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Work and Energy
Work done by an external agency to move a charge :
W  QV b  V a 
 Work done is path independent
Electrostatic force is conservative
Dr. Champak B. Das (BITS, Pilani)
Work done to bring a charge from

infinity to r :

W  QV r 
Potential is the work required
to create the system (potential
energy) per unit charge
Dr. Champak B. Das (BITS, Pilani)
Energy of a Point
Charge Distribution
Ex: Case of assembling three point charges
W1  0
q1

r1

r2
q2
rs12
 q1
1
W2 
q2 
4 πε0  rs12




Dr. Champak B. Das (BITS, Pilani)
 q1 q2
1
W3 
q3 

4 πε0  rs13 rs23




q3

r3
rs23
rs13
q1

r1
rs12

r2
q2
Dr. Champak B. Das (BITS, Pilani)
Energy of a Point
Charge Distribution
Work necessary to assemble
n number of point charges
1
W 
8πε0
n
n
 
i 1 j 1, j  i
qi q j
rsij

1
 W   qiV ri 
2 i 1
n
Dr. Champak B. Das (BITS, Pilani)
Energy of a Continuous Charge
Distribution
1
W   ρVdτ
2
ε0
W 
2
 
 
 
ε 
 W     E  V dτ   VE  da 
2
 
  E V dτ
0

τ
ε0
W 
2
s

 E dτ
2
all space
Dr. Champak B. Das (BITS, Pilani)
ELECTROSTATIC ENERGY

1
W   qiV ri 
2 i 1
n
(can be +ve/-ve)
ε0
W 
2
E
d
τ

2
all space
(always +ve)
Dr. Champak B. Das (BITS, Pilani)
ELECTROSTATIC ENERGY
•Energy of a point charge is infinite !
•Energy is stored in the field/charge ?
•Doesn’t obey superposition principle !
Dr. Champak B. Das (BITS, Pilani)
Prob. 2.32 (a) :
Find the energy stored in a uniformly charged
solid sphere of radius R and charge q using:
1
W   ρVdτ
2
Where
q 1 
r2 
 3  2 
V r  
4πε0 2R 
R 
1 3q

W 
4πε0  5 R
2
Ans (a):



Dr. Champak B. Das (BITS, Pilani)
Prob. 2.32 (b) :
Find the energy stored in a uniformly charged
solid sphere of radius R and charge q using:
ε0
W 
2
where
Ans (b):
2
E
 dτ
all space
1 q
E
4 πε0 r 2
for r  R
1 qr

4 πε0 R 3
for r  R
1  3 q2 


W 
4πε0  5 R 
Dr. Champak B. Das (BITS, Pilani)
CONDUCTORS
Conductor:
charges free to move within the material.
Electrostatic Equilibrium:
there is no net motion of charge
within the conductor.
Dr. Champak B. Das (BITS, Pilani)

E = 0 inside a conductor.
The existence of electrostatic equilibrium is
consistent only with a zero field in the conductor.
When an external field is applied ?
Dr. Champak B. Das (BITS, Pilani)
A conductor in an electric field:
e-
Electrons move
upward in response
to applied field.
Dr. Champak B. Das (BITS, Pilani)
A conductor in an electric field: (contd.)
• Electrons accumulate
on top surface.
E0
• Induced charges set
up a field E in the
interior.
Dr. Champak B. Das (BITS, Pilani)
A conductor in an electric field: (contd.)
Two surfaces of a conductor: sheets of charge
interior : Net
 

E  E0  E '
(magnitudes ) :E  E0  E '
Dr. Champak B. Das (BITS, Pilani)
A conductor in an electric field: (contd.)
Field of induced charges
tends to cancel off the
original field
 E0 must move enough
electrons to the surface
such that, E = E0
Dr. Champak B. Das (BITS, Pilani)
In the interior of the conductor
NET FIELD IS ZERO.
The process is Instantaneous
Dr. Champak B. Das (BITS, Pilani)

 = 0 inside a conductor.

 
 ρ  ε0   E


E 0 ρ0
same amount of positive and negative charges
NET CHARGE DENSITY IS ZERO.
Dr. Champak B. Das (BITS, Pilani)

Any net charge resides on the
surface
Dr. Champak B. Das (BITS, Pilani)

A conductor is an equipotential.
For any two points, a and b:
V a   V b  0
V
E
R
r
R
r
Dr. Champak B. Das (BITS, Pilani)

E is  to the surface, outside a
conductor.
E
E=0
Else, the tangential component
would cause
charges to move
Dr. Champak B. Das (BITS, Pilani)
A justification for surface distribution
of charges in a conductor :
 go for a configuration to minimize the potential energy
Example : Solid sphere carrying charge q
 q2 

Wσ  
 8πε0R 
2

6
q 

W ρ  
5  8πε0R 
Wσ  W ρ
Dr. Champak B. Das (BITS, Pilani)
Induced Charges
Conductor
+q
Induced charges
Dr. Champak B. Das (BITS, Pilani)
A cavity in a conductor
+q
Gaussian surface
If +q is placed in the cavity, -q is induced
on the surface of the cavity.
Dr. Champak B. Das (BITS, Pilani)
Prob. 2.35:
A metal sphere of radius R, carrying charge q is
surrounded by a thick concentric metal shell. The shell
carries no net charge.
(a) Find the surface charge density at R, a and b
Answer:
a
q
R
q
σR 
2
4 πR
q
σ


a
2
b
4 πa
q
σb 
4 πb 2
Dr. Champak B. Das (BITS, Pilani)
Prob. 2.35(b):
Find the potential at the centre, using
infinity as the reference point.
a
q
R
b
Answer:
1 q q q
V 0 
   
4πε0  b R a 
Dr. Champak B. Das (BITS, Pilani)
Surface charge on a conductor
Recall electrostatic boundary condition:


σ
E above  E below  nˆ
ε0
=> Field outside a conductor:
 σ
E  nˆ
ε0
Dr. Champak B. Das (BITS, Pilani)
The surface charge density :


σ  ε0 E  nˆ

OR
σ   ε0 V n 
Knowledge of E or V just outside the conductor
 Surface charge on a conductor
Dr. Champak B. Das (BITS, Pilani)
Force on a conductor
Dr. Champak B. Das (BITS, Pilani)
Forces on charge distributions
Force on a charge element dq placed in
an external field E(e) : 
 e 

F  dq E  dq E
On a volume charge distribution :


F   ρ E dτ
τ
Dr. Champak B. Das (BITS, Pilani)
Prob. 2.43:
Find the net force that the southern hemisphere
of a uniformly charged sphere exerts on the
northern hemisphere.
Z
R
X
 r
Q
Ans:
Y

F
2
1 3Q ˆ
k
2
4πε0 16R
Dr. Champak B. Das (BITS, Pilani)
Forces on charge distributions
Force on a charge element dq placed in
an external field E(e) : 
 e 

F  dq E  dq E
On a volume charge distribution :


F   ρ E dτ
τ
On a surface charge distribution :


F   σ E da
s
Dr. Champak B. Das (BITS, Pilani)
Forces on surface charge distributions
“ E is discontinuous across the distribution ”

 dq  o
Ea  Ea  E
below
n̂
da
above

 dq  o
Eb  Eb  E


 1 

The force per unit area : f  σ E above  E below
2 Dr. Champak B. Das (BITS, Pilani)
Force on a conductor
Force (per unit area) on the
conductor surface:

1 2
f 
σ nˆ
2 ε0
Outward Pressure on the
conductor surface :
1
2
P  ε0 E
2
The direction of the force is
“outward” or “into the field”…..
whether  is positive or negative
Dr. Champak B. Das (BITS, Pilani)
Prob. 2.38:
A metal sphere of radius R carries a total charge Q.
What is the force of repulsion between the northern
hemisphere and the southern hemisphere?
Z
Ans:
R

Y
X

F
2
1 Q ˆ
k
2
4πε0 8R
Q
Dr. Champak B. Das (BITS, Pilani)
CAPACITORS
Potential difference between two
conductors carrying +Q and –Q charge:

    
V  V  V    E  dl   E  dl

E  Q

V  Q
Q
C
V
Dr. Champak B. Das (BITS, Pilani)
Capacitance :
• Is a geometrical property
• Units: Farad (= coulomb/volt)
Different possible geometries:
• Planer
• Spherical
• Cylindrical
Dr. Champak B. Das (BITS, Pilani)
Plates are very large and very close
ε0 A
C
d
Dr. Champak B. Das (BITS, Pilani)
A Spherical capacitor
Dr. Champak B. Das (BITS, Pilani)
Cross section of a spherical capacitor
ab
C  4πε0
ba
Dr. Champak B. Das (BITS, Pilani)
A cylindrical capacitor
L
Dr. Champak B. Das (BITS, Pilani)
Cross section of a cylindrical capacitor
Prob 2.39 :
Capacitance per unit length of a cylindrical capacitor
1
C  2πε0
lnb a 
Dr. Champak B. Das (BITS, Pilani)
Work done to charge a capacitor
At any instant,
q
V 
C
2
q
 
1Q
dW   dq  W 
C 
2 C
1
2
W  CV
2
Dr. Champak B. Das (BITS, Pilani)