Shortcut Methods and Tricks for Solving Numericals on the Mirror Equation:

1. Focal length shortcuts:

• For concave mirrors: (f = \frac{R}{2}), where R is the radius of curvature.
• For convex mirrors: (f = -\frac{R}{2}).

2. Quick magnification formula: (m=\frac{-d_o}{d_i}) where (d_o) is the object distance and (d_i) is the image distance.

3. Mirror equation shortcut: (\frac{1}{f}=\frac{1}{d_o} + \frac{1}{d_i} )

Solved Examples:

1. A concave mirror has a focal length of 15 cm. An object is placed 10 cm in front of the mirror.

Solution:

• Using the mirror equation shortcut: (\frac{1}{15} = \frac{1}{10} + \frac{1}{d_i}).
• (d_i =30\ cm)

2. A convex mirror of 30 cm focal length has an object placed 10 cm in front. Find the size and nature of the image.

Solution:

• Using the magnification shortcut (m=\frac{-d_o}{d_i},) where (d_o) is the object distance and (d_i) is the image distance.
• Calculate the image distance (d_i), by the mirror equation (\frac{1}{f}=\frac{1}{d_o} + \frac{1}{d_i} )
• (d_i = -7.5\ cm).
• Magnification (m = -\frac{d_o}{d_i} = - \frac{10}{-7.5} = \frac{2}{3}).
• Since (m) is positive, the image is virtual and diminished.

3. A plane mirror is placed at a distance of 10 cm from an object.

• Object distance (d_o = -10) cm (negative sign signifies object distance in front of the mirror)
• Applying the mirror equation: ( \frac{1}{f}=\frac{1}{d_o} + \frac{1}{d_i}) provides: ( d_i = - 20) cm.
• Negative image distance represents the image formed behind the mirror.

4. A concave mirror produces an image that is twice the size of the object. If the object is placed 10 cm from the mirror, find the focal length of the mirror.

Solution:

• Magnification (m = 2)
• Use the magnification shortcut (m=\frac{-d_o}{d_i} ) to find image distance (d_i).
• Substitute (m=2) and (d_o = -10) cm ((d_o)is negative because the object is in front of the mirror).
• Solve for (d_i = -20) cm
• Now apply the mirror equation: (\frac{1}{f}=\frac{1}{d_o} + \frac{1}{d_i} ). You will find that (f = 15) cm.

Remember that these shortcut methods and tricks can simplify complex calculations and save time while solving numericals.