EMF Handout 9: Electromagnetic Induction ELECTROMAGNETIC INDUCTION (Y&F Chapters 30, 3; Ohanian Chapter 3) This handout coers: Motional emf and the electric generator Electromagnetic Induction and Faraday s Law Lenz s Law Induced electric field Inductance Magnetic energy The Electric and magnetic fields are inter-related The electric and magnetic fields are not independent. In fact: A changing magnetic field induces an electric field The reerse is also true: a changing electric field induces a magnetic field To see how this occurs, we first consider MOTIONAL emf Motional emf Consider a wire moing with elocity in a magnetic field B. Assume for simplicity that and B are perpendicular The electrons in the wire experience a force F = e( B) B (out of page)....
EMF Handout 9: Electromagnetic Induction This produces a separation of charge with an excess of negatie + + + b charge at one end and positie charge at the other. There is therefore an INDUCED emf between the two ends. L x B F = -e( x B) a - - - Induced emf E = work done to moe a unit positie charge from a to b against the magnetic force F = QB = B B (out) Work done = (Force)(Distance) E = BL MOTIONAL emf The electric generator Assume that the ends of the wire are connected up through some external circuit (which we represent here by a simple resistor). Current I flows and dissipates electrical power in R. B (out of page) Where does this power come from? The current is due to electrons moing in the wire with some drift elocity electron. I R The electrons therefore experience a magnetic force F = e( electron B) This force OPPOSES the motion of the wire through the field. B (out) F = -e( electron x B) electron
EMF Handout 9: Electromagnetic Induction 3 The oerall force on the wire is F mag = ΣF for all the electrons. Recall: Force on a current-carrying wire in a magnetic field is F mag = BIL. So, the power deliered to the circuit comes from the effort needed to PUSH the wire through the magnetic field against this force. The magnetic field thus acts as a mediator in the conersion of MECHANICAL ENERGY into ELECTRICAL ENERGY. This is the principle of the ELECTRIC GENERATOR. Exercise: Show that the power needed to moe the wire through the magnetic field is equal to the power dissipated in the resistor. Motional emf and magnetic flux Recall: E = BL In a time, the conductor moes a distance. The area swept out is therefore ds = L The magnetic flux passing through this area is dψ = Bds = BL Therefore dψ = BL E =.... dψ B (out of page) L The induced emf is equal to the rate of sweeping out of magnetic flux Note:. This applies to ANY shape of conductor in ANY magnetic field.. The same equation applies to a stationary conductor in a changing magnetic field.
EMF Handout 9: Electromagnetic Induction 4 Faraday's law of induction The induced emf around any closed path in a magnetic field is equal to minus the rate of change of the magnetic flux intercepted by the path: E = Lenz's law dψ Why the negatie sign? This is represented by the negatie sign in the aboe equation for the induced emf. The induced emf is in such a direction as to OPPOSE the change in magnetic field that produces it. B (increasing) If ψ is increasing then E causes current to flow so as to generate a magnetic field which OPPOSES ψ. Current due to induced emf B induced opposes B If ψ is decreasing then E causes current to flow so as to generate a magnetic field which SUPPORTS ψ. Current due to induced emf B (decreasing) B induced supports B
EMF Handout 9: Electromagnetic Induction 5 Induced electric field Consider again a wire moing in a magnetic field. Look at the situation from the point of iew of someone who moes along with the wire: = 0 there is no magnetic force Therefore, this obserer interprets the force acting on the electrons in the wire as being due to an INDUCED ELECTRIC FIELD....... B (out of page).... E L...... E d L around the closed path shown 0. The induced electric field is not conseratie. Faraday's Law is usually written in this form: E dl = dψ FARADAY'S LAW MAXWELL'S 3rd EQUATION In words: The line integral of the electric field around a closed path is equal to minus the rate of change of magnetic flux through the path
EMF Handout 9: Electromagnetic Induction 6 Mutual inductance Faraday's Law a changing B induced emf Consider two nearby circuits: Changing current in Circuit Circuit Circuit I B Changing magnetic field through Circuit Induced emf in Circuit Current flows in Circuit Let I (t) create magnetic field B (t). Let ψ be the flux through circuit due to I Clearly, B I, so ψ I DEFINITION: The constant of proportionality between ψ and I is called the MUTUAL INDUCTANCE: M = Ψ I Magnetic flux through circuit (due to) Current in circuit From Faraday s Law, the induced emf is gien by di E = -M
EMF Handout 9: Electromagnetic Induction 7 Note: M depends on the shape, size, numbers of turns and relatie positions of the two circuits M = M so we need only use M as the symbol for mutual inductance Ohanian uses L for mutual inductance In the SI system, inductance is measured in Henrys (H) H = Inductance that produces an emf of Volt for a rate of change of current of A s - H V s A - The usual unit for µ o is H m - Self inductance Een a single circuit produces a magnetic field that passes through the circuit itself. So, if I changes B changes Ψ changes induced emf Definition: SELF INDUCTANCE L = Ψ I Magnetic flux (due to) Current through a circuit in the circuit itself Inductance and Lenz s law M or L = Ψ I E = -M di di or -L Recall: The negatie sign represents Lenz s Law: the emf causes current to flow so as to oppose the change in flux that produces it.
EMF Handout 9: Electromagnetic Induction 8 INCREASING If ψ is then E will cause current to flow so as to create DECREASING OPPOSES a magnetic field that ψ. SUPPORTS Circuit Circuit i (t) B (t) i (t) B (t) i (t) increases ψ increases Induced emf in circuit dries current i (t) I (t) generates B ( t) OPPOSES B ( t) which Circuit Circuit i (t) decreases i (t) i (t) B (t) B (t) ψ decreases Induced emf in circuit dries current i (t) I (t) generates ( t) SUPPORTS B ( t) B which Procedure for finding inductance. Assume that a current I flows in (one of the) circuit(s). Find the magnetic field (e.g., use Ampere s Law) 3. Find the magnetic flux linked 4. Put M or L equal to ψ/i Examples:. Mutual inductance of two long coaxial solenoids. Self inductance of a long solenoid See lecture notes
EMF Handout 9: Electromagnetic Induction 9 Energy storage in inductors Recall: Putting a charge on a capacitor Storing Doing potential U elec = ½ CV work energy Similarly: Making current flow in an inductor Doing work Storing potential energy U mag =? Because: changing current changing B induced emf which opposes attempt to change current (called a back emf ) To find the energy stored in an inductor: Let current in the inductor i = 0 initially Apply external emf current increases at rate This leads to a back emf di di E b = -L To make current flow, the required external emf is The externally applied power is therefore di E ext = L di P = E ext i = Li The work done to increase the current from 0 to I is therefore U = t I P = Lidi U = ½ LI 0 0 Energy stored in an inductor charged with current I
EMF Handout 9: Electromagnetic Induction 0 Where is the stored energy? It is in the MAGNETIC FIELD. Recall: For a solenoid of length a, radius R, n turns per unit length: L = πr µ o n a B = µ o ni U = ½ LI = [ ] o o n B a n R µ µ π = [ ] a R B o π µ Therefore U = [ ] the solenoid Volume of B o µ The ENERGY DENSITY OF THE MAGNETIC FIELD is therefore µ = o B u This holds generally, for any magnetic field.