# Synopsis by nuhman10

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```									These notes are a brief synopsis of the electricity and magnetism course based on
the Physics for Scientists and Engineers by Serway and Beichner book.

Magnetism:
A magnetic field is in many ways similar to an electric field. It has a strength and
direction. It is measured in Teslas T. Its force dies off with the square of the
distance r. A magnetic field can come from a permanent magnet or it can be generated
by a moving charge.
The magnetic field is measured by the force it generates on a moving charge. The
force depends on the velocity of the charge and its magnitude and its angle of motion
relative to the field.

FB  q vBsin

N     N
1T           
C  m s A m

The magnetism created by a moving charge i.e. current flowing in a wire is called
electromagnetism. Electromagnetism is the basis of many modern technologies such
as microphones, loudspeakers, motors, generators and transformers.

One of the key points to remember is that magnetism is caused by the movement of
electric charges which in turn is caused by electric fields. Conversely moving
magnetic fields will cause electric fields to be generated under the right
circumstances.

The strength and direction of a magnetic field caused by a current in a wire can be
calculated using the Biot-Savart law.

o Ids  r
ˆ         0  4  10 7 T.m / A
dB 
4 r 2

This law by itself tells us the strength and direction of the field at some point near a
current carrying conductor due to the current flowing in a very small segment ds. It
does not tell us about the effect of the current in the whole wire. To find this effect
we need to integrate the contributions to the field from all of the segments of the wire.
In the general case this is fairly difficult but there are a few specific cases where the
integration can be done analytically. These cases are the straight wire and the circular
segment.

For the straight wire the equation becomes:

0 I
B        cos1  cos 2 
4a
where a is the perpendicular distance from the wire to the point of interest and 1 2
are the angles between the ends of the wire and the point of interest. Remember to
always measure these angles clockwise from 0. The direction of the field is given by
the right hand rule. Grasp the wire in your right hand with your thumb pointing in the
direction of the current flow and your fingers will point in the direction of the field.
Field directions are indicated with X to indicate it is going away from you (imagine
the tail flight of an arrow) and a dot to indicate the field is coming toward you
(imagine the point of the arrow) Field lines point from North poles to South poles.
In the case of a very long straight wire the equation above reduces to
0 I
B
2a
because the angles 1 2 tend towards 0 and 180 respectively.

In the case of the circular loop the field at the centre of the loop is given by :

0 I
B        
4R
Where  is the portion of the full circle in radians.
For a full circle loop therefore the equation becomes :
0 I
B
2R

The Biot-Savart law applies to the general case but can be very difficult to apply at
times. Ampere’s Law can be used to simplify many situations. Ampere’s name was
given to the definition of the Amp which is the unit of current. Symbol I.
When two parallel conductors carry a current they both create a magnetic field. The
current in one will experience a force due to the field of the other and vice versa. The
size of this force is given by:
 II
F  o 1 2 l This force is used as in the definition of the Amp
2a

When the force per unit length between two long conductors separated by 1m is
2 x10-7 N/m the current in the conductors is 1 amp.
Ampere’s law then goes on to show that the line integral of the magnetic field along
any closed path is dependant only on the total current passing through the area
enclosed by that path and is given by oI.

This law can be used to deduce the magnetic field inside and outside a conductor. It
can also be used to deduce the field inside a toroid and a solenoid.
Inside the ring there is no field because no current is enclosed. Outside the whole
device there is equal current flowing into and out of the plane of the page so the total
is zero also. Along any line within the body of the toroid the otal current enclosed is
given by the current I x the number of turns N.

Inside an ideal solenoid the magnetic field is uniform in strength and direction. The
strength is given by:
B   0 nI
Where n is the number of turns per unit length of the solenoid. An ideal solenoid is
one which is much longer than it is wide and where each turn is exactly touching the
previous one.

Magnetic Flux:
This is the term used to describe the total amount of magnetic field passing through a
surface. It is measured in Webers Symbol W and is Tesla per meter squared.
Flux depends on the strength of the field and the area of interest and on the cos of the
angle between the area and the direction of the field thus:
B  BAcos

Magnetism in Materials:
When a magnetic field interacts with materials other than vacuum the strength of the
field will be altered by a certain amount. This amount depends on the magnetic
susceptibility of the material. The material can enhance the field if it is
Paramagnetic or decrease the field if it is diamagnetic. This effect works by
changing the 0 term in the equation. m = 0(1+susceptibility).
For ferrous materials the effect is thousands of times stronger and it is then called
ferromagnetism. Ferromagnetism is not a linear effect however because the
contribution to the field strength will saturate when the strength gets too high.

Faraday discovered that a changing magnetic flux will induce a voltage in a nearby
conductor. The size of this voltage or emf depends on the rate of change of flux and
also on the number of turns of wire involved in the case of a coil.

dB
E  N
dt
This law becomes the basis for many other effects. Lenz’s Law, self induction, mutual
induction and transformer action all spring directly from this.
Lenz’s law says that a changing magnetic flux will induce a current in a loop in such
a sense as to oppose the changing flux.

d B
 E  ds       dt

This also gives rise to another idea which says that a changing magnetic flux will
produce a circular electric field around itself even when there is no material present.
This will become an important part of electromagnetism later on.

The changing flux around a coil in which the current is changing will create a voltage
in that coil which opposes the change in current. This voltage is called a back emf
and is the basic cause of self inductance L.

L                     Vs
L                   1H  1
dI dt                    A

Resistors:
Resistors in series add. Rtotal = R1 +R2 +R3…
Resistors in parallel add as their inverses. Rtotal = 1/(1/ R1 + R2 + R3 + …)
For pairs of resistors the Product/Sum rule can be used as a short cut for calculation.
Resistors are measured in Ohms symbol 
Resistor values are usually tens, hundreds or thousands of Ohms.
The abbreviations used are k for thousands(103) and M for millions (106).
Sometimes simply k or M are used without the  symbol.
Sometimes the k or M are used in place of a decimal point so 1k5 stands for 1500
Ohms.

Power in a resistor is measured in Watts or Joules per second. This can be calculated
in several ways. P = VxI P = I2R P = V2/R where I stands for current and V stands
for voltage.
In a circuit containing a mixture of parallel and series resistors you can calculate the
total resistance by reducing each parallel combination to a single resistor first and
then combining the result with the series components.

Capacitors:

Capacitors in series add as their inverses. See parallel resistors above.
Capacitors in parallel add.
Capacitance is measured in Farads F
Capacitance values are usually very small, much less than a Farad. Typical
abbreviations used are milli mF 10-3, micro F 10-6, nano nF 10-9, pico pF 10-12.

Capacitors act as energy stores and the energy is measured in Joules and is given by:
1
U  CV 2
2
The voltage on a capacitor does not rise to its final value instantly. Current must flow
into it through a resistor before the voltage on the plates rises. The charge or voltage
on a capacitor at any time after the switch has been closed can be calculated by:
t

q( t )  Q(1  e       RC
) where Q is the maximum charge = C x V and R is in Ohms
and C is in Farads.

Maximum current flows in at switch on and is simply V/R. The value RC is called
the time constant of the circuit and is measured in seconds. The capacitor is said to be
fully charged after 5 time constants i.e. 5RC.

Capacitors will allow an alternating current to pass through. Because the voltage is
constantly changing the capacitor is continually charging and discharging so a current
is passing.
The size of the current is determined by a property of the capacitor called the
reactance XC . This is measured in Ohms like a resistor and to all intents a capacitor
acts like a resistor for AC. Capacitive reactance XC = 1/2FC = 1/C. It falls with
rising frequency.
The current that flows in a capacitor in an AC circuit is 90o out of phase with the
voltage. The current leads the voltage.
Inductors:
Inductors in series add. Inductors in parallel add as their inverses. See resistors
above.
The unit of inductance is the Henry symbol L. Milli, micro etc. apply as for
capacitors.
Inductors can be used as energy stores. The energy is stored in the magnetic field
around a current carrying inductor. The energy is measured in Joules and is given by:
1
U  LI 2
2
The current through an inductor does not rise immediately to its maximum value.
Work must be done to change the current against the back emf. The current flowing
in an inductor at any time t after switch on can be calculated by:
t

L
I t   I (m ax)e         R
where I(max) is the maximum current that will flow in the
circuit and is given by V/R. L is in Henries and R is in Ohms and t is in seconds. The
current rises in exactly the same way as the charge in the capacitor. (See above). The
term L/R is called the time constant for the circuit and the current will reach its
maximum value in 5L/R.

Inductors will oppose the flow of AC current and behave like resistors in this respect.
They have a property called inductive reactance XL which is measured in Ohms and
is given by XL = 2FL = L. XL rises with rising frequency. The current in an
inductor in an AC circuit will be out of phase with the applied voltage. The current in
an inductor lags the applied voltage by 90o

RCL Circuits:
When R,C and L are combined we generally use phasor diagrams to calculate the
overall opposition to the flow of AC current. The term used for the combination of
Reactance and resistance is called impedance. Impedance will have a value measured
in Ohms and a phase term measured in degrees or radians.

Impedance is given by: Z  R 2  ( XL  XC) 2
X  XC
And the phase term is given by :   tan 1 ( L     )
R

The result of these ideas leads to resonance of an LC circuit. There are two ways to
imagine resonance in this case. 1) Energy stored in a capacitor is analogous to the
potential energy in a pendulum displaced from its equilibrium position and energy
stored in an inductor is analogous to the kinetic energy of a pendulum as it moves
through its arc. Energy swaps back and forth between the two components and
gradually dissipates due to resistance (friction). In either case if we add small bursts
of energy at the correct rate then the strength of the oscillation will build up. The
1
natural resonance frequency for an LC circuit is given by  0           .
LC
The other way to think of resonance is that when XC and XL are equal they cancel
out and the only component of the impedance is the resistance R. At the frequency
where they cancel current in the circuit will rise to a maximum.

Transformers:

If you pass an AC current through a coil then you will produce a changing magnetic
field in its vicinity. If this field intersects a second coil it will induce an AC voltage
in that coil. The size of the induced voltage depends on the rate of change of flux and
on the number of turns in the second coil. This arrangement is called a transformer
and is used to convert voltages from one level to another.

I1V1  I 2 V2

N2
V2         V1
N1

The interesting point here is that the power (VI) on both sides is the same. So if the
voltage goes down the current goes up and vice versa.
When power is transmitted electrically some is wasted as heat in the resistance of the
connecting wires. Maximising the voltage minimises the current and so minimises
the power loss. This is why very high voltage is used to transmit power around the
country on the national grid. Transformers are used in your local area to reduce this
voltage to a safe level of 230VAC for use in your house.

These Notes should be used as a guide to study of the Physics text by Serway.

Sample Questions are available on the Experimental Physics web site.

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