Question:

A concave mirror with a focal length of 20 cm is placed on a table. An object is placed 40 cm away from the mirror along its principal axis. Determine the position, nature, and magnification of the image formed by the mirror. Consider that all distances are measured from the mirror's surface.

Answer:

Given:

Focal length, f = -20 cm (negative sign indicates concave mirror) Object distance, d_o = 40 cm

To determine the position, nature, and magnification of the image formed by the mirror, we will use the mirror equation:

1/f = 1/d_i + 1/d_o

Here, d_i represents the image distance. Since the focal length f is negative for a concave mirror, we need to keep the sign conventions consistent. Therefore, d_o will also be negative.

Substituting the given values:

1/-20 cm = 1/d_i + 1/-40 cm

Let's solve for d_i:

1/-20 cm - 1/-40 cm = 1/d_i

Simplifying:

-2/-40 cm - 1/-40 cm = 1/d_i

-2/-40 cm = 1/d_i

Taking the reciprocal:

d_i = -40 cm / -2

d_i = 20 cm

Therefore, the image distance d_i is equal to 20 cm. Since d_i is positive, the image will be formed on the same side as the object (real image).

Next, we can determine the magnification of the image using the magnification formula:

magnification (m) = -d_i / d_o

Substituting the values:

m = -20 cm / 40 cm

Simplifying:

m = -0.5

Therefore, the magnification of the image is -0.5. The negative sign indicates an inverted image.

Hence, the image is formed 20 cm to the left of the concave mirror, it is a real image, and it is inverted with a magnification of -0.5.