Chapter 25, Problem 57P

To determine

Part(a)To determine:

The magnification of the parallel X ray beams.

Answer to Problem 57P

Solution:

The magnification of a parallel beam of X rays is +1 x.

Explanation of Solution

Given:

Parallel beam of X rays falls on the object.

When a parallel beam of X rays falls on an object, the rays pass through the object in straight lines and cast a shadow on the film placed behind the object.

If the shadow is considered as the image of the object, it can be seen from the above diagram, the size of the image is equal to the size of the object.

Since magnification m is given by,

$m=\frac{h_i}{h_o}$

The magnification is positive and equal to 1.

To determine

Part (b)To determine:

The magnification produced by a point source of X rays.

Answer to Problem 57P

Solution:

The magnification produced varies from +1 x to +2.7 x.

Explanation of Solution

When a diverging beam of X rays from a point source falls on a person, the ray diagram can be drawn as shown below.

The diverging beams form two similar triangles for the extreme positions.

The ratio of the heights of the image and the object $\frac{h_i}{h_o}$ is equal to the ratio of the distances of the front and the back of the person $\frac{d_i}{d_0}$ from the source.

Hence magnification is given by,

$m=\frac{h_i}{h_o}=\frac{d_i}{d_0}$

If the person’s thickness is t, then,

$d_i=d_0+t$

Therefore,

$m=\frac{d_0+t}{d_0}$

At the centre, however, the beam passes without diverging and hence the magnification of the central regions remain at +1.

Given:

Distance between the person and the source $d_0=15\text{ cm}$

The person’s thickness $t=25\text{ cm}$

Formula:

$m=\frac{d_0+t}{d_0}$

Calculation:

Substitute the given values in the equation and solve.

$\begin{array}{c}m=\frac{d_0+t}{d_0}\\ =\frac{\left(15\text{ cm}\right)+\left(25\text{ cm}\right)}{\left(15\text{ cm}\right)}\\ =2.7\end{array}$