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.
      
      

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

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.
   
   
   
   

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).
   
   

1. Since Φ_B = ∫ 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.
   
   

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 
   
   

1. In Fig. 2b above, the magnetic flux Φ_B = BA cos B,A.
   
   
2. Let B,A = Θ =  ωt, where ω equals the angular velocity of the rotating coil.  Then Φ_B = BA cos ωt  and  ε = - NdΦ_B/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.
   
   
   
   

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

B(2 πr) = µ₀e₀ dΦ_E /dt              (Equation 2)