CHAPTER 5: MAGNETISM 5.1 MAGNETS AND MAGNETIC FIELDS ✓ Magnet always comes in a pair (North & South) ✓ Magnet itself produces and surrounded by its the magnetic field. ✓ The magnetic field lines originate from the NORTH pole to SOUTH pole. ✓ Like poles repel each other ✓ Unlike poles attract each other ✓ Magnetic Force = F SI Unit = Newton, N ✓ Magnetic Field = B SI Unit = Tesla, T
F Magentic force, N
v B sin 90 0 F max
q Charge, C Velocity, m/s B Magnetic field, T
v || B sin 0 0 0
Relationship between ELECTRIC & MAGNETISM? When an electric current passes through a wire, a magnetic field is formed. (Electric field surrounds the charges). Direction of B depends on the direction of current in wire (RHR-2)
THE CIRCULAR TRAJECTORY The F is always remains to v and B and is directed toward the centre of the circular path Determine the RADIUS of the circle:
Moving charges (current) produce magnetic field, B.
F B F C
When a charged particle is in motion in magnetic field,
it will experience magnetic force, F.
5.2 MAGNETIC FORCE ON MOVING CHARGES Moving charge experience F when placed in B .
F q vB sin
q vB m r
v
v B sin F
2
r
mv
q B F C Centripetal force, N FB Magnetic force, N m mass of the object, Kg v speed of the object, m/s r radius, m 5.3 FORCE ON A CURRENT-CARRYING WIRE IN A MAGNETIC FIELD Current-carrying wire experience F when placed in B
F IlBSin B IlBSin I Current in wire, A l Length of conductor, m
B Magnetic Field, T 0 I B sin 90 F max
I || B sin 0 0 0
0 I B sin F
5.4 MAGNETIC FIELDS PRODUCED BY CURRENTS Currents in infinitely long & straight wire produce magnetic field
B
0 I
2 r 0 permeability of free space (4 10 -7 T.m/A ) r distance of pa partic rticle le from wire (m) I Current in wire (A)
5.5 FORCE BETWEEN TWO PARALLEL WIRES A current-carrying wire will exert force on another
nearby F attract for wires with parallel currents. F repel for wires with opposite currents.
d
MAGNETIC FIELD PRODUCED BY THE CURRENTS IN WIRE 1 & 2
B1
0 I 1
B2
2 d
0 I 2
2 d
MAGNETIC FORCE EXPERIENCED BY WIRE 2
0 I 1 0 I 1 I 2 l d d 2 2
F 2 I 2lB1 I 2l
I B sin 90 0 F max I || B sin 0 0 0 I B sin F 5.6 CURRENT LOOPS AND MAGNETIC TORQUES ➢ If a loop of wire is suspended in a magnetic field, the magnetic force produces a torque that tends to rotate the loop. MAGNETIC FORCE EXPERIENCE BY VERTICAL SEGMENT TORQUE ON EACH VERTICAL SEGMENT Two forces resulted from the direction of current The loop experience a net torque that tends to have the same magnitude but point in opposite rotate it in a clockwise. directions. So the loop experiences zero net force. w sin IhB Force at one side: 2 F .r sin w Width (m) r length of moment arm
F IhB
F IhB
moment arm
w
2
sin
SI Unit N.m
TORQUE ON BOTH VERTICAL SEGMENTS (TOTAL τ) Torque max = normal of horizontal with B Torque 0 = normal of horizontal || with B
total IhB
w
w
sin IhB sin 2 2 IBhwsin SI Unit N.m
TORQUE IN GENERAL hw is also the area, A of the rectangular loop,
hence Torque in general:
IBA sin
SI Unit N.m TORQUE WITH N TURNS
NIAB sin
Magnetic moment, NIA
5.7 APPLICATIONS: SOLENOIDS AND ELECTROMAGNETS ➢
Current –carrying wire formed into a circular loop and produced magnetic field, B ➢ Current flow clockwise: Left side is SOUTH & Right side is NORTH ➢ Current flow counterclockwise: Left side is NORTH & Right side is SOUTH MAGNETIC FIELD INSIDE SOLENOIDS
N I 0 nI L L length or (width) (m) h height (m)
B 0
n number of loops per unit length
N L
Into
Out
CHAPTER 6: ELECTROMAGNETIC INDUCTION AND FARADAY’S LAW
When an electric current is passed through a coil of wire wrapped around a metal core, a very strong magnetic field is produced. This is called electromagnet. • EM induction is the process by which the coil of wire is used to create or i nduce an EMF if moved in a magnetic field. 6.1 INDUCED EMF AND INDUCED CURRENT 6.1 MAGNETIC FLUX A BAR MAGNET & COIL OF WIRE Measure of the number of magnetic field lines that ➢ Magnet moves towards coil or away from coil: cross a given cross-sectional area. Changing magnetic field of magnet at the coil produces emf & current. Φ BA cos ➢ The emf & current produced in the coil are called SI Unit Weber (Wb) induced emf & induced current as it is generated 1 T.m 2 1Wb (induced) by the changing field. ➢ Both magnet and coil of wire were held stationary: No N loop B cos 90 0 0 change of B, hence produce no emf & current . •
N loop || B cos 0 0 max N loop B cos
6.3 FARADAY’S LAW OF INDUCTION ➢ Changing flux induce an EMF. ➢ THE INDUCED EMF, Ε is proportional to the RATE OF CHANGE OF THE MAGNETIC FLUX WITH TIME and is in such a direction as to oppose the change in the magnetic flux.
N
final initial N t final t initial t
N No.of turns (loop) MAGNITUDE OF ε
N
final initial N t final t initial t
6.4 LENZ’S LAW ➢ The Induced EMF and Current oppose the change of flux. ➢
Lenz’s law is a physical way of expressing the meaning of the minus sign in Faraday’s law of induction. ➢ It states: Induced current is directed to oppose change in magnetic flux taking place. The induced emf is in opposite of the applied voltage across the coil. ➢ Changing field induce EMF & I. The induced I flowing in the wire will generate magnetic field called induced magnetic field. The direction of I should create the induced B that opposite to the change of flux.
6.5 EMF INDUCED IN A MOVING CONDUCTOR (MOTIONAL EMF) When a conducting rod moves through a constant magnetic field, magnetic flux will change which after that induced emf & I in the rod. Charges within the conductor rod move with velocity as the rod moved in the uniform magnetic field. The velocity of the rod is to B.
➢ ➢ ➢
➢
When the rod moves in a DECREASING area Conductor rod move in decreasing area. = BA through the loop decreases. To oppose this decrease, Lenz’s law states that the induced current must flow in a direction that strengthen the B within the loop, hence increase the . The induced B by the induced I must go into the page or additive to the existing B.
INCREASE IN MAGNETIC FLUX ➢ Increase loop’s area, increase the ɸ
Φ B A Bvlt A (vt )l B uniform d vt A distance it travel x Length of rod
INDUCED CURRENT FOR SINGLE LOOP
I
R vB 2 L2 B 2 L2v 2 v Fv R R
F q
➢
INDUCED EMF FOR SINGLE LOOP
N
B A Bvlt BA N N 1 Blv t t t t
B uniform d vt
A distance it travel x Length of rod vtl MAGNETIC FORCE EXERTED ON THE ROD
ELECTRICAL POWER
6.6 RELATION BETWEEN B AND E FIELDS Since charges move in the conductor, there is an electric field induced by the changing magnetic flux in the moving conductor.
E
➢
2 2
MECHANICAL POWER
Pmechanical
➢
vBl B l v F IlB lB R R
Blv
R
➢
When the rod moves in the INCREASING area Conductor rod move in increasing area. = BA through the loop increase. To oppose this increase, Lenz’s law states that the induced current must flow in a direction that against the direction of B, in order to decrease the . The induced B by the induced I must going into the page to decrease the flux.
qvB q
vB
2
Pelectrical
2 2 2
B L v Bvl I R R R R 2
6.7 ELECTRIC GENERATOR A device that apply the concept of motional emf (induction) to convert the mechanical energy (rotation of the coil) into electrical energy. One side: Blv Blv sin Both sides:
2 Blv 2 Blv sin 0 sin
6.7 COUNTER (OR BACK) EMF ➢ There are two ways of inducing an emf in a coil: Mutual inductance: The effect in which a changing current in one circuit induces an emf in another circuit This physical phenomenon is used in transformers. Self-inductance: The effect in which a changing current in a circuit induces an emf in the same circuit. The induced emf is proportional to the current changing in the coil. Such a coil is known as an inductor. RL CIRCUIT (RESISTOR & INDUCTOR) ➢ When the switch is closed, currents flows through a circuit or coil. As the I increases with time, magnetic flux through the circuit loop also increases with time ➢ According to Faraday’s law, the changing magnetic flux will create and induce an emf in the coil. ➢ The direction of this induced emf is determined according to Lenz’s law in such that its direction is opposite the direction of the battery. Because of the direction of the induced emf, it is called back emf.
SELF-INDUCTANCE The effect of back emf is called self-induction because the changing flux through the circuit and induce emf arise from the circuit itself. The inductance of the coil prevents the I in the circuit from increasing or decreasing instantaneously.
L N
Φ I
L Inductance SI Unit Henry (H)
Φ Faraday' s Law t I L Self - induced EMF t N
INDUCTANCE OF A SOLENOID
L N
Φ I
N N 0 I A Φ f Φi N BA 0 l N I f I i I I 0 2
Let I0 = 0, If = I 0 =
B in solenoid:
0 N A B 0
l 0 n 2 Al n number of turns per unit length Al volume of solenoid
N l
0, f = BA cos θ
N I 0 nI l
ENERGY STORED IN AN INDUCTOR ➢ A coil or solenoid is referred as an inductor used to store energy in the magnetic field. ➢ The energy is required to do work to create/initiate a current in an inductor. ➢ To derive an expression for the energy stored in an inductor, we consider increasing the current in the inductor of inductance L. ➢ The current increase linearly with time: I i = 0 at t i = 0, If = I at t f = t EMF
AVERAGE POWER
L
I I 0 LI L t 0 t t
AVERAGE ENERGY U av Pav t 12 LI 2
Pav
u B
magnetic energy 1 B 2 volume 2 0
1 2
I
1 2
LI
2
t
ENERGY STORED IN SOLENOIDS MAGNETIC FIELD U
U ENERGY DENSITY
I av
1 2
LI
1 2 0
2
2
1 2
n Al I
B Al
2
0
2
CHAPTER 7: ALTERNATING CURRENT (AC) 7.1 MUTUAL INDUCTANCE AND TRANSFORMER MUTUAL INDUCTANCE Changing current in one circuit induces an emf in the second circuit.
MUTUAL INDUCTANCE, MS OF SECONDARY COIL WITH RESPECT TO PRIMARY COIL
P
M S N S
I P
SI Unit Henry ( H ) INDUCED EMF IN SECONDARY COIL BY PRIMARY COIL
S N S
ΔΦ P Δt
Faraday' s Law
I M S P N S N S
Mutual inductance
Δt
S M S
ΔI P Δt
TRANSFORMER AC transformer apply the concept of mutual inductance. A device that transmit electric power from the electrical generating plants to the consumers of
electricity. It is used to step-up or step-down the voltage. The primary coil connected to the input alternating-voltage source (AC generator) and has N P turns, and the secondary coil has N S turns and connected to the load. Primary coil
P N P
Secondary coil
P t
s N S
Since
S t
P =
S
P N P S N S
Induced ε in Secondary coil
Average power in Secondary coil
N S P S N P
I S I P
N P N S
7.2 ALTERNATING CURRENT (AC) CIRCUITS Direct current: Current flows in one direction from positive terminal to negative terminal
meanwhile electron flows in opposite direction. Alternating current: Current from a generator reverses direction many times per second (AC changes over time in an oscillating repetition).
7.2.1
ALTERNATING VOLTAGES AND CURRENTS
AC GENERATOR (POWER SOURCE OF AC CIRCUIT) Mechanical energy to electrical energy. The generator consists of a coil of wire that is rotated by some external mechanical means in a
uniform field. Derive expression for the loops:
Vertical side : BLv
BLv sin
Both sides: 2 BLv 2 BLv sin With N turns: 2 NBLv sin Angular speed,
v r
W 2
t
so,
t
Change the v with respect to angular speed. The induced EMF in the loops:
2 NBlv sin
W 2 NBl sin t 2 NB (lW ) sin t NBA sin t MAXIMUM EMF
ZERO EMF
max NBA
NBA 0
wt 900 or 2700
wt 0 or 180 0
0
RESISTOR IN AC CIRCUIT An AC circuit consists of the circuit elements (resistor) and AC power source that provides an
alternating voltage ∆v and alternating current I vary sinusoidally with time ANGULAR FREQUENCY OF THE AC VOLTAGE
2 f
2 T
frequency f, VOLTAGE IN RESISTOR
CURRENT IN RESISTOR
V t V max sin t
I (t )
Vmax = Voltage amplitude, Max output voltage
1 T
(Hz or s -1 )
MAX CURRENT IN RESISTOR
V max sin t R R I (t ) I max sin t V (t )
I max
V max R
7.2.2 ROOT MEAN SQUARE VALUES (RMS) Since Vav and Iav have zero value in AC, the useful type of average or mean is the root mean square (rms) values. RMS CURRENT
I rms
7.2.3
I max
2
RMS VOLTAGE
V rms
0.707 I max
V max
2
0.707V max
POWER IN AC CIRCUIT WITH RESISTOR (Watt, P)
INSTANTANEOUS POWER IN RESISTOR
Pt I R 2
2 I max R
2
sin t
AVERAGE RESISTOR
POWER
DELIVERED
2 I rms R 2 sin t 2 2 2 12 V max V rms 1 V max Pav 2 R R P(t ) IV I max sin t V max sin t I maxV max sin 2 t R
P (t ) R R 2
V
2 V max
2 Pav 12 I max R
Pav
I maxV max
2
2 1 I R max 2
I max V max
TO
2
2
I rmsV rms
7.2.4
PHASORS DIAGRAM FOR AC CIRCUIT WITH RESISTOR
The phasor diagram represents the current & voltage values in a rotating arrow with angular
speed, ω. The length is proportional to the max value of the I & V which projected onto x-y-axis. The phasor rotates counterclockwise.
Both Vmax & Imax located at the same position. Phase difference is 0.
For a sinusoidal applied voltage, iR in resistor is always in phase with vR across the resistor.
7.3 AC CIRCUITS AND IMPEDANCE 7.3.1 CAPACITOR IN AC CIRCUIT An AC circuit consists of the circuit elements (capacitor) and AC power source that provides an
alternating voltage ∆v and alternating current I vary sinusoidally with time. RMS CURRENT CAPACITOR
I rms
MAX VOLTAGE AT CAPACITOR
V max,C I max X C
V rms X C
CAPACITIVE REACTANCE How much rms current exits in capacitor in response to a given rms voltage across capacitor.
X C
1 C
1 2 fC
SI Unit 7.3.2 PHASOR DIAGRAM FOR AC CIRCUIT WITH CAPACITOR
➢
IC leads VC by 90° ➢ VC lags IC by 90° ➢ VC and IC are out of phase.
Vmax located at 90ͦ from the origin & I max at 0. The phase difference between voltage and
current is 90 where current leads the voltage.
7.3.3 RC CIRCUITS IN AC
VR = voltage across resistor VC =Voltage across the capacitor V = total voltage across the RC series circuit
MAX VOLTAGE ACROSS RESISTOR
PHASOR DIAGRAM ➢ Voltage drop in resistance, V R=ImaxR taken in phase with the current vector. ➢ Voltage drop in capacitive reactance, V C=ImaxXC is drawn 90° behind the current vector, as I leads V by 90°. ➢ The vector sums of the 2 voltage drops is equal to the applied voltage, V TOT=ImaxZ RMS TOTAL VOLTAGE IMPEDANCE FOR RC CIRCUIT V
V max, R I max R ➢
Total voltage ≠the sum of the Vmax across resistor and Vmax across capacitor as they are not in phase
TOTAL VOLTAGE OF RC 2 2 V max, T V max, R V max, C
I rms
I max R I max X C 2
2
rms
Z
Z R 2 X C 2
SI Unit
I max R 2 X C 2 I max Z POWER FACTOR
cos
➢
I max R
R
I max Z Z
R R 2 X C 2
Between the total voltage & current
7.3.4 POWER CONSUMED BY AN RC CIRCUIT No power consumed by the capacitor. Total power of the circuit is the power dissipated by the resistor. AVERAGE POWER OF RC CIRCUIT
Pav I rms R I rms I rms R 2
V I rms rms R Z
Pav I rmsV rms cos
Power factor cos
7.3.5 INDUCTORS IN AC CIRCUIT An AC circuit consists of the circuit elements (inductor) and AC power source that provides an
alternating voltage ∆v and alternating current I vary sinusoidally with time. RMS CURRENT AT INDUCTOR
I rms
MAX VOLATGE AT INDUCTOR
V max, L I max X L
V rms X L
INDUCTIVE REACTANCE How much rms current exits in inductor in response to a given rms voltage across inductor.
X L 2 fL L
SI Unit 7.3.6 PHASOR DIAGRAM FOR AC CIRCUIT WITH INDUCTOR
➢
VL leads IC by 90° ➢ IC lags VC by 90° ➢ VC and IC are out of phase.
Imax located at 90ͦ from the origin & V max at 0. The phase difference between voltage and
current is 90 where voltage leads the current. 7.3.7 RL CIRCUIT IN AC
VR = voltage across resistor VL =Voltage across the inductor V = total voltage across the RL series circuit TOTAL VOLTAGE OF RL
V max
I max R 2 I max X L 2
I max R 2 X L2 I max Z AVERAGE POWER
Pav I rmsV rms cos
Power factor cos
IMPEDANCE FOR RL
POWER FACTOR
Z R 2 X L2
cos
I max R
R
I max Z Z
R R X L 2
2
7.3.7 RLC CIRCUIT IN AC Resistor, inductor & capacitor connected in series circuit.
❖ The phasors are combined on a single set of axes. ❖ The total voltage V max (resultant) makes an angle ɸ with Imax
TOT. VOLTAGE OF INDUCTOR & CAPACITOR
TOTAL VOLTAGE OF RLC
V LC V L V C
V max
V LC I max X L I max X C
V R 2 V L V C 2 2
I max R I max X L I max X C I max R 2 X L X C
2
2
I max Z PHASE ANGLE ɸ
IMPEDANCE OF RLC
Z R 2 X L X C
2
1 R 2 L C
tan
R Z
I max R
2
POWER FACTOR
cos
I max X L X C
MAXIMUM CURRENT
I max
X L X C
V max Z
R
7.3.7 RESONANCE IN RLC CIRCUIT
R 2
R 2 X L X C
X L X C is ve, VTOT leads I X C X L is ve, I leads VT OT AVERAGE POWER
Pav I rmsV rms cos
Power factor cos
RESONANCE IS WHEN XL = XC: ➢ I & VTOT are in phase ➢ Z is minimum ➢ I is maximum ➢ Phase angle, ɸ is 0 ➢ Circuit is purely resistive
X C X L 1 2 fC f
2 fL
1
1
2π LC
RESONANCE FREQUENCY
Vrms, C Vrms, L I rms X C I rms X L
1 I rms L C
I rms
1 LC
or f
1
1
2 LC
