Magnification and Image Characteristics of Concave Mirror

When an object of 1 cm is placed at a distance of 15 cm from a concave mirror with a focal length of 10 cm, the image formed is real, inverted, and diminished. This article will delve deeper into the nature, position, and size of the image using the mirror formula and coordinate method.

Mirror Equation and Image Characteristics

The mirror formula is given by:

1/f 1/v - 1/u

where f is the focal length of the mirror, u is the distance of the object from the mirror, and v is the distance of the image from the mirror.

Problem 1: Object of 1 cm at 15 cm from a Concave Mirror with Focal Length 10 cm

Given:

u -15 cm (since the object is to the left of the mirror) f -10 cm (since it's a concave mirror)

Using the mirror formula:

1/-10 1/-15 - 1/v

Solving for v (image distance):

1/v 1/15 - 1/10 -1/30

v -30 cm

The negative sign indicates that the image is real and inverted.

The size of the image is given by:

size of the image -v/u -(-30)/(-15) * 1 2 cm

Problem 2: Object of 2 cm at 15 cm from a Concave Mirror with Focal Length 10 cm

Given:

u -15 cm f -10 cm h_o 2 cm (object height)

Using the mirror formula to find v (image distance):

1/-10 1/-15 - 1/v

Solving for v (image distance):

1/v 1/15 - 1/10 -1/30

v -30 cm

The negative sign indicates that the image is real and inverted.

Now, to find the height of the image:

magnification, m -v/u -(-30)/(-15) -2

h_i m * h_o -2 * 2 -4 cm

The negative sign indicates that the image is inverted, and the size of the image is 4 cm.

Co-ordinate Method

The equation of the ray ED or EF can be found as follows:

Slope of ED, m_1 OE/OF 2/10 1/5

y (1/5)x 2

The equation of the ray BO (where BO is the principal axis):

x 0

Substituting x 0 into the equation of the ray ED:

y 2

The intersection point of ED and BO (image formation) gives the coordinates of the image.

Conclusion

Understanding the nature, position, and size of images formed by a concave mirror is crucial in optics. This article has covered the use of the mirror formula and the coordinate method to determine these characteristics for different scenarios. Proper application of these methods ensures accurate image analysis in various optical systems.