DAY NINETEEN
Electrostatics
Learning & Revision for the Day
u Electric Charge u Motion of A Charged Particle u Electric Potential Energy
u Coulomb’s Law of Forces in An Electric Field u Equipotential Surface
between Two Point Charges u Electric Dipole u Conductors and Insulators
u Superposition Principle u Electric Flux (φE ) u Electrical Capacitance
u Electric Field u Gauss Law u Capacitor
u Continuous Charge Distribution u Electric Potential
If the charge in a body does not move, then the frictional electricity is known as static
electricity. The branch of physics which deals with static electricity is called
electrostatics.
Electric Charge
Electric charge is the property associated with matter due to which it produces and
experiences electric and magnetic effects.
Conservation of Charge
We can neither create nor destroy electric charge. The charge can simply be transferred
from one body to another. There are three modes of charge transfer:
(a) By friction (b) By conduction (c) By induction
Quantisation of Charge
Electric charge is quantised. The minimum unit of charge, which may reside
independently is the electronic charge e having a value of 1.6 × 10 –19 C, i.e. Q = ± ne,
where, n is any integer.
Important properties of charges are listed below
l
Like charges repel while opposite charges attract each other.
l
Charge is invariant i.e. charge does not change with change in velocity.
l
According to theory of relativity, the mass, time and length change with a change in
velocity but charge does not change.
l
A charged body attracts a lighter neutral body.
l
Electronic charge is additive, i.e. the total charge on a body is the algebraic sum of all
the charges present in different parts of the body. For example, if a body has different
charges as + 2q, + 4q, − 3 q, − q, then the total charge on the body is + 2 q.
,Coulomb’s Law of Forces between Properties of electric field lines are given below
Electric field lines come out of a positive charge and go into
Two Point Charges l
the negative charge.
l
If q 1 and q2 be two stationary point charges in free space l
No two electric field lines intersect each other.
separated by a distance r, then the force of attraction /
repulsion between them is
l
Electric field lines are continuous but they never form a
closed loop.
K |q 1||q2| 1 |q 1||q2| 1
F = = ⋅ K = 4πε l
Electric field lines cannot exist inside a conductor. Electric
r2 4πε 0 r2 0
shielding is based on this property.
9 × 10 9 × | q 1|| q2|
= [K = 9 × 10 9 N-m2 /c2 ]
r2
l
If some dielectric medium is completely filled between the
Continuous Charge Distribution
given charges, then the Coulomb’s force between them The continuous charge distribution may be one dimensional,
becomes two dimensional and three-dimensional.
1 q 1q2 1 q 1q2 ε 1. Linear charge density (λ ) If charge is distributed along a
Fm = = Q ε = ε r or k line, i.e. straight or curve is called linear charge
4πε r 2 4πε 0ε r r 2 0 distribution. The uniform charge distribution q over a
1 q q length L of the straight rod.
= ⋅ 1 2
4πkε 0 r 2 Then, the linear charge density, λ =
q
L
Its unit is coulomb metre −1 (Cm–1).
Superposition Principle 2. Surface charge density (σ) If charge is distributed over a
It states that, the net force on any one charge is equal to the surface is called surface charge density, i.e. σ = q / A
vector sum of the forces exerted on it by all other charges. If Its unit is coulomb m–2 (Cm–2 )
there are four charges q 1, q2 , q3 and q 4, then the force on q 1 3. Volume charge density (ρ) If charge is distributed over
(say) due to q2 , q3 and q 4 is given by F1 = F12 + F13 + F14,
the volume of an object, is called volume charge density,
where F12 is the force on q 1 due to q2 , F13 that due to q3 and q
F14 that due to q 4. i. e., p = . Its unit is coulomb metre −3 (Cm–3 ).
V
1. Electric Field due to a Point Charge
Electric Field l
Electric field at a distance r from a point charge q is
The space surrounding an electric charge q in which another 1 q
charge q 0 experiences a force of attraction or repulsion, is E = ⋅ 2
4πε 0 r
called the electric field of charge q. The charge q is called the
source charge and the charge q 0 is called the test charge. The l
If q 1 and q2 are two like point charges, separated by a distance
test charge must be negligibly small so that it does not modify r, a neutral point between them is obtained at a point distant
the electric field of the source charge. r
r1 from q 1, such that r1 =
q2
Intensity (or Strength) of Electric Field (E) 1 + q
1
The intensity of electric field at a point in an electric field is l
If q 1 and q2 are two charges of opposite nature separated by
the ratio of the forces acting on the test charge placed at that a distance r, a neutral point is obtained in the extended line
point to the magnitude of the test charge. joining them, at a distance r1 from q 1, such that,
F r
E= , where F is the force acting on q 0. r1 =
q0 q2
Electric field intensity (E) is a vector quantity. q − 1
1
The direction of electric field is same as that of force acting on
the positive test charge. Unit of E is NC−1 or Vm −1. 2. Electric Field due to Infinitely Long Uniformly
Charged Straight Wire
Electric Field Lines Electric field at a point situated at a normal distance r,
An electric field line in an electric field is a smooth curve, from an infinitely long uniformly charged straight wire
tangent to which, at any point, gives the direction of the having a linear charge density λ, is
electric field at that point. λ
E =
2 πε 0 r
, 3. Electric Field due to a Charged Cylinder Fx = 0
l
For a conducting charged cylinder of linear charge density Y E
λ and radius R, the electric field is given by
λ P (x, y)
E = , for r > R,
2 πε 0 r
λ O
u
X
E = , for r = R
2 πε 0 R
∴Acceleration of the particle along Y-axis is given by
and E = 0, for r < R
Fy qE
l
For a non-conducting charged cylinder, for r ≤ R, ay = =
m m
λr
E = The initial velocity is zero along Y-axis (u y = 0).
2 πε 0R2
∴The deflection of charged particle along Y-axis after time t
λ 1
and E = , for r > R is given by y = u y t + a y t 2
2 πε 0 r 2
qE 2
E Emax E Emax = t
2m
r>R Along X -axis there is no acceleration, so the distance
R
covered by particle in time t along X -axis is given by x = ut
r<
r>R
r
E ∝ 12
∝
r Eliminating t , we have
E
O qE
O r=R r r=R r y= x2
2 mu2
(a) Variation of electric field (b) Variation of electric field
with distance for conducting with distance for y ∝ x2
cylinder non-conducting cylinder
This shows that the path of charged particle in
perpendicular field is a parabola.
4. Electric Field due to a Uniformly Charged
Infinite Plane Sheet
Electric field near a uniformly charged infinite plane sheet
Electric Dipole
having surface charge density σ is given by An electric dipole consists of two equal and opposite charges
σ separated by a small distance.
E=
2 ε0 A p B
–q +q
5. Electric Field due to a Uniformly Charged 2a
Electric Dipole
Thin Spherical Shell
For a charged conducting sphere/ E The dipole moment of a dipole is defined as the product
r<R of the magnitude of either charges and the distance
shell of radius R and total charge
Q, the electric field is given by between them. Therefore, dipole moment
Case I E = 0, for r < R p = q(2a)
Q O
Case II E = , for r = R R Electric Field due to a Dipole
4πε 0R 2 Variation of electric field
Q with distance for uniformly l
At a point distant r from the centre of a dipole, along its
Case III E = , for r > R charge spherical shell 1 2 pr
4πε 0r 2 axial line E = ⋅ 2
4πε 0 (r − a2 )2
[direction of E is the same as that of p]
Motion of a Charged Particle in For a short dipole, E =
1 2p
⋅ [r > > a]
4πε 0 r 3
an Electric Field l
At a point distant r from the centre of a dipole, along its
Let a charged particle of mass m and charge q, enters the
equatorial line
electric field along X -axis with speed u. The electric field E
1 p
is along Y-axis is given by E=− ⋅
F y = qE 4πε 0 (r 2 + a2 )
and force along X -axis remains zero, i.e. [direction of E is opposite to that of p]
Electrostatics
Learning & Revision for the Day
u Electric Charge u Motion of A Charged Particle u Electric Potential Energy
u Coulomb’s Law of Forces in An Electric Field u Equipotential Surface
between Two Point Charges u Electric Dipole u Conductors and Insulators
u Superposition Principle u Electric Flux (φE ) u Electrical Capacitance
u Electric Field u Gauss Law u Capacitor
u Continuous Charge Distribution u Electric Potential
If the charge in a body does not move, then the frictional electricity is known as static
electricity. The branch of physics which deals with static electricity is called
electrostatics.
Electric Charge
Electric charge is the property associated with matter due to which it produces and
experiences electric and magnetic effects.
Conservation of Charge
We can neither create nor destroy electric charge. The charge can simply be transferred
from one body to another. There are three modes of charge transfer:
(a) By friction (b) By conduction (c) By induction
Quantisation of Charge
Electric charge is quantised. The minimum unit of charge, which may reside
independently is the electronic charge e having a value of 1.6 × 10 –19 C, i.e. Q = ± ne,
where, n is any integer.
Important properties of charges are listed below
l
Like charges repel while opposite charges attract each other.
l
Charge is invariant i.e. charge does not change with change in velocity.
l
According to theory of relativity, the mass, time and length change with a change in
velocity but charge does not change.
l
A charged body attracts a lighter neutral body.
l
Electronic charge is additive, i.e. the total charge on a body is the algebraic sum of all
the charges present in different parts of the body. For example, if a body has different
charges as + 2q, + 4q, − 3 q, − q, then the total charge on the body is + 2 q.
,Coulomb’s Law of Forces between Properties of electric field lines are given below
Electric field lines come out of a positive charge and go into
Two Point Charges l
the negative charge.
l
If q 1 and q2 be two stationary point charges in free space l
No two electric field lines intersect each other.
separated by a distance r, then the force of attraction /
repulsion between them is
l
Electric field lines are continuous but they never form a
closed loop.
K |q 1||q2| 1 |q 1||q2| 1
F = = ⋅ K = 4πε l
Electric field lines cannot exist inside a conductor. Electric
r2 4πε 0 r2 0
shielding is based on this property.
9 × 10 9 × | q 1|| q2|
= [K = 9 × 10 9 N-m2 /c2 ]
r2
l
If some dielectric medium is completely filled between the
Continuous Charge Distribution
given charges, then the Coulomb’s force between them The continuous charge distribution may be one dimensional,
becomes two dimensional and three-dimensional.
1 q 1q2 1 q 1q2 ε 1. Linear charge density (λ ) If charge is distributed along a
Fm = = Q ε = ε r or k line, i.e. straight or curve is called linear charge
4πε r 2 4πε 0ε r r 2 0 distribution. The uniform charge distribution q over a
1 q q length L of the straight rod.
= ⋅ 1 2
4πkε 0 r 2 Then, the linear charge density, λ =
q
L
Its unit is coulomb metre −1 (Cm–1).
Superposition Principle 2. Surface charge density (σ) If charge is distributed over a
It states that, the net force on any one charge is equal to the surface is called surface charge density, i.e. σ = q / A
vector sum of the forces exerted on it by all other charges. If Its unit is coulomb m–2 (Cm–2 )
there are four charges q 1, q2 , q3 and q 4, then the force on q 1 3. Volume charge density (ρ) If charge is distributed over
(say) due to q2 , q3 and q 4 is given by F1 = F12 + F13 + F14,
the volume of an object, is called volume charge density,
where F12 is the force on q 1 due to q2 , F13 that due to q3 and q
F14 that due to q 4. i. e., p = . Its unit is coulomb metre −3 (Cm–3 ).
V
1. Electric Field due to a Point Charge
Electric Field l
Electric field at a distance r from a point charge q is
The space surrounding an electric charge q in which another 1 q
charge q 0 experiences a force of attraction or repulsion, is E = ⋅ 2
4πε 0 r
called the electric field of charge q. The charge q is called the
source charge and the charge q 0 is called the test charge. The l
If q 1 and q2 are two like point charges, separated by a distance
test charge must be negligibly small so that it does not modify r, a neutral point between them is obtained at a point distant
the electric field of the source charge. r
r1 from q 1, such that r1 =
q2
Intensity (or Strength) of Electric Field (E) 1 + q
1
The intensity of electric field at a point in an electric field is l
If q 1 and q2 are two charges of opposite nature separated by
the ratio of the forces acting on the test charge placed at that a distance r, a neutral point is obtained in the extended line
point to the magnitude of the test charge. joining them, at a distance r1 from q 1, such that,
F r
E= , where F is the force acting on q 0. r1 =
q0 q2
Electric field intensity (E) is a vector quantity. q − 1
1
The direction of electric field is same as that of force acting on
the positive test charge. Unit of E is NC−1 or Vm −1. 2. Electric Field due to Infinitely Long Uniformly
Charged Straight Wire
Electric Field Lines Electric field at a point situated at a normal distance r,
An electric field line in an electric field is a smooth curve, from an infinitely long uniformly charged straight wire
tangent to which, at any point, gives the direction of the having a linear charge density λ, is
electric field at that point. λ
E =
2 πε 0 r
, 3. Electric Field due to a Charged Cylinder Fx = 0
l
For a conducting charged cylinder of linear charge density Y E
λ and radius R, the electric field is given by
λ P (x, y)
E = , for r > R,
2 πε 0 r
λ O
u
X
E = , for r = R
2 πε 0 R
∴Acceleration of the particle along Y-axis is given by
and E = 0, for r < R
Fy qE
l
For a non-conducting charged cylinder, for r ≤ R, ay = =
m m
λr
E = The initial velocity is zero along Y-axis (u y = 0).
2 πε 0R2
∴The deflection of charged particle along Y-axis after time t
λ 1
and E = , for r > R is given by y = u y t + a y t 2
2 πε 0 r 2
qE 2
E Emax E Emax = t
2m
r>R Along X -axis there is no acceleration, so the distance
R
covered by particle in time t along X -axis is given by x = ut
r<
r>R
r
E ∝ 12
∝
r Eliminating t , we have
E
O qE
O r=R r r=R r y= x2
2 mu2
(a) Variation of electric field (b) Variation of electric field
with distance for conducting with distance for y ∝ x2
cylinder non-conducting cylinder
This shows that the path of charged particle in
perpendicular field is a parabola.
4. Electric Field due to a Uniformly Charged
Infinite Plane Sheet
Electric field near a uniformly charged infinite plane sheet
Electric Dipole
having surface charge density σ is given by An electric dipole consists of two equal and opposite charges
σ separated by a small distance.
E=
2 ε0 A p B
–q +q
5. Electric Field due to a Uniformly Charged 2a
Electric Dipole
Thin Spherical Shell
For a charged conducting sphere/ E The dipole moment of a dipole is defined as the product
r<R of the magnitude of either charges and the distance
shell of radius R and total charge
Q, the electric field is given by between them. Therefore, dipole moment
Case I E = 0, for r < R p = q(2a)
Q O
Case II E = , for r = R R Electric Field due to a Dipole
4πε 0R 2 Variation of electric field
Q with distance for uniformly l
At a point distant r from the centre of a dipole, along its
Case III E = , for r > R charge spherical shell 1 2 pr
4πε 0r 2 axial line E = ⋅ 2
4πε 0 (r − a2 )2
[direction of E is the same as that of p]
Motion of a Charged Particle in For a short dipole, E =
1 2p
⋅ [r > > a]
4πε 0 r 3
an Electric Field l
At a point distant r from the centre of a dipole, along its
Let a charged particle of mass m and charge q, enters the
equatorial line
electric field along X -axis with speed u. The electric field E
1 p
is along Y-axis is given by E=− ⋅
F y = qE 4πε 0 (r 2 + a2 )
and force along X -axis remains zero, i.e. [direction of E is opposite to that of p]
