A beam of gamma rays is incident on a block of lead. The intensity of gamma rays passing through a uniform medium varies with depth x, in a manner given by the expression I(x) = 1, e-ux Here, I, is the intensity of the radiation at the surface of the material (at x = 0) and u is called the linear absorption coefficient. A) Find an expression for the thickness of material that would absorb half of the gamma rays. B) Find an expression for the thickness of material that would reduce the radiation intensity to a fraction of the initial intensity.

A. x = -㏑(1/2)/μ B. x = -㏑(I/I₀)/μ

Explanation:

A. Since the intensity I = I₀exp(-μx) where I₀ = intensity at x = 0.

When I = I₀/2,  

I = I₀exp(-μx)

I₀/2 = I₀exp(-μx)

dividing through by I₀, we have

1/2 = exp(-μx)

taking natural logarithm of both sides, we have

㏑(1/2) = ㏑[exp(-μx)]

㏑(1/2) = -μx

dividing both sides by -μ

x = -㏑(1/2)/μ

where x is the thickness of the material that would absorb half of the gamma rays

B. Since the intensity I = I₀exp(-μx) at thickness x, where I₀ = intensity at x = 0

I = I₀exp(-μx)

dividing through by I₀

I/I₀ = exp(-μx)

taking natural logarithm of both sides, we have

㏑(I/I₀) = ㏑exp(-μx)

㏑(I/I₀) = -μx

dividing both sides by -μ

x = -㏑(I/I₀)/μ

where x is the thickness of the material that would reduce the radiation intensity to a fraction of the initial intensity.

this is an example of "committed" time.

committed time, as contracted time, takes need over essential and available time since it is seen as profitable work. it alludes to the time allotted to keep up a home and family. at the point when a man is commuting using committed time this individual may feel that the drive is an obligation to family, for example, strolling youngsters to class or driving a mate to work. contracted and committed time clients may feel that their commute could really compare to the commute of important or leisure time clients on the grounds that their commute is profitable work. in this manner, they might be more disposed to pick a mechanized method of movement.