INTEGRATION USING ALGEBRAIC SUBSTITUTIONS 395
H
- The electrostatic potential on all parts of a
conducting circular disc of radiusris given
by the equation:
V= 2 πσ
∫ 9
0
R
√
R^2 +r^2
dR
Solve the equation by determining the
integral.
[
V= 2 πσ
{√
(9^2 +r^2 )−r
}]
- In the study of a rigid rotor the following
integration occurs:
Zr=
∫∞
0
(2J+1)e
−J(J+1)h^2
8 π^2 Ik T dJ
DetermineZr for constant temperatureT
assumingh,Iandkare constants.[
8 π^2 IkT
h^2
]
- In electrostatics,
E=
∫π
0
⎧
⎪⎨
⎪⎩
a^2 σsinθ
2 ε
√(
a^2 −x^2 −2axcosθ
)dθ
⎫
⎪⎬
⎪⎭
where a,σandεare constants,xis greater
than a, andxis independent ofθ. Show that
E=
a^2 σ
εx
