1.3 Solutions
33
1.98 The alpha ray activity of a material is measured after equal successive intervals (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 = 14.5 C/M
1.101 Make the best fit for the parabola y = a 0 + a 1 x + a 2 x
2 , with the given pairs
of values for x and y.
x −2 −1 0 +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 concentric cylinder with radii a and b, (b > a) is C =
2πε0
ln(b/a)
. If a = 10 ± 1 mm and
b = 20 ± 1 mm, with what relative precision is C measured?
[University of Cambrdige Tripos 2004]
1.103 If f (x) is the probability density of x given by f (x) = xe
−x/λ over the
interval 0 < x < ∞, find the mean and the most probable values of x.
1.2.14 Numerical Integration
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 Vector Calculus
1.1 ∇Φ =
ˆ
i
∂
∂ x
+ ˆ j
∂
∂ y
+ ˆ
k
∂
∂z
(x
2
+ y
2
+ z
2 )
−1/2
=
−
1
2
.2x ˆ
i −
1
2
.2y ˆ j −
1
2
.2z ˆ
k
(x
2
+ y
2
+ z
2 )
−3/2
= −(x ˆ
i + y ˆ j + z ˆ
k)(x
2
+ y
2
+ z
2 )
−
3
2 = −
r
r 3
33
1.98 The alpha ray activity of a material is measured after equal successive intervals (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 = 14.5 C/M
1.101 Make the best fit for the parabola y = a 0 + a 1 x + a 2 x
2 , with the given pairs
of values for x and y.
x −2 −1 0 +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 concentric cylinder with radii a and b, (b > a) is C =
2πε0
ln(b/a)
. If a = 10 ± 1 mm and
b = 20 ± 1 mm, with what relative precision is C measured?
[University of Cambrdige Tripos 2004]
1.103 If f (x) is the probability density of x given by f (x) = xe
−x/λ over the
interval 0 < x < ∞, find the mean and the most probable values of x.
1.2.14 Numerical Integration
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 Vector Calculus
1.1 ∇Φ =
ˆ
i
∂
∂ x
+ ˆ j
∂
∂ y
+ ˆ
k
∂
∂z
(x
2
+ y
2
+ z
2 )
−1/2
=
−
1
2
.2x ˆ
i −
1
2
.2y ˆ j −
1
2
.2z ˆ
k
(x
2
+ y
2
+ z
2 )
−3/2
= −(x ˆ
i + y ˆ j + z ˆ
k)(x
2
+ y
2
+ z
2 )
−
3
2 = −
r
r 3