1
CHAPTER
Electric Charges and Fields
7. Charge cannot exist without mass, while mass can exist
ELECTROSTATICS without charge. e.g. neutron, neutrino, antineutrino all are
The branch of physics which deals with properties of charges at neutral particles having mass.
rest is called electrostatics. SI Unit: coulomb (C)
[1 coulomb = 1 ampere × 1 second]
ELECTRIC CHARGE C.G.S. unit: stat coulomb or franklin
1 coulomb = 3 × 109 stat coulomb
Charge is scalar physical quantity associated with matter due to
which it produces and experiences electrical and magnetic effects. 1
1 coulomb = 3 × 109 esu of charge = emu of charge
The excess or deficiency of electrons in a body gives it a net 10
[esu = electrostatic unit]
charge. A negatively charged body has excess of electrons while a [emu = electromagnetic unit]
positively charged body has deficiency of electrons. 1 esu of charge = 1 franklin
Properties of Electric Charge Practical Units: 1amp × hr = 3600 coulomb and
1. Charges interact with each other i.e., they exert force on each 1 faraday = 96500 coulomb (charge on 1 mole of electrons)
other. Like point charges repel each other while unlike point
charges attract each other. METHODS OF CHARGING
2. Charge is of two kinds: Positive and negative.
3. Total charge of an isolated system is conserved Charging by Conduction
(Conservation of charge). Charging by conduction refers to the technique of charging an
4. Charge is quantised: Charge is an integral multiple of uncharged body by bringing it in contact with some other charged
electronic charge i.e., Q = Ne, where e = 1.6 × 10–19 C and N material.
is an integer. After conduction both the bodies acquire same type of charge if
5. Charge can be transferred: Charge can be transferred from one of the bodies involved is uncharged. However, if both bodies
one body to another. This occurs due to transfer of electrons are charged then final type of charge on both bodies will be of that
from one body to another. One of the common example of whose magnitude is larger.
transfer of charge is charging by friction. There are three types of materials in nature:
e – (i) Conductor: Conductors are the material in which the outer
+ – most electrons are very loosely bound to the nucleus, so they
Rubbing + Transfer –
A B
+ – are free to move (flow). So in a conductor, there are large
+ – number of free electrons.
Neutral Neutral Example: Metals like Cu, Ag, Fe, Al.
(ii) Insulator or Dielectric or Non-conductor: Non-conductors
Frictional Electricity: When two bodies are rubbed with
each other, they are found to attract each other. This is so are the materials in which outer most electrons are very
because, on rubbing, transfer of electrons takes place from tightly bound to the nucleus, so that they cannot move
one body to another. One of them acquires a positive charge (flow). Hence in a non-conductor there are no free electrons.
and other acquires a negative charge. Example: plastic, rubber, wood etc.
6. Charge is invariant: Charge of a particle is independent of (iii) Semiconductor: Semiconductors are the materials which
its speed. have free electrons but very few in number.
, Now lets see how the charging is done by conduction. In this Step 2: Bring a charged rod near it. Due to the charged rod,
method, we take a charged conductor ‘A’ and an uncharged charges will induce on the conductor.
conductor ‘B’. When both are connected, some charge will –– +
–– +++
flow from the charged body to the uncharged body. If both ++ ++ + +
++ ++ + + –––
––– +
+
the conductors are identical and kept at large distance and –––+++
connected to each other, then charge will be divided equally
Step 3: Connect another neutral conductor with it. Due to attraction
in both the conductors otherwise they will flow till their
of the rod, some free electrons will move from the right conductor
electric potential becomes same. Its detailed study will be
to the left conductor and due to deficiency of electrons positive
done later.
charges will appear on right conductor. On the left conductor, there
+
+ + ++
++ ++++ will be excess of electrons due to transfer from right conductor.
++ + + ++ +
+ +
A B
Charged Uncharged ++
++
body body ++ ++ ++
A B + Step 4: Now disconnect the connecting wire and remove the
rod. One body becomes negatively charged while second body
Charging by Friction becomes positively charged.
When two bodies are rubbed together, electrons are transferred
from one body to the other. This makes one body positively
charged while the other negatively charged, e.g., when a glass
rod is rubbed with silk the rod becomes positively charged while
the silk becomes negatively charged. Clouds are also charged by Method-II
friction. Charging by friction is in accordance with conservation of Step 1: Take an isolated neutral conductor.
charge. The positive and negative charges appear simultaneously
in equal amounts due to transfer of electrons from one body to
the other.
Charging by friction is based on difference in work function (f) of Step 2: Bring a charged rod near it. Due to the charged rod,
the bodies that are being rubbed. We cannot charge two bodies by charges will induce on the conductor.
rubbing that are made up of same material. –– +
++ ++ + + ––– +++
– +
Charging by Induction ––
++ ++ + +
–– +
–––+++
If a charged body is brought near a neutral body, the charged body
will attract opposite charge and repel similar charges present in Step 3: Connect the conductor to the earth (this process is
the neutral body. One side of the neutral body becomes positively called grounding or earthing). Due to attraction of the rod, some
charged while the other side becomes negative. free electrons will move from earth to the conductor, so in the
conductor there will be excess of electrons due to transfer from the
Important Points earth, so net charge on conductor will be negative.
Inducing body neither gains nor loses charge. –– +
––– +++
The nature of induced charge is always opposite to that of
+ electron
+
+ ++ +
+ –– + transfer
+ –
–– +
inducing charge. + ++
+ –––+++
Induced charge can be lesser or equal to inducing
charge (but never greater) and its maximum value is Step 4: Now disconnect the connecting wire. Conductor becomes
1 negatively charged.
q′ = −q 1 − [only when electric field is uniform]
K
q = the inducing charge
K = the dielectric constant of the material of the uncharged body
For metals, K = ∞, so q′ = –q
A body can be charged by induction in the following two ways:
GOLD LEAF ELECTROSCOPE
Method-I Gold leaf electroscope is a device which is used to detect the
charge on a body.
Step 1: Take an isolated neutral conductor.
A gold-leaf electroscope is defined as a type of electroscope that
consists of two gold leaves and is used for detecting the electrical
charge of the body and for the classification of its polarity.
P
2 W JEE (XII) Module-1 PHYSICS
,Construction of Gold Leaf Electroscope 1 q1 q2
Then F ∝ q1 q2 & F ∝ ∴F∝
A gold leaf electroscope consists of a brass rod with a brass disk r2 r2
at the top, and at the bottom, there are two thin gold leaves in the 1 q1q2
form of foils. In order to keep the rod in place, the rod travels F =
through the insulator. The charges move from the disc to the leaves 4π ∈0 r 2
through the rod. At the lower portion of the jar, a thin aluminium q1 q2
foil is connected. The aluminium foil is grounded with the help r
of a copper wire so that the leaves are protected from external
electrical disruptions. 1
= 9 × 109 Nm²/C²
Brass disc 4πε0
[in electrostatic unit (esu) constant of proportionality = 1]
Insulator ε0 = 8.85 × 10–12 C²/Nm² = permittivity of free space or
plug vacuum
Glass bottle
Brass rod
Gold leaf EFFECT OF MEDIUM
Metal foil
The dielectric constant of a medium is the ratio of the electrostatic
force between two charges separated by a given distance in
Earth vacuum to electrostatic force between same two charges separated
by same distance in that medium.
Applications of Gold Leaf Electroscope
The following are the applications of gold leaf electroscope: 1 q1q2 1 q1q2
Fvacuum = and Fmedium =
Detection of charge.
4πε0 r 2 4πε0ε r r 2
Identification of the nature of the charge. Fmedium 1 1
⇒ = =
Identification of the body as a conductor or an insulator Fvacuum ε r K
Detection of Charge εr or K = dielectric constant or relative permittivity or specific
For the detection of charge, the object that needs to be tested is inductive capacity of medium.
touched with the metal cap. If the leaves diverge, the body is Permittivity: Permittivity is a measure of the ability of the medium
said to be charged, and if there is no change in the leaves of the surrounding electric charges to allow electric lines of force to pass
electroscope, then the body is uncharged. through it. It determines the forces between the charges.
Identification of the Nature of the Charge Relative Permittivity: The relative permittivity or the dielectric
constant (εr or K) of a medium is defined as the ratio of the
To identify the nature of the charge, let’s consider an example. A
positively charged body is brought near the metal cap. Then an permittivity ε of the medium to the permittivity ε0 of free space
unknown body is brought near the metal cap. If the leaves diverge ε
i.e. εr or K =
further, we can conclude that the unknown body has a positive ε0
charge. If the leaves come closer to each other, then the charge of Dimensions of permittivity
the unknown body is negative. [M–1 L–3 T4 A2]
Identification of Body as a Conductor or an Insulator The dielectric constants of different mediums
To identify if a body is a conductor or an insulator, two gold leaf Medium Vacuum Air Water Mica
electroscopes are taken. One gold leaf electroscope is charged so
that the leaves will diverge. Then the other gold leaf electroscope ε 1 1.00059 80 6
is connected to the first one through the body. If the leaves of the Teflon Glass PVC Metal
other electroscope diverge, then the body is a conductor, and if 2 5-10 4.5 ∞
there is no change in the leaves, the body is an insulator.
COULOMB’S LAW COULOMB’S LAW IN VECTOR FORM
Coulomb's law states that the force of attraction or repulsion A charge q1 is placed at A whose position vector is r1 .
between two stationary point charges is directly proportional to
Another charge q2 is placed at B whose position vector is r2 , such
the product of charges and inversely proportional to the square of
distance between them. This force acts along the line joining the that AB =| r2 − r1 | = r .
point charges.
The magnitude of force is given by
If q1 and q2 are charges
k q1 q2
r = the distance between them F=
F = the force acting between them 4πε0 r 2
Electric Charges and Fields 3
, Force on q2 due to q1, in vector form, is given by
1 q1q2
F21 = AB , where AB is a unit vector along line joining
4πε0 r 2
A and B, pointing from A to B.
Example 1: Two equal point charges (10–3 C) are placed 1 cm
apart in medium of dielectric constant K = 5.
1 q1q2 ( r2 − r1 ) 1 q1q2
⇒ F21
= =
2 3 ⋅ ( r2 − r1 ) (a) Find the interaction force between the point charges.
4πε0 r r2 − r1 4πε0 r
(b) Net force on any of the charge.
y q1 q2 Sol. (a) Interaction force between point charges
AB
( )
2
10−3
F F21 1 q1q2
12 A B F = 9 × 10
= 9
9 × 107 N
=
4π ∈0 r 2
( )
2
r1 10−2
r2
(b) Net force on any one charge
x
( )
−3 2
Similarly force on q1 due to q2 is given by, 1 q1q2 9 × 109 10
F=' = = 18 × 106 N
1 q1q2 ( r1 − r2 ) 1 q1q2
4π ∈0 K r 2 5 10 −2 2
( )
= F12 =
2 3 ⋅ ( r1 − r2 )
4πε0 r r1 − r2 4πε0 r Example 2: Two small balls each of mass m and charge q
on each of them are suspended through two light insulating
Note that F12 = − F21
strings of length l from a point. Find the expression for
Both forces are action reaction pair. angle q made by any of the string with vertical when under
static equilibrium.
PRINCIPLE OF SUPERPOSITION O
Consider a system of n stationary charges q1, q2, q3, ... qn in
vacuum. What is the force on q1 due to q2, q3 ... qn? Coulomb’s law
q q
is not enough to answer this question. Force on any charge due to a
m m
number of other charges is the vector sum of all the forces on that
Sol. Angle of any string with vertical is q as shown.
charge due to the other charges taken one at a time. The individual
forces are unaffected due to the presence of other charges. This is o
termed as the ‘principle of superposition’. (Note that the principle
of superposition is independent of Coulomb’s law and all of T
T
electrostatics is basically a consequence of Coulomb’s law and Fe Fe
superposition principle together). mg mg
Now, the force on q1 can be found out by calculating separately For equilibrium in horizontal direction
the forces F12 , F13 ……..F1n exerted by q2, q3 ......qn, respectively 1 q2
on q1 and then adding these forces vectorially. =
Fe = T sin θ ...(i)
4π ∈0 ( 2l sin θ )2
F1n q2 For equilibrium in vertical direction
q1 T cos θ = mg ...(ii)
F13 q3 Dividing (i) by (ii)
F12 Fe
tan θ =
mg
qn
Example 3: For the following system to be in equilibrium,
So,=
F1 F 12 + F13 +……+ F 1n what should be the value of charge Q?
q1 q2 rˆ12 q3 rˆ13 qn rˆ1n Q
=∴ F1 2 + 2 +……. + 2 q a a q
4πε0 r12 r13 r1n
Kq 2 K |Q|q q
q n
qi rˆ1i r1 − ri Sol. = ⇒ Q=–
= 1
4πε0
∑ r12i
r1 − ri and rˆ1i =
where r1i =
| r1 − ri |
(2a ) 2
a2 4
i =2
P
4 W JEE (XII) Module-1 PHYSICS
CHAPTER
Electric Charges and Fields
7. Charge cannot exist without mass, while mass can exist
ELECTROSTATICS without charge. e.g. neutron, neutrino, antineutrino all are
The branch of physics which deals with properties of charges at neutral particles having mass.
rest is called electrostatics. SI Unit: coulomb (C)
[1 coulomb = 1 ampere × 1 second]
ELECTRIC CHARGE C.G.S. unit: stat coulomb or franklin
1 coulomb = 3 × 109 stat coulomb
Charge is scalar physical quantity associated with matter due to
which it produces and experiences electrical and magnetic effects. 1
1 coulomb = 3 × 109 esu of charge = emu of charge
The excess or deficiency of electrons in a body gives it a net 10
[esu = electrostatic unit]
charge. A negatively charged body has excess of electrons while a [emu = electromagnetic unit]
positively charged body has deficiency of electrons. 1 esu of charge = 1 franklin
Properties of Electric Charge Practical Units: 1amp × hr = 3600 coulomb and
1. Charges interact with each other i.e., they exert force on each 1 faraday = 96500 coulomb (charge on 1 mole of electrons)
other. Like point charges repel each other while unlike point
charges attract each other. METHODS OF CHARGING
2. Charge is of two kinds: Positive and negative.
3. Total charge of an isolated system is conserved Charging by Conduction
(Conservation of charge). Charging by conduction refers to the technique of charging an
4. Charge is quantised: Charge is an integral multiple of uncharged body by bringing it in contact with some other charged
electronic charge i.e., Q = Ne, where e = 1.6 × 10–19 C and N material.
is an integer. After conduction both the bodies acquire same type of charge if
5. Charge can be transferred: Charge can be transferred from one of the bodies involved is uncharged. However, if both bodies
one body to another. This occurs due to transfer of electrons are charged then final type of charge on both bodies will be of that
from one body to another. One of the common example of whose magnitude is larger.
transfer of charge is charging by friction. There are three types of materials in nature:
e – (i) Conductor: Conductors are the material in which the outer
+ – most electrons are very loosely bound to the nucleus, so they
Rubbing + Transfer –
A B
+ – are free to move (flow). So in a conductor, there are large
+ – number of free electrons.
Neutral Neutral Example: Metals like Cu, Ag, Fe, Al.
(ii) Insulator or Dielectric or Non-conductor: Non-conductors
Frictional Electricity: When two bodies are rubbed with
each other, they are found to attract each other. This is so are the materials in which outer most electrons are very
because, on rubbing, transfer of electrons takes place from tightly bound to the nucleus, so that they cannot move
one body to another. One of them acquires a positive charge (flow). Hence in a non-conductor there are no free electrons.
and other acquires a negative charge. Example: plastic, rubber, wood etc.
6. Charge is invariant: Charge of a particle is independent of (iii) Semiconductor: Semiconductors are the materials which
its speed. have free electrons but very few in number.
, Now lets see how the charging is done by conduction. In this Step 2: Bring a charged rod near it. Due to the charged rod,
method, we take a charged conductor ‘A’ and an uncharged charges will induce on the conductor.
conductor ‘B’. When both are connected, some charge will –– +
–– +++
flow from the charged body to the uncharged body. If both ++ ++ + +
++ ++ + + –––
––– +
+
the conductors are identical and kept at large distance and –––+++
connected to each other, then charge will be divided equally
Step 3: Connect another neutral conductor with it. Due to attraction
in both the conductors otherwise they will flow till their
of the rod, some free electrons will move from the right conductor
electric potential becomes same. Its detailed study will be
to the left conductor and due to deficiency of electrons positive
done later.
charges will appear on right conductor. On the left conductor, there
+
+ + ++
++ ++++ will be excess of electrons due to transfer from right conductor.
++ + + ++ +
+ +
A B
Charged Uncharged ++
++
body body ++ ++ ++
A B + Step 4: Now disconnect the connecting wire and remove the
rod. One body becomes negatively charged while second body
Charging by Friction becomes positively charged.
When two bodies are rubbed together, electrons are transferred
from one body to the other. This makes one body positively
charged while the other negatively charged, e.g., when a glass
rod is rubbed with silk the rod becomes positively charged while
the silk becomes negatively charged. Clouds are also charged by Method-II
friction. Charging by friction is in accordance with conservation of Step 1: Take an isolated neutral conductor.
charge. The positive and negative charges appear simultaneously
in equal amounts due to transfer of electrons from one body to
the other.
Charging by friction is based on difference in work function (f) of Step 2: Bring a charged rod near it. Due to the charged rod,
the bodies that are being rubbed. We cannot charge two bodies by charges will induce on the conductor.
rubbing that are made up of same material. –– +
++ ++ + + ––– +++
– +
Charging by Induction ––
++ ++ + +
–– +
–––+++
If a charged body is brought near a neutral body, the charged body
will attract opposite charge and repel similar charges present in Step 3: Connect the conductor to the earth (this process is
the neutral body. One side of the neutral body becomes positively called grounding or earthing). Due to attraction of the rod, some
charged while the other side becomes negative. free electrons will move from earth to the conductor, so in the
conductor there will be excess of electrons due to transfer from the
Important Points earth, so net charge on conductor will be negative.
Inducing body neither gains nor loses charge. –– +
––– +++
The nature of induced charge is always opposite to that of
+ electron
+
+ ++ +
+ –– + transfer
+ –
–– +
inducing charge. + ++
+ –––+++
Induced charge can be lesser or equal to inducing
charge (but never greater) and its maximum value is Step 4: Now disconnect the connecting wire. Conductor becomes
1 negatively charged.
q′ = −q 1 − [only when electric field is uniform]
K
q = the inducing charge
K = the dielectric constant of the material of the uncharged body
For metals, K = ∞, so q′ = –q
A body can be charged by induction in the following two ways:
GOLD LEAF ELECTROSCOPE
Method-I Gold leaf electroscope is a device which is used to detect the
charge on a body.
Step 1: Take an isolated neutral conductor.
A gold-leaf electroscope is defined as a type of electroscope that
consists of two gold leaves and is used for detecting the electrical
charge of the body and for the classification of its polarity.
P
2 W JEE (XII) Module-1 PHYSICS
,Construction of Gold Leaf Electroscope 1 q1 q2
Then F ∝ q1 q2 & F ∝ ∴F∝
A gold leaf electroscope consists of a brass rod with a brass disk r2 r2
at the top, and at the bottom, there are two thin gold leaves in the 1 q1q2
form of foils. In order to keep the rod in place, the rod travels F =
through the insulator. The charges move from the disc to the leaves 4π ∈0 r 2
through the rod. At the lower portion of the jar, a thin aluminium q1 q2
foil is connected. The aluminium foil is grounded with the help r
of a copper wire so that the leaves are protected from external
electrical disruptions. 1
= 9 × 109 Nm²/C²
Brass disc 4πε0
[in electrostatic unit (esu) constant of proportionality = 1]
Insulator ε0 = 8.85 × 10–12 C²/Nm² = permittivity of free space or
plug vacuum
Glass bottle
Brass rod
Gold leaf EFFECT OF MEDIUM
Metal foil
The dielectric constant of a medium is the ratio of the electrostatic
force between two charges separated by a given distance in
Earth vacuum to electrostatic force between same two charges separated
by same distance in that medium.
Applications of Gold Leaf Electroscope
The following are the applications of gold leaf electroscope: 1 q1q2 1 q1q2
Fvacuum = and Fmedium =
Detection of charge.
4πε0 r 2 4πε0ε r r 2
Identification of the nature of the charge. Fmedium 1 1
⇒ = =
Identification of the body as a conductor or an insulator Fvacuum ε r K
Detection of Charge εr or K = dielectric constant or relative permittivity or specific
For the detection of charge, the object that needs to be tested is inductive capacity of medium.
touched with the metal cap. If the leaves diverge, the body is Permittivity: Permittivity is a measure of the ability of the medium
said to be charged, and if there is no change in the leaves of the surrounding electric charges to allow electric lines of force to pass
electroscope, then the body is uncharged. through it. It determines the forces between the charges.
Identification of the Nature of the Charge Relative Permittivity: The relative permittivity or the dielectric
constant (εr or K) of a medium is defined as the ratio of the
To identify the nature of the charge, let’s consider an example. A
positively charged body is brought near the metal cap. Then an permittivity ε of the medium to the permittivity ε0 of free space
unknown body is brought near the metal cap. If the leaves diverge ε
i.e. εr or K =
further, we can conclude that the unknown body has a positive ε0
charge. If the leaves come closer to each other, then the charge of Dimensions of permittivity
the unknown body is negative. [M–1 L–3 T4 A2]
Identification of Body as a Conductor or an Insulator The dielectric constants of different mediums
To identify if a body is a conductor or an insulator, two gold leaf Medium Vacuum Air Water Mica
electroscopes are taken. One gold leaf electroscope is charged so
that the leaves will diverge. Then the other gold leaf electroscope ε 1 1.00059 80 6
is connected to the first one through the body. If the leaves of the Teflon Glass PVC Metal
other electroscope diverge, then the body is a conductor, and if 2 5-10 4.5 ∞
there is no change in the leaves, the body is an insulator.
COULOMB’S LAW COULOMB’S LAW IN VECTOR FORM
Coulomb's law states that the force of attraction or repulsion A charge q1 is placed at A whose position vector is r1 .
between two stationary point charges is directly proportional to
Another charge q2 is placed at B whose position vector is r2 , such
the product of charges and inversely proportional to the square of
distance between them. This force acts along the line joining the that AB =| r2 − r1 | = r .
point charges.
The magnitude of force is given by
If q1 and q2 are charges
k q1 q2
r = the distance between them F=
F = the force acting between them 4πε0 r 2
Electric Charges and Fields 3
, Force on q2 due to q1, in vector form, is given by
1 q1q2
F21 = AB , where AB is a unit vector along line joining
4πε0 r 2
A and B, pointing from A to B.
Example 1: Two equal point charges (10–3 C) are placed 1 cm
apart in medium of dielectric constant K = 5.
1 q1q2 ( r2 − r1 ) 1 q1q2
⇒ F21
= =
2 3 ⋅ ( r2 − r1 ) (a) Find the interaction force between the point charges.
4πε0 r r2 − r1 4πε0 r
(b) Net force on any of the charge.
y q1 q2 Sol. (a) Interaction force between point charges
AB
( )
2
10−3
F F21 1 q1q2
12 A B F = 9 × 10
= 9
9 × 107 N
=
4π ∈0 r 2
( )
2
r1 10−2
r2
(b) Net force on any one charge
x
( )
−3 2
Similarly force on q1 due to q2 is given by, 1 q1q2 9 × 109 10
F=' = = 18 × 106 N
1 q1q2 ( r1 − r2 ) 1 q1q2
4π ∈0 K r 2 5 10 −2 2
( )
= F12 =
2 3 ⋅ ( r1 − r2 )
4πε0 r r1 − r2 4πε0 r Example 2: Two small balls each of mass m and charge q
on each of them are suspended through two light insulating
Note that F12 = − F21
strings of length l from a point. Find the expression for
Both forces are action reaction pair. angle q made by any of the string with vertical when under
static equilibrium.
PRINCIPLE OF SUPERPOSITION O
Consider a system of n stationary charges q1, q2, q3, ... qn in
vacuum. What is the force on q1 due to q2, q3 ... qn? Coulomb’s law
q q
is not enough to answer this question. Force on any charge due to a
m m
number of other charges is the vector sum of all the forces on that
Sol. Angle of any string with vertical is q as shown.
charge due to the other charges taken one at a time. The individual
forces are unaffected due to the presence of other charges. This is o
termed as the ‘principle of superposition’. (Note that the principle
of superposition is independent of Coulomb’s law and all of T
T
electrostatics is basically a consequence of Coulomb’s law and Fe Fe
superposition principle together). mg mg
Now, the force on q1 can be found out by calculating separately For equilibrium in horizontal direction
the forces F12 , F13 ……..F1n exerted by q2, q3 ......qn, respectively 1 q2
on q1 and then adding these forces vectorially. =
Fe = T sin θ ...(i)
4π ∈0 ( 2l sin θ )2
F1n q2 For equilibrium in vertical direction
q1 T cos θ = mg ...(ii)
F13 q3 Dividing (i) by (ii)
F12 Fe
tan θ =
mg
qn
Example 3: For the following system to be in equilibrium,
So,=
F1 F 12 + F13 +……+ F 1n what should be the value of charge Q?
q1 q2 rˆ12 q3 rˆ13 qn rˆ1n Q
=∴ F1 2 + 2 +……. + 2 q a a q
4πε0 r12 r13 r1n
Kq 2 K |Q|q q
q n
qi rˆ1i r1 − ri Sol. = ⇒ Q=–
= 1
4πε0
∑ r12i
r1 − ri and rˆ1i =
where r1i =
| r1 − ri |
(2a ) 2
a2 4
i =2
P
4 W JEE (XII) Module-1 PHYSICS
