1.3 Solutions 33
1.98 The alpha ray activity of a material is measured after equal successive inter-
vals (hours), in terms of its initial activity as unity to the 0.835; 0.695; 0.580;
0.485; 0.405 and 0.335. Assuming that the activity obeys an exponential
decay law, find the equation that best represents the activity and calculate
the decay constant and the half-life.
[Osmania University 1967]
1.99 Obtain the interval distribution and hence deduce the exponential law of
radioactivity.1.100 In a Carbon-dating experiment background counting rate=10 C/M. How
long should the observations be made in order to have an accuracy of 5%?
Assume that both the counting rates are measured for the same time.
(^14) C+background rate= 14 .5C/M
1.101 Make the best fit for the parabolay=a 0 +a 1 x+a 2 x^2 , with the given pairs
of values forxandy.
x − 2 − 10 + 1 + 2 + 3
y 9.1 3.5 0.5 0.8 4.6 11.0
1.102 The capacitance per unit length of a capacitor consisting of two long concen-
tric cylinder with radiiaandb,(b>a)isC=ln(^2 πεb/a^0 ).Ifa= 10 ±1 mm and
b= 20 ±1 mm, with what relative precision isCmeasured?
[University of Cambrdige Tripos 2004]
1.103 If f(x) is the probability density ofxgiven by f(x) =xe−x/λover the
interval 0<x<∞, find the mean and the most probable values ofx.
1.2.14 NumericalIntegration
1.104 Calculate
∫ 10
1 x(^2) dx, by the trapezoidal rule.
1.105 Calculate
∫ 10
1 x(^2) dx, by the Simpson’s rule.
1.3 Solutions..................................................
1.3.1 VectorCalculus....................................
1.1 ∇Φ=
(
iˆ∂
∂x+ˆj∂
∂y+kˆ∂
∂z)
(x^2 +y^2 +z^2 )−^1 /^2=
(
−
1
2
. 2 xˆi−
1
2
. 2 yˆj−
1
2
. 2 zˆk
)
(x^2 +y^2 +z^2 )−^3 /^2=−(xiˆ+yjˆ+zkˆ)(x^2 +y^2 +z^2 )−(^32)
=−
r
r^3