Question

A concave mirror has a radius of curvature of 60.0 cm. Calculate the image position and magnification of an object placed in front of the mirror at distances of (a) 90.0 cm and (b) 20.0 cm. (c) Draw ray diagrams to obtain the image characteristics for each case.

Answer 1

Answer:

the following ray diagram is as above

Answer 2

Answer: a. position of image is  -45cm from concave mirror and magnification is -0.5

b. position of  image is 60 cm from concave mirror and magnification is 3.

• The relationship that exists between a spherical mirror's focal length (f) and its radius of curvature (R) is that the focal length is equivalent to half of the radius of curvature, which can be written as f = R/2.
• 1/f= 1/u + 1/v. The "mirror formula" is the name given to this particular equation. The formula works equally well for convex as it does for concave mirrors.

• Magnification is also equal to the negative of the ratio of the image distance(v) to object distance(u). m=−v/u

Explanation:

Step 1: Given Data

Radius of curvature, R = 60 cm

position of object 1, $u_1 = -90 cm$

position of object 2, $u_2 = -20 cm$

Let the position of images of the objects be $v_1 \hspace{4}and \hspace{4} v_2$ respectively from the mirror.

Step 2: Find the focus of Concave Mirror

focus f = $f = \frac{R}{2}\\f = \frac{60}{2} = -30 \hspace{4}cm$

Therefore focus f of the mirror is 30 cm.

Step 3: Find the position of of the objects

a. for object 1,

Mirror Formula,

$\frac{1}{f}=\frac{1}{u_1}+\frac{1}{v_1}\\\frac{1}{v_1}=\frac{1}{f}-\frac{1}{u_1}}\\\frac{1}{v_1}=-\frac{1}{30}+\frac{1}{90}\\v_1 = \frac{90X 30}{-90+30}\\v_1 = - 45\hspace{4} cm$

Magnification.

$m_1 =\frac{-v}{u}= -\frac{-45}{-90} = -0.5$

Image formed is real and inverted and half the original size

b. for object 2

Mirror Formula,

$\frac{1}{f}=\frac{1}{u_2}+\frac{1}{v_2}\\\frac{1}{v_2}=\frac{1}{f}-\frac{1}{u_2}}\\\frac{1}{v_2}=-\frac{1}{30}+\frac{1}{20}\\v_2 = \frac{20X 30}{-20+30}\\v_2 = 60\hspace{4} cm$

Magnification.

$m_2 =\frac{-v}{u}= -\frac{60}{-20} = 3$

Image formed is virtual and erect and three times the original size of object

Step 4. Ray diagram for each case

Ray diagram is given in a.png file

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