My lecture slides are posted at http://www.physics.ohio-state.edu/~humanic/

Information for Physics 1201 Midterm I Wednesday, February 20

1) Format: 10 multiple choice questions (each worth 5 points) and two show-work problems (each worth 25 points), giving 100 points total. 2) Closed book and closed notes. 3) Equations and constants will be provided on the midterm 4) Covers the material in Chapters 16, 17, 18, 19, and 20

Chapter 21

Electromagnetic Induction

Induced Emf and Induced Current

There are a number of ways a magnetic field can be used to generate an electric current.

It is the changing field that produces the current.

So far we have dealt only with currents and magnetic fields which are constant in time, and we have shown that currents (i.e. moving charges) produce magnetic fields. Can magnetic fields produce currents? --> yes, if the magnetic field going through a loop of wire is changing in time!

Induced Emf and Induced Current

The current in the coil is called the induced current because it is brought about by a changing magnetic field. Since a source emf (electromotive force) is always needed to produce a current, the coil behaves as if it were a source of emf. This emf is known as the induced emf.

Induced Emf and Induced Current

An emf can be induced by changing the area of a coil in a constant magnetic field

In each example, both an emf and a current are induced because the coil is part of a complete circuit. If the circuit were open, there would be no induced current, but there would be an induced emf.

The phenomena of producing an induced emf with the aid of a magnetic field is called electromagnetic induction.

Motional Emf

THE EMF INDUCED IN A MOVING CONDUCTOR

Each charge within the conductor is moving and experiences a magnetic force

qvBF =

The separated charges on the ends of the conductor give rise to an induced emf, called a motional emf.

downward force on electrons

Motional Emf

Derive the motional emf, E, when v, B, and L are mutually perpendicular

Magnetic force on a charge in the rod FM = qvB In equilibrium, the electric force of repulsion from charge buildup at the ends, qE (where E is the electric field), balances FM qE = qvB è E = vB Gradient equation è E = |ΔV/Δs| = E/L E/L = vB è E = vBL

FM

Motional Emf

Example: Operating a Light Bulb with Motional Emf Suppose the rod is moving with a speed of 5.0 m/s perpendicular to a 0.80-T magnetic field. The rod has a length of 1.6 m and a negligible electrical resistance. The rails also have a negligible electrical resistance. The light bulb has a resistance of 96 ohms. Find (a) the emf produced by the rod, (b) the current induced in the circuit, (c ) the electric power delivered to the bulb, and (d) the energy used by the bulb in 60.0 s.

Motional Emf

(a) ( )( )( ) V 4.6m 6.1T 80.0sm0.5 === vBLE

A 067.096

V 4.6=

Ω==

RI E

(b)

E

E

(c ) P = IE = (0.067 A)(6.4 V) = 0.43 W

(d) Energy = Pt = (0.43 W)(60.0 s) = 26 J

Motional Emf

MOTIONAL EMF AND ELECTRICAL ENERGY

In order to keep the rod moving at constant velocity, the force the hand exerts on the rod must balance the magnetic force on the induced current which acts opposite the direction of Fhand

Fhand = F = ILB Using the numbers from the last example,

Fhand = (0.067 A)(1.6 m)(0.80 T) = 0.086 N and the work done by the hand in 60 s is Whand = Fhandx = Fhandvt =(0.086 N)(5 m/s)(60 s) = 26 J = Energy

Motional Emf

The direction of the current in this figure would produce a force which violates the principle of conservation of energy since it accelerates the rod thus creating energy out of nothing.

F

The direction of the current in this figure gives an induced force consistent with the conservation of energy è the direction of the induced current tends to oppose the applied motion -- it decelerates the rod once the applied force is removed (i.e. it takes energy to light the bulb).

Motional Emf

Conceptual Example: Conservation of Energy A conducting rod is free to slide down between two vertical copper tracks. There is no kinetic friction between the rod and the tracks. Because the only force on the rod is its weight, it falls with an acceleration equal to the acceleration of gravity. Suppose that a resistance connected between the tops of the tracks. (a) Does the rod now fall with the acceleration of gravity? (b) How does the principle of conservation of energy apply?

Magnetic Flux

RELATIONSHIP BETWEEN MOTIONAL EMF AND MAGNETIC FLUX

( ) ( )o

o

o

o

o

o

o

o

ttBABA

BttAA

BttLxxL

BLttxx

vBL−

−=""

#

$%%&

'

−

−=""

#

$%%&

'

−

−=""

#

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'

−

−==E

è magnetic flux BA=Φ

E

In time t0 an area A0 is swept out. In time t an area A is swept out.

ttt o

o

Δ

ΔΦ=

−

Φ−Φ=E

The motional emf equals the change of the magnetic flux per time

E

Magnetic Flux

GENERAL EXPRESSION FOR MAGNETIC FLUX

φcosBA=Φ

Units of magnetic flux: T m2 = Weber = Wb

è  depends on the angle at which the B-field crosses the area

Magnetic Flux

φcosBA=Φ

Example. A rectangular coil of wire is situated in a constant magnetic field whose magnitude is 0.50 T. The coil has an area of 2.0 m2. Determine the magnetic flux for the three orientations φ = 0o, 60o, and 90o, as shown.

φ = 0o Φ = (0.50)(2.0)cos 0o = 1.0 Wb φ = 60o Φ = (0.50)(2.0)cos 60o = 0.50 Wb φ = 90o Φ = (0.50)(2.0)cos 90o = 0 Wb

Magnetic Flux

GRAPHICAL INTERPRETATION OF MAGNETIC FLUX

The magnetic flux is proportional to the number of field lines that pass through a surface.

Faraday’s Law of Electromagnetic Induction

FARADAY’S LAW OF ELECTROMAGNETIC INDUCTION The average emf induced in a coil of N loops is

tN

ttN

o

o

Δ

ΔΦ−=$$

%

&''(

)

−

Φ−Φ−=E

SI Unit of Induced Emf: volt (V)

the minus sign reminds us that the induced emf will oppose the change in Φ à Lenz’s Law

E

Faraday’s law states that an emf is generated if the magnetic flux changes for any reason. Since Φ = BA cos φ, any change of B, A, or φ will induce an emf.

(The motional emf-Φ relation we derived is a special case of this.)

Faraday’s Law of Electromagnetic Induction

Example: The Emf Induced by a Changing Magnetic Field A coil of wire consists of 20 turns each of which has an area of 0.0015 m2. A magnetic field is perpendicular to the surface. Initially, the magnitude of the magnetic field is 0.050 T and 0.10 s later, it has increased to 0.060 T. Find the average emf induced in the coil during this time.

( )( ) ( )

V 100.3

s 0.10T 050.0T 060.0

0cosm 0015.020cos

coscos

3

2

−×−=

−−=#

$

%&'

(Δ

−−=

Δ

−−=

Δ

ΔΦ−=

tBB

NA

tABBA

Nt

N

o

o

φ

φφEE

Lenz’s Law

LENZ’S LAW The induced emf resulting from a changing magnetic flux has a polarity that leads to an induced current whose direction is such that the induced magnetic field opposes the original flux change.

Reasoning Strategy for applying Lenz’s Law to finding the induced current direction 1.  Determine whether the magnetic flux that penetrates the coil is increasing or decreasing. 2.  Find what the direction of the induced magnetic field must be so that it can oppose the change in flux by adding or subtracting from the original field. 3. Use RHR to determine the direction of the induced current.

Lenz’s Law

Conceptual Example: The Emf Produced by a Moving Magnet A permanent magnet is approaching a loop of wire. The external circuit consists of a resistance. Find the direction of the induced current and the polarity of the induced emf.

Since the applied magnetic field in the loop is increasing and pointing to the right, Lenz’s law says an induced current will be created in the loop to try to oppose this change by creating an induced magnetic field to the left.

Lenz’s Law

Conceptual Example: The Emf Produced by a Moving Copper Ring. There is a constant horizontal magnetic field directed into the page in the shaded region. The field is zero outside the shaded region. A copper ring is dropped vertically through the region. For each of the five positions, determine whether an induced current exists and, if so, find its direction. Is the acceleration of the ring the same as it drops through the five positions?