Download Example 12. Find electric field a distance h above the center of a (i

Example 12. Find electric field a distance h above the center of a (i) circular loop of radius R,
which carries a uniform line charge λ, (ii) flat circular disc of radius R, which carries a uniform
surface charge σ. Explore the limit h → ∞ in both cases and R → ∞ in the case of flat disc.
~ is a special kind of vector – it is irrotational,
Electrostatic potential: Electric field E
1
1
~r
= ∇ 3 × ~r + 3 ∇ × ~r
∇×
3
r
r
r
3
1
= − 5 ~r × ~r + 3 · 0
r
r
= 0
~ r ) = 0.
⇒ ∇ × E(~
(8)
~ = 0, from
Please keep in mind, the curl is calculated at the field point. Because ∇ × E
~
Stoke’s theorem it follows that line integral of E around any closed loop is zero
I
~ · d~l = 0
E
and it is independent of path, which in turn implies we can define a function,
Z ~r
~ · d~l.
E
V (r) ≡ −
(9)
O
Here O is some agreed standard reference point. The scalar function V (~r) is called the
electrostatic potential. The potential difference between points p~1 and p~2 is defined as,
Z p~1
Z p~2
~ · d~l
~ · d~l +
E
E
V (p2 ) − V (p1 ) = −
O
p
~2
= −
= −
Z
O
Z p~2
O
O
~ · d~l −
E
Z
p
~1
~ · d~l
E
(10)
p
~1
Now, since V (p2 ) − V (p1 ) ≡ dV = ∇V · d~l,
−
Z
p
~2
~ · d~l = −
E
Z
p
~2
∇V · d~l.
(11)
p
~1
p
~1
Since this is true for any p~1 and p~2 , we have
~ r) = −∇V (r)
E(~
I
~ · d~l.
~
E
∇×E = 0 ⇒
(12)
(13)
~ is a conservative vector field and points downhill in V i.e. high potential
Electrostatic field E
to low potential.
From vector calculus, we know that
1 q
1
r̂
1 q
~
r̂ = E
∇
= − 2 ⇒ −∇
=
r
r
4πǫ0 r
4πǫ0 r 2
1
and this leads to,
V (r) =
1
1 q
⇒ V (r) =
4πǫ0 r
4πǫ0
Z
ρdτ
etc.
r
(14)
Work done to move unit charge in the field of source charge q from a to b,
V (r) =
Z
b
~ r ) · d~l
E(~
a
b
1 q
r̂ · (r̂dr + θ̂rdθ + φ̂r sin θdφ)
2
a 4πǫ0 r
Z b
q
1
dr
=
4πǫ0 a r 2
q
q
1
−
= −
4πǫ0 rb ra
=
Z
implying the previously stated result that
R
~ · d~l is independent of path.
E
~ 1 = k[xy î + 2yz ĵ + 3xz k̂], E
~2 =
Example 13. Which one can represent a valid electric field: E
2
2
k[y î + (2xy + z )ĵ + 2yz k̂].
Electrostatic energy: The potential difference between two points a and b defined above,
as we have seen, is equal to the work done per unit charge required to carry it from a to b.
The work done in transporting charge Q is
W = Q[V (b) − V (a)] = Q[V (b) − V (∞)] = QV (b)
(15)
In this definition, one of the points is usually taken to be ∞, where we set the zero of
the potential. In this sense potential is the potential energy (the work it takes to create
the system bringing charges from infinity) per unit charge. The electrostatic energy is the
potential energy stored in an electric field is, therefore, work done to assemble it.
2