How far should one hold an object from a concave mirror of focal length 40 cm so as to get an image twice the size of the object?

To determine how far to hold an object from a concave mirror to get an image that is twice the size of the object, we can use the mirror formula and magnification concepts.

Key Concepts

• Focal Length (f): The distance from the mirror's surface to its focal point. In this case, f = -40 cm (negative for concave mirrors).
• Magnification (m): The ratio of the height of the image (h') to the height of the object (h). Here, m = -2 (since the image is inverted and twice the size).

Mirror Formula

The mirror formula is given by:

1/f = 1/v + 1/u

Where:

• f = focal length
• v = image distance
• u = object distance

Magnification Formula

The magnification can also be expressed as:

m = -v/u

Calculating Object Distance

Since we know the magnification is -2, we can set up the equation:

-2 = -v/u

This simplifies to:

v = 2u

Now, substitute v in the mirror formula:

1/(-40) = 1/(2u) + 1/u

Finding a common denominator gives:

1/(-40) = (1 + 2)/(2u)

1/(-40) = 3/(2u)

Cross-multiplying leads to:

2u = -120

Thus, u = -60 cm.

Final Result

The object should be placed 60 cm in front of the concave mirror to produce an image that is twice the size of the object.