Week 8 Faraday’s Law Inductance Energy

 The law states that the induced electromotive force or EMF in any closed circuit is equal to the time rate of change of the magnetic flux through the circuit: where ◦ Ɛ is the electromotive force (EMF) in Volts ◦ Φ B is the magnetic flux through the circuit in Webers

 The magnetic flux Φ B through a surface S, is defined by an integral over a surface:

 If the loop enters the field at a constant speed, the flux will increase linearly and EMF will be constant.  What happens when the entire loop is in the field?

 A conducting rod of length L moves in the x- direction at a speed va x as show below.  The wire cuts into a magnetic field B a z.  Calculate the induced EMF.

 Compute the total flux Φ B Φ B = B A = B L (x o + vt)  Calculate the rate of change Ɛ = - dΦ B /dt = - vBL  Since the magnetic flux through the loop increases linearly the EMF induced in the loop must be constant.

 An EMF is induced on the rectangular loop of the generator.  The loop of area A is rotating at an angular rate ω (rads).  The magnetic field B is held constant.

 Assume B = B o a x  the square loop rotates so its normal unit vector is a n = a x cos ωt + a y sin ωt  Determine the flux as a function of time and the induced EMF

 Inductance is the property in an electrical circuit where a change in the current flowing through that circuit induces an EMF that opposes the change in current.

 The term 'inductance' was coined by Oliver Heaviside. It is customary to use the symbol L for inductance, possibly in honor of Heinrich Lenz.  Other terms coined by Heaviside ◦ Conductance G ◦ Impedance Z ◦ Permeability μ ◦ Reluctance R

 The electric current produces a magnetic field and generates a total magnetic flux Φ acting on the circuit.  This magnetic flux tends to act to oppose changes in the flux by generating an EMF that counters or tends to reduce the rate of change in the current.  The ratio of the magnetic flux to the current is called the self- inductance which is usually simply referred to as the inductance of the circuit.

 Derive an expression for the inductance of a solenoid Since B = μNI/ℓ Φ = BA = μNIA/ℓ thus L = NΦ/I = μN 2 A/ℓ

 Derive an expression for the inductance of a toroid. Recall that B = μNI 2πr

 The pickup from an electric guitar captures mechanical vibrations from the strings and converts them to an electrical signal.  The vibration from a string modulates the magnetic flux, inducing an alternating electric current.

 Taking the time derivative of the flux linkage λ = NΦ = Li yields = v This means that, for a steady applied voltage v, the current changes in a linear.

 Let the current through an inductor be i(t) = I o cos(ωt). Determine v(t) across the inductor.

 In practice, small inductors for electronics use may be made with air cores.  For larger values of inductance and for transformers, iron is used as a core material.  The relative permeability of magnetic iron is around 200.

 Here, L 11 and L 22 are the self-inductances of circuit one and circuit two, respectively. It can be shown that the other two coefficients are equal: L 12 = L 21 = M, where M is called the mutual inductance of the pair of circuits.

 Consider for example two circuits carrying the currents i 1, i 2.  The flux linkages of circuits 1 and 2 are given by L 12 = L 21 = M

 Describe what will happen when the switch is on.

 The energy density of a magnetic field is u = B 2 2μ  The energy stored in a magnetic field is U = ½ LI 2

 In a certain region of space, the magnetic field has a value of T, and the electric field has a value of 2x10 6 V/m.  What is the combined energy density of the electric and magnetic fields?

 Read Sections 6-1, 6-2, 6-4, and 6-5.  Solve end-of-chapter problems 6.2, 6.5, 6.10, and 6.12.