You can also see this by bringing up two magnets at the same speed of one or bringing up one more rapidly (Fig. 4c above).

      4. When the magnet is at rest, there is no change in magnetic flux and no induced current (Fig. 4d above).

      5. When the North pole is moved away from the right end of the coil the current in the coil reverses. The end of the coil nearest the magnet acts like a South pole and you must do work to remove the North pole of the magnet and induce a current in the coil (Fig. 4e above).

    5. Sample problems in 106 Problem Set for Induction: 1, 2.

  2. Faraday's Law

    1. Induced emf = the rate of change of magnetic flux.

      1. Since ΦB = ∫ B . dA, you can change the magnetic flux by changing the magnetic field, the area through which the field exists or the angle between the magnetic field and the area.

      2. Lenz's Law states that the direction of the induced current is such to oppose the change that produced it. Lenz's Law is really a statement of conservation of energy. In order to get an induced emf or current, you must do work.

    2. Motional emf

      1. As conductor in Fig. 5 moves to the right, the electrons in the conductor move with it velocity v and experience a magnetic force down equal to qvB, where q is the charge of the electron. They accumulate at the bottom of the conductor leaving the top of the conductor with an excess of positive charge.

      2. This separation of charge produces an electric field from a to b that results in an electric force up on the electron equal to qE, where q is the charge of the electron.

      3. When the electric force equals the magnetic force,  qE = qvB,  the motion of the electron ceases and the electric field in the conductor E = vB. The potential difference between the ends of the conductor
      4. The motional.  In Fig. 5,  B was perpendicular to v.
        If the magnetic field is not perpendicular to the velocity, the motional 

    3. Alternating Current Generator

      1. In Fig. 2b above, the magnetic flux ΦB = BA cos angle symbolB,A.

      2. Let angle symbolB,A = Θ =  ωt, where ω equals the angular velocity of the rotating coil.  Then ΦB = BA cos ωt  and  ε = - NdΦB/dt = N ωBA sin ωt,
        where N = the number of turns of the coil.

      3. Figure 6a and 6b below are plots of the magnetic flux and the electromotive force as a function of time t, respectively.

    4. Sample problems in 106 Problem Set for Induction: 3-14.

  3. The Production of an Electric Field by a Changing Magnetic Flux

    1. To describe what happens in Fig. 7, we can say the motion of charge in the wire is produced by an induced current due to the change in magnetic flux through the loop. We can also explain it as the production of an electric field that produces a motion of charge.

    2. To put this statement into a mathematical form, we recall that electromotive force ε equals the energy per unit charge or work done per unit charge and the electric field equals the force per unit charge. The work done per unit charge around the loop of distance 2 πr equals E(2 πr),  or

      ε = E(2 πr) = -dΦB /dt.         (Equation 1)

  4. Inductance L

    1. ε = - dΦ/dt = L dI/dt.
      The unit of inductance is the Henry (H) = 1 volt/(A/s).

    2. Inductance is the third circuit element.
      By comparison, resistance R = Vab/I and capacitance C = q/Vab.

    3. Energy associated with inductor U = 1/2 LI2.

  5. Maxwell's Suggestion

    1. Since Nature is amazing symmetrical, Maxwell predicted that a changing electric flux should produce a magnetic field. The actual equation is similar to Eq.1 above, but it is only dimensionally correct if
      B(2 πr) = µ0e0 E /dt              (Equation 2)
    2. While it is above the level of Physics 106, we can tell you that these changing electric and magnetic fields are propagated as a wave. The use of Maxwell's Equations (there are really four) can be put together to get a wave velocity which must equal (1/µ0e0)1/2 in a vacuum or in air. Let's try it:

      µ0 = 4 πx 10-7 N/A2
      1/µ0 = 1/(4 πx 10-7 N/A2
      1/ ε0 = 4 πk = 4 π(9 x 109 N-m2/C2) (1/µ0e0)
             = (1/(4 πx10-7 N/A2)(4 πx 9 x 109 N-m2/C2)
             = 9 x 1016 (A/C)2m2
             = 9 x 1016 (m/s)2 (1/µ0e0)1/2
             = 3 x 108 m/s
             = the speed of light in a vacuum.

    3. As the late Professor Feynman of Cal Tech suggested, Genesis should be changed to read, "Let there be electricity and magnetism and there will be light."

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Susan D. Kunk
Phyllis J. Fleming
August 8, 2002
April 23, 2003