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INTRODUCTION
Electric charge
The word electricity or electric is derived from Greek word electron for amber
...

There are two type of charge;
POSITIVE CHARGE
NEGATIVE CHARGE
Two (or more) positive charges repel each other
Similarly two (or more) negative charges repel each other
However a positive charge attracts a negative charge
Structure of atom and electric charge
The atom is composed of three particles
 The negatively charged electron
 The positively charged proton
 Uncharged (neutral) neutron
The proton and neutron together form a dense core called the NUCLEUS
...


Nucleus

Proton
Electron
Neutron

The electrons are hels within the atom by the attractive electric forces exerted by
the positively charged nucleus
...

If one or more electron is removed from an atom, the atom becomes positively
charge and becomes a positive ion
...

The process of gain or loss of an electron is called IONIZATION
Two references can be made from discussion so far
...

 The magnitude of charge on an electron or proton is a natural unit of
charge
...
e every observable amount of charge is always an integer multiple of this basic
unit
...
e the charge q on a body is

Where n is a positive or negative integer and ‘e’ is the natural unit of charge
Charging by friction and by induction
The method of charging bodies by rubbing them together is called charging by
friction
...
Charging by induction is the process by
which a charged body gives another body a charge of opposite sign without losing
any of its own charge
...

He observed that for two point charges q1 and q2 separated by a distance r, the
force F, of attraction or repulsion is
(i)

Proportional to

(ii)

Proportional to the product of q1q2 of the charges
...

i
...
988 X 109 Nm2/C2
K is also given as

if the charges are in vacuum or air

Where ℮o = 8
...

Thus

Point charges are one whose dimensions are small compared to the distance r
between them
...


𝑞
𝐹

𝑞

𝐹

𝑜𝑛

Opposite charges

𝐹

𝑜𝑛

𝑜𝑛

r
𝑞

𝐹

Charges of the same sign
𝑞

Example:

(1) (a) Two charges of equal magnitude repel each other with a force of 0
...

Solution
, F 0
...
5 m

჻
√

√
-6

C

(b) what will be the magnitude of the charge if they are placed in a medium
whose permittivity is ten times that of vacuum?
Solution
Let eo be permittivity of the medium
Therefore

𝑜𝑛

Therefore

√

√

Vector form of coulomb’s law
In vector form, the force acting on a charge
by;

due to presence of

𝐹

+

𝑟

𝑟

𝑞

is given
−

𝑞

+

,

𝑞

−
𝑞

𝐹

r12 denotes the vector (position) originating from

and

and

and ending as position

are position vectors of q1 and q2 respectively
...
Determine the force on

Solution
Y
𝑗

𝑞

𝑟
𝑖

𝑞

𝑟

= 𝑗

The force on

,

𝑟

=

X

𝑖

due to

−
=

−

Thus
√

+ −

and
−

−

Therefore

(

−

)

,

(

−

)

−
Principle of superposition
When two or more charges exert forces on another charge, the total force on that
charge is said to be the vector sum of the forces due to each of the other charges
on it
...
Thus the total force F on a charge q
due to n other charges is
∑
=

Example:
Two equal charges =
= 2
...
3 m and -0
...
4 m by and ?
Solution
Two charges q1 and q2 are located along the y-axis and the third q is located
along x-axis i
...
0ℳ C

𝑞 = 4
...
e

𝑞

𝐹𝑦
𝐹

𝐹𝑥

⍺
⍺

𝐹𝑦
𝑞

𝐹𝑥
𝐹

x

And

=

sin

=

sin

−

pointing opposite

Since
|=

=>

sin

sin

Therefore the total (resultant) component along y axis is
−
sin −

sin

The x component of F1 and F2 are

And

=

os

=

os

And
Therefore the total (resultant) components along x-axis is
+
os +

os
+
os

Thus the total force exerted on q due to q1 and q2 is 0
...
Two charges +88micro C and + 77 micro C are placed 1
...
75 m)
between them
...

Ans
...


2
...

Ans
...

ELECTRIC FIELD (INTENSITY) E
- Electric charges create fields around themselves
- Generally electric field is a region or space where force of electrical origin is
experienced
- Experimental test for the existence of an electric field at a point P is to
place a charged body (called a +ve test charge) at the point
...

- Electric field E is a vector and is defined simply as the ratio of the force F,
exerted on a positive test charge q to the magnitude q’ of the test charge
placed at the point i
...


E
E

𝑃
𝑞

𝑞

Test charge
𝑟

𝑟

𝑞
𝑎

𝑞
𝑏

The direction of electric field E (a) due to a +ve charge q (b) due to a –ve charge q
The force F’ exerted on the test charge q’ due to a charge of interest q is

Thus the electric field due to q is

i
...

The number of field lines drawn per unit cross-sectional area is
proportional to the magnitude of E
- Electric field lines (lines of force):
Originate from the +ve charge and terminate on –ve charge
They do not intersect or cross
Where the lines are
(i)
Close together, the field is strong
(ii)
Far apart, the field is weak
(iii) Parallel and equally spaced, the field is uniform
...

Example
1
...
0 m from a point
charge of 4
...

Solution
r= 2
...
0n C= 4
...
A point charge q = - 8n C is located at the origin of a Cartesian coordinate
...
2 m and y= -1
...
0 nC

1
...
6 mj

Sketching the situation, P is the point given by x = 1
...
6 m
...
Determine the electric field vector due to an electric dipole at a field point
P at a distance r along the perpendicular bisector of the line joining the
charges
...
e

+𝑞

+𝑎𝑗
r+

𝑎
r

𝑎

ri
𝑎

𝐸

r’

𝐹
𝐸

−𝑞 −𝑎𝑗

The position vector r of P from the + q
−
And the position vector of r- of P from –q is
− −
+
The electric field vector at P due to +q is

+

where

√

and

+

and

+

−
+

√

−
+
⁄

√

−

+

+
and the field vector at P due to – q is
−
+ −

where
+

,

√

+

+

− −

−
−

−

+
√ +
Therefore
+

−
+
−

√
⁄

+
+

+
Thus the total field at P
+

+

⁄

+

⁄

+

⁄

−

(( −

+(

−
+

) + (− −

⁄

+

)

))

Exercise
1
...
75 X 103 N/C
...
Calculate the electric field E (in magnitude and direction) at a point P
shown below
+𝑞

𝑃

𝑎

+ 𝑞

Ans
...
e

The total electric field E at the distance r is then the vector sum of the
contributions of all the charges
; i
...

Determine the electric field at a point P distance x from the centre along the line
perpendicular to the plane of the ring through its centre
...


Thus the total a field due to the ring
+
+

⁄

+

Motion of a point charge in a uniform electric field
Consider a positive charge , mass , projected into a uniform field created by a
positive and a negative parallel plates separated by a distance with a velocity

+

+

𝑞

-

+

+

+

+

+

𝑈𝑥

-

-

𝑑

-

-

-

-

𝐿

- The force on the charge

is

- The downward acceleration

of the charge

- The horizontal distance

travel after time is

Or

The time

required for the charge to cover the length

- The downward velocity

- The vertical distance

(

of the plates

of the charge at time is

moved after time is

)

The
from the equation shown the path taken by the charge in between the
parallel plate is parabolic
...
1 m with upper plate –ve and lower
plate +ve with a velocity 3 X 106 m/s
...

ii
...

iv
...
: (i) 3
...
3 X 10-8 s (iii) 1
...
17 X 106 m/s
Electric flux ( )
Earlier it was stated that electric field E can be represented by lined of force
The number of lines of force crossing any surface (area) depends on:
 The field strength
 The surface area
 The orientation of surface area relative to the field
Electric flux ( ) is defined as the product of the electric field E at a point and the
area A perpendicular to point of consideration
...

- Gauss law states, for any closed surface

Is the net flux through any (real or imaginary) closed
surface is directly proportional to the net electric
field within the surface
...


Application: To apply Gauss law;
(i)
(ii)

(iii)

Sketch the situation
Imagine a closed surface to be constructed over the body which is
appropriate to the symmetry of the body (this imaginary surface is the
GAUSSIAN SURFACE)
Apply the law

Example
Determine the electric field of a closed hollow sphere of radius R and charge q
(i)
(ii)

Inside the sphere (rOutside the sphere (r>R)

Solution
(i)

For (r
𝑟

𝑟<𝑅

𝑅

𝑞
- Draw the sphere
- Consider an imaginary Gaussian surface of radius r within the sphere
- Apply Gauss law

Q = 0 since the sphere is hollow it contains no charge
...
find the expression for the electric field at a distance R above an infinite line of charge
...

Solution
Draw the line of charge
Infinite line of charge
Gaussian surface

R

L

L

Draw Gaussian surface of the same symmetry as line of charge in this case a cylindrical Gaussian surface
of length L distance R from line of charge
...
Show that the electric
field (using Gaussian law)

(i)

Inside the solid sphere (r
(ii)

Outside the solid sphere (r>R) is

Electric potential:
Electric potential V is the work done W per unit charge q’ in bringing the unit charge q’ from infinity to a
particular point in the field
...
e
-

The above definition is also referred to as absolute potential
...
d) between two points A and B VAB is the work done WAB against
electric energy in carrying (moving) a +ve charge q from A to B
...
d
−

-

Electric potential is a scalar quantity
Note: for a potential rise, V will be +ve and the electric potential energy will increase if q is +ve
...

Practical zero potential is that of the earth
...

Electron volt: the work done in carrying (moving) an electron through a potential rise of 1 volt is
defined as one electron volt (1eV)

Thus;

Examples:
How much word is done in carrying an electron from a positive terminal of a 12 v battery to the
negative terminal?
Solution
From the +ve to the –ve terminal of the battery, the electron passes through a potential rise is
+
Thus

−
−

2
...
what is its speed?
Solution
Word done by the electron, through potential rise is

+

This work appears as K
...
Thus

√

3
...
67 X 10 -27 kg) falls through a potential drop of 3 X 105 V
...
A nucleus has a charge + 50e
(i)
Find the absolute potential V at a radius of 10-12 m from the nucleus
(ii)
If a proton is release from this point, how fast will it be moving when it is at a point r =
from the nucleus
...
71 X 106 m/s
2
...
2 X 10-19 C
...

Ans; 2
...
9 V, 4
...

When no current (charge) flows in a conductor, the surface of the conductor is an equipotential
...

Electric field lines

Equipotential

Surface of the conductor

Equipotential Surfaces

Equipotential Surfaces in
surfaces of
neighbourhood of
opposite charges

Equipotential Surfaces

Electric field lines

Equipotential Surfaces

Equipotential Surfaces in surfaces of
neighbourhood of like charges
Potential gradient:
-

Potential gradient relates electric field intensity to the electric potential
Suppose an electric force, F is exerted on a charge, q makes the charge moves a distance dr,
The work dw done on the charge is
−̅
− ̅

−̅

or

but

 −̅

̅

−

r

is called the electric potential gradient
-

If the electric potential gradient is a function of x, y and z (Cartesian coordinate)

r

−(

+

+

)

The rectangular components of the electric field are given by the partial derivatives

,

,

Example:
1
...
3
X 104 N/C
...
d between the plates and the energy gained in moving an electron from
the positive to the negative plate
...
For a ring of charge with radius ‘a’ and total charge Q, the potential at a point P distance ‘x’ from
the centre of the ring on a line through the centre and perpendicular to the plane of the ring is

√

+

+

)

Determine the electric field at the point x
...
The electric potential inside a uniform spherical distribution of charge

−

(

)

Find the electric field inside the distribution
...
The potential due to a point charge is

,

√

+

+

Determine the x, y and z components of the electric field
...
e
∫

-

Example
1
...
The distance
from the centre of the disk to point P is y
...

Solution
Assume (imagine) a very small circular ring of radius x and thickness dx within the disk
Since is the charge density, the charge dq on the ring of radius x and thickness dx is

𝑃

𝑦

R
𝑑𝑥

The potential on the small element

And the total charge on the ring is thus
∫

∫

+

√

But

∫

+
∫
+
[

*

+
+

]
−

+

Exercise:
A circular ring of radius R has a charge per unit length
...
1 CAPACITORS AND CAPACITANCE
DEFN: - CAPACITOR OR CONDENSER; Any device used for storing
electric charge
...
1
...
1
...

*Note- The two conductors are connected to the two terminals of
a battery when they acquire equal and opposite charges
...


 A typical capacitor is capable of storing large amount of charge
(hence large amount of energy) in a small space
...
2)
Note: - By definition, capacitance is always a positive quantity
...
(1
...

DEFN: - CAPACITANCE: - Is a measure of the capacitor’s ability to store
charge and electric potential energy
...
2), the S
...
3)
F)

: Picofarads,”pF” (pF =
F)
...

: - Electric field confined in the space between the two plates is
uniform
...


1
...

 The capacitance of a giving pair of conductors depends on the
geometry of the conductors
...

The surface charge density,

on either of the plates is given as

= ⁄

(1
...
5) From (1
...
I
units is 8·85 ×
C2/N
...
6) From (1
...


Substituting EQ (1
...
2), we obtain
C=

=

=> C =

(1
...


Where A is the area of the plate and d is the separation between the
plates
...

 Other types of capacitors in everyday use are:
(a) Cylindrical capacitor
...

1·3 COMBINATIONS OF CAPACITORS:
 Two or more capacitors are often combined for use in electrical
circuits
...
Two most common methods of
capacitor combinations are [i] capacitors in SERIES and [ii]
capacitors in PARALLEL
...
3
...
When the capacitors are charged in series, each capacitor
acquires the same charge
...
The total potential difference Ѵ, across any number of
capacitor connected in series is equal to the sum of the
potential drop across the individual capacitors (The applied
potential difference is showed among all capacitors)
Thus, for the combination of three capacitors shown;

Ѵ=

+

+

(1
...
2),

= ,

=

,

=

(1
...
8) Becomes
Ѵ=

+

+

=(

+

+

If

=

+

+

)Q

(1
...
10a)

Where Ceq = Equivalent capacitance for the combination, then Eq
...
10) becomes
Ѵ=

Q
...
10b)

If EQ (1·10a) B extended to n numbers of capacitors connected in
series, then
=

+

+

+…
...
11)

DEFN: - The equivalent capacitance of a series combination of
capacitors is always less than any individual capacitance in the
combination
...
3
...


=

The figure above shows that:

1
...

Each of the capacitors have different amount of charges, such that
2
...
i
...
=>
Q=
+
+
(1
...
(1
...
13)
= C1V,
Thus Eq
...
12) becomes
Q = C1V + C2V + C3V
=(

+

+

)Ѵ

If Ceq =

+

+

(1
...
15)

Where Ceq equivalent capacitance of the combination then Eq
...
1)
becomes
Q = CeqV

(1
...
15) is extended n number of capacitors connected in parallel
then
Ceq =

+

+

+…
...
17)

DEFN: - The equivalent capacitance of a parallel combination of
capacitors is greater than any of the individual capacitances
...
4 ENERGY STORED IN A CHARGED CAPACITOR
...

 Consider a parallel plate capacitor that is initially uncharged (i
...
d across the plates = 0)
...
d that is built up between the plates

when connected to a battery
...

NOTE: this energy is independent of the charge transfer process
Let q be the charge on the capacitor at a given instant of time when the
p
...
18)
...
19) from (1
...
20)

This work done (Eq 1
...
e
...
21) from eq
...
18)
NOTE: - Eq
...
21) applies to any capacitor irrespective of its
geometry
...
(1
...
This is why capacitors are always
labeled with a maximum operating voltage
...
5 DIELECTRICS
...

DEFN: -A DIELECTRIC- Any no conducting material, (an insulator)
such as rubber, glass, wood, paper or waxed paper
...

 Consider a parallel –plate capacitor whose plates are separated by
air or vacuum, having charge Q˳ and capacitance C˳
...
22) from (1
...
23) K have value
greater than 1; WHY??
 Where K = DIELECTRIC CONSTANT, a property of the dielectric
material
...
24)

(1
...
22)

 Thus from Eq
...
24)
(1
...

 K Is a dimensionless factor, which is a property of the dielectric
material (insulator) and varies from one dielectric material to
another
...
(1
...

 For a parallel- plate capacitor, with air (vacuum) in between the
plates,
(1
...
(1
...
27)
Where

is the permittivity of the dieletric
...
28) is the PERMITTIVITY

of the dielectric
...

DEFN: - The maximum electric field that can exist in a dielectric without
electric field in the dielectric exceeds the dielectric strength, the
insulating proportion, and break down and the dielectric begins to
conduct
...
Increases the capacitance of a capacitor
2
...

3
...


 Various materials have them breakdown voltage or dielectric
strength as shown in the table:
MATERIAL
DIELECTRIC
DIELECTRIC
CONSTANT( )
STRENTH( ⁄ )
Air (dry)
Bakelite
Fused Quartz
Neoprene Rubber
Nylon

1
...
9
3
...
7
3
...
7
Poly styrene
2
...
4
Porcelain
6
Pyrex glass
5
...
5
Strontium titan ate
233
Teflon
2
...
000 0
Water
80
2·0 CURRENT AND RESISTANCE

16×
24×
40×
12×
14×
15×
8×
60×
---------

2·1 ELECTRIC CURRENT:
 Whenever a material is connected to a voltage supply (source),
an electric field is established inside the material which result in a
net flow of charges inside the material as shown:

FIG
...
1 Charges in motion through a giving area A
...
There the giving region
...


 Let the amount of charge that passes through the giving area in a
time interval t be DQ, then the average current I have is given to
be
Iave =

(2
...
2)

NOTE: - Although I have a sense of direction, it is not a vector
quantity but a scalar quantity
...
1 can either be positive
(+), negative (-) or both conventionally, the direction of current
flow is the same as the direction of flow of positive charge
...

Consider the current in a conductor of cross- sectional area A as
shown;

 The volume of a section of the conductor of length × is given to
Volume = A
(2
...


 If n is the number of mobile charge carries per unit volume (the
charge carriers density), then the number of carriers in the
volume is
(2
...
5)
Where q is the charge on each carrier
...
6)
Thus Eq
...
5) can be written as
Q = (nA
t) q
(2
...
8)
is an average speed called the DRIFT SPEED
...
(2
...

For the conductor of cross-sectional area A, carrying a current I,
DEFN: - CURRENT DENSITY (J): - The current per unit crosssectional area
...
e
(2
...
8)
S
...

 CURRENT density is a vector quantity- i
...
2 RESISTIVITY, RESISTANCE AND OHM’S LAW
...

DEFN: - A current density J, and an electric field E are always
established in a conductor whenever a potential difference Ѵ is
maintained across the conductor
...
Thus:
DEFN: - The current density is directly proportional to the electric
field, i
...
10)
where the constant of proportionality,
the conductor
...
(2
...

DEFN:OHM’S LAW: - For many materials, the ratio of the current density
to the electric field is a constant, , that is independents of the
electric field producing the current
...
2
...

DEFN:OHM’S LAW:- For many materials the ratio of the current density to
the electric field is a constant, , that is independent of the electric
field producing the current
...
10) are called OHMIC MATERIAL
Now: -

Consider a segment of a straight conductor (wire) of uniform crossseminal area A and length L as shown below:
L

I

A

𝑉𝑏

𝑉𝑎

FIG : 2
...

The potential difference that is maintained across the wire, thereby
create the electric field and the current in the wire is,
−

(2
...
12)
Thus Eq
...
10) can be expressed as
(2
...
(2
...
(2
...
14)
is called the RESISTANCE, R of the conductor
...
e
(2
...
I unit Resistance is
(

)

 From Eq
...
15),
NOW,
 The inverse of conductivity
Thus, from Eq
...
10)

is called the RESISTIVITY,
...
(2
...
16)
S
...
m) = ohm- meters
...
(2
...
(2
...
17)
Eq
...
17) shows that the resistance of any material samples depends
on;
(i) the resistivity and (ii) the geometry of the material
...
(2
...

Meaning: - If the length of the wire is doubled, its resistance is also
doubled
...
3 RESISTANCE AND TEMPERATURE
Every Ohmic material has a characteristic resistivity that depends on
the property of the material and on TEMPERATURE
...
range) according to the expression:
+
Where

−

(2
...


P= Resistivity at some reference temperature T˳ (˚C)
= Temperature coefficient of RESISTIVITY
...
(2
...
18b)
–

Where

=

-

–
,

−
−

˳ = change in Resistivity

T = T-T˳ = Temperature interval
...
I unit of
Also,
From Eq
...
17), Resistance is proportional to Resistivity
...

(2
...
19)

2
...

 Wherever a battery (A source of Emf) is used to establish an
electric current in a conductor, the chemical energy stored in the
battery is continuously transform into kinetic energy of the charge
carriers
...

 Consider a simple circuit consisting of a battery whose terminals
are connected to a Resistor as shown;
𝐼

𝑐

𝑏
𝑉

𝑅

𝑎
𝑑

𝐼

 The potential difference Ѵ across the Resistor R carrying a current
I is
−
In an interval of time t, a quantity of charge Q ( Q =
)
Passing through the Resistor experiences a charge in potential
energy, given by
(2
...

(2
...
(2
...

This energy loss by the charge will appear as the internal energy of the
load
...
I unit of power

= watts

2
...
4 from Fig 2:4, the potential energy loss
by the moving charges appears as THERMAL ENERGY in the
Resistor
...

DEFN: - JOOLE HEATING: - The process by which the potential
energy is being dissipated in the Resistor
...
(2
...
(2
...
22)
 Eq
...
22) is known as JOOLE’S LAW; and this power loss is called
the
LOSS OR JOULE’S HEATING LOSS
...
6 ELECTROMOTIVE FORCE (Emf)
...


Q? => How is steady current maintained in a circuit?
 To maintain a steady direct current, the potential difference must
be maintained at a constant rate within the circuit
...

DEFN: - that influence that makes current to flow from a lower
potential to a higher potential is called ELECTROMOTIVE FORCE
(emf) (see Eq
...
23)
 S·I unit of emf is the volt (Ѵ); where =
 Emf is represented mathematically as
DEFN: - A source of emf is equal to the work done in carrying 1 coulomb
of charge through the source
...
23a)

Sources of emf include:
1
...

2
...

3
...

 Now, consider the circuit given below
...




𝐼
𝐸
𝑎
𝐸

𝐹𝑛

𝑉𝑎𝑏

𝐸

𝐹𝑒

𝐼

𝑏
𝐸

𝐼

The potential difference between the ends of the wire is giving to be
(2
...
(2
...

Meaning: - when a positive charge q flows around the circuit, the
POTENTIAL RISE, , as it passes through the ideal source is numerically
equal to the potential drop
as it passes through the
remaining part of the circuit
...
23b) because of the
INTERNAL RESISTANCE, r of the source itself
...

 Thus the final potential difference between the ends of the wire is
now

−

(2
...
(2
...

 Eq
...
24) shows that the terminal voltage is always less than the
emf, by the value which represents the potential drop cross the
internal resistance of the source
...

 DEFN: - For a real source of emf, the terminal voltage equals the
emf only if no current is flowing through the source
...

Thus,
From Eq
...
23b) and (2
...
25)

 Current in the circuit = the source emf divided by the total
resistance in the circuit
...
on u tor with n gligibl r sist n
𝑅

𝜀

= Resistor
...

𝑉

𝑂𝑅

𝜀

voltm t r m sur s th pot nti l

i

r n

b tw n th t rmin l
𝐴

mm t r m sur s th

urr nt through th

ir uit

NOTE: - For the source, the long vertical line represents the positive
terminal which is usually the terminal with higher potential
...
0: DIRECT – CURRENT CIRCUITS
...
Example includes
the flash lights and automobile wiring systems
...

NOTE: - Most circuits to be analyzed here are assumed to be in
steady state, i
...

3
...
1
...

Let three Resistors , , be connected end- to- end as
shown:
𝑉
𝑅

𝑉

𝑉

𝑅

𝑅
𝜀

𝑎

B

A

=

I

𝜀

𝑏

 The basic characteristics of a series circuit is that the same
Current I runs through each of the resistors
...
1)

Applying Ohm’s law (Eq
...
15) each Resistor, we have
,

,

(3
...
(3
...
1) gives
+

+

+

+

+

+

Thus
(3
...
(3
...
16)
...
(3
...
4)

DEFN: For resistors, connected in series, the equivalent resistance equals the
sum of their individual resistances
...


If source of emf are connected in series, as shown below, then the
single source of emf which is equivalent to several sources of emf
connected in series is given to be

𝜀

𝜀

𝜀
𝐹𝐼𝐺

+

+

+

(3
...
1
...

Let three resistors

,

be connected as shown:

,

𝐼
𝐼

𝐼

𝑅

𝐼

𝑅𝑒𝑞

𝐼

𝑅

V

=

𝑅

𝐼

V

𝑎

𝑏
FIG
...
3)

 The basic characteristic future describing the parallel connection
of resistors is that, the current through each resistor need not be
the same, the potential difference across each of the resistors is
the same
...
6)
But from Ohm’s law,
,

,

(3
...
(3
...
8)

Following the definition of equivalent Resistance Req, (Fig 3
...
8),
+

+

Generalizing to any number of resistors connected in parallel,
then;
+

+

+

(3
...

 The equivalent resistance is always less than any individual
Resistances
...
2 MULTILOOP CIRCUITS (KIRCHHOFF’S RULES)
...
A JUNCTION: - This is a point in a circuit where three or more
conductor meet (Also called a BRANCH)
...
A LOOP: - Any closed conducting path
...
The junction are:- Points A and B
...
The loops are:
...

Q? => Why are points E, F, C, D not classified as junctions!
KIRCHHOFF’S RULES: JUNCTION RULES: - The algebraic sum of the current at any junction OR
branch point is equal to zero
...
10)
(∑ )

*The sum of electric currents entering any junction must equal the sum
of electric current leaving that junction
...
LOOP RULE: - The algebraic sum of the potential differences
encountered in going round a closed loop is equal is ZERO
...

That is:
∑ + ∑

(3
...

 The loop rule is a direct consequence of the law of conservation of
energy
...
If a resistor is traversed in the same direction as the current,
the potential difference across the resistor is negative (i
...

2
...
e, + IR)
...
If a source of emf (assumed to have Zero internal Resistance) is
traversed in the direction of the emf (from negative to
positive), the potential difference is positive (i,e + )
...
If a source of emf (assumed to have Zero internal resistance) is
traversed in the direction opposite the emf (from opposite to
negative), the potential difference is negative, (i
...

BRAIN CHECK

Identify the number of (a) loops and (junctions in the circuit above
...
3 R-C CIRCUITS
...
3
...

 A circuit containing a series combination of resistor and a
capacitor is called an RC circuit
...
5a) = The capacitor is uncharged since no current flows while
the switch S is opened
...
5b) = At t= 0, charge begins to flow while the switch is
closed, thereby setting up a current in the circuit and the
capacitor begins to charge
...

 Once the capacitor is fully charged, (maximum charge is reached),
the current in the circuit becomes Zero because the potential
difference across the capacitor balances that supplied by the
battery
...
5a):- pd across the capacitor = Ѵbc = 0 at t = 0
...
Therefore, the current, through the resistor
is giving by ohm’s law as:
(3
...
Thus the entire emf of the battery
appears across the capacitor – i
...

 If the charge on the capacitor is q and the covert I in the circuit at
a time t after the switch is closed, then,
(3
...
14)
−
−

−

(

)
(3
...
16)

Integrating Eq
...
16)
∫

∫

−
−
ln (
−

)

−

⁄
( −
)
Where the total charge Q = C on the capacitor
...
(3
...
17)
, then

⁄

Therefore,

⁄

(3
...
(3
...
18) shows that both the charge and the current
are exponential function of time
...
3
...
(3
...
18)
is called the time constant
...

Meaning: At a time t = ,
At a time t = 2 ,

 Likewise, the time constant , it the time at which the charge or the
capacitor increases from Zero to (1 ) of it’s final value
...
e
At a time t = ,
−
Thus, the quantity RC is a measure of how quickly the capacitor
charges
...
19)

3
...
3 DISCHARGING A CAPACITOR:
Consider the circuit below:

−𝑄

−𝑞

𝑅

𝐶

𝑅

𝐶
+𝑞

+𝑄
𝑆

𝑆

𝑏 t>0

𝑎 t<0

𝐹𝐼𝐺

Discharging a capacitor
...
6a) => A charged capacitor, carrying a maximum charge Q
connected to a resistor and a switch which is opened at t < 0
...
6b) => The switch is closed, a current that decreases in magnitude
with time is set up in the direction shown and the charge on the
Capacitor decreases exponentially with time
...
Since there is no voltage supply in the
circuit, Eq
...
14) becomes

− −

(3
...
21)
(where q= Q at t = o)
Differentiating Eq
...
21) w
...
t time gives the INSTATENOUS CURRENT
as a function of time
...
e
(
−

)
(3
...
23)
The negative sign in Eq
...
22) indicates that the direction of current as
the capacitor discharge is opposite the current direction as the
capacitor was being charged
...




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Title: Physics questions and answers
Description: A comprehensive and significant questions and answers in electric charge in physics.

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