Tuesday, April 13, 1999
Announcements:

Lecture notes:
Explanations of Last Lecture's Demonstrations:

1. 1. Eddy Current

• When the magnetic flux through a conductor changes, induced current appears in the conductor in the direction that opposes the charge (Lenz's Law)
1.  2. Eddy Current

•     The magnet falls very much slower than a nonmagnetic object inside a metal tube.
•     Why?
• Induced current and therefore induced magnetic field are set up inside the tube to oppose the falling down of the magnet
1. 3. Flying Ring Experiment

• As the switch is rapidly closed: B and F_B in the iron core increases (rapidly) into B pointing up.
• This induced a current in the ring ( with B_induced  pointing down)
• This is similar to the case with two magnets, with north poles pointing at each other---Hence the repulsive forcce causing the ring to  fly up.
• Further explanation of the Flying wheel, induced current and induced electric field...
• Top view of the metallic ring:

• F = BA;  dF/dt = A dB/dt
• There is an induced current flowing around the ring. Therefore, there must be an induced Electric Field, E, that acclerates the charges.
• I(integral) E_induced ( Dot Product) ds = x_induced = - dF/ dt
• The changing magnetic field causes an induced electric field along the tangent of the ring. The changing B creates an induced E that is perpendicular to B
• x_induced = I_induced R; so therefore.... B_induced = m_oI_induced / 2r.... where R is resistance and r is the radius of the ring
• Even if the loop of wire is not there, there would still be an induced E, where E is perpendicular to B.  Consider the case of ionized charged particles in space. The induced E would accelerate these charges into circular orbit
• Induced electric field is very different form that due to static electric charges in that induced electric fields are not conservative.

• As the switch is closed, the current, i, in the circuit increases; it produces a magnetic field in the coil which is also increasing in magnitude. This increasing B field induces an emf; according to Lenz's Law that opposes further growth in B and i. Such an emf is sometimes calles self-induced emf or back emf : (Coil is often called a choke; symbol is L)
• According to Faraday's law, x_L = - dF_B/ dt (for a coil with N turns)
• Now the flux is proportional to B which is proportional to i
• Therefore  x_L = - N(dF_B/ dt) = - L (di/dt)
• L = NF/i = inductance of the coil
• Unit of L = vs/A = Tm²/A ( in umits of Henry)
• Let's consider a coil in the shape of a solenoid where the length of the coil is l and the cross sectional area is A
• B = m_o n i =  m_o (N/l) i
• F = BA =  m_o (N/l)A i
• L = NF/i (for N turns)
• L = m_o (N²/l)A i = m_o n²lA i = m_o n² (Volume)
• Therefore L does not depend on i, only on the geometry!
• Let us consider the circuit again:

• As the switch is closed, how would the current, i , build up?
• Qualitatively:
• Assume that L is an ideal coil with no resistance.

• Quantitatively:
• Note the similarity of this problem with that of charging a capacitor.
• 
• At t = 0 the switch is closed.
• x - iR - q/C = 0            x - R dq/dt - q/C =0
• q = Cx( 1 - e^-t/t)
• The same procedure can be followed for an RL circuit
• x = - L (di/dt)
• Treat this as another emf, in the opposite direction of the emf due to the battery
• From Kirchhoff's Loop rule
• x - iR - L di/dt = 0