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The magnification formula is a crucial concept when dealing with mirrors, especially concave mirrors. In essence, it helps us understand the relationship between the size of the image and the object. The formula is:\[ m = \frac{h_i}{h_o} = \frac{-v}{u} \]Here,

• \( m \) is the magnification, indicating how much larger or smaller the image is compared to the object.
• \( h_i \) is the height of the image, while \( h_o \) is the height of the object.
• \( v \) is the distance from the mirror to the image and \( u \) is the distance from the mirror to the object.

The negative sign in the formula \( \left(-\frac{v}{u}\right) \) comes from the convention used in optics, which defines the distances in a particular way for mirrors. When solving problems, always check if the values are consistent, like using the same units. This formula lets us find one unknown if we know the others, as we did for the object distance in the original problem.

The mirror formula is another fundamental principle when studying mirrors. It relates the object and image distances with the focal length of the mirror. The formula is:\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]In this equation,

• \( f \) represents the focal length of the mirror, a critical property that defines how the mirror focuses light.
• \( v \) is the image distance from the mirror.
• \( u \) is the object distance from the mirror.

The process involves substituting the known values into the formula, solving for the unknown. In this exercise, we applied known values of \( v = 500 \) cm and \( u = -200 \) cm to determine the focal length. Understanding how to manipulate this formula is essential in optics, allowing you to transition between these distances easily.

Calculating the focal length of a mirror is often a key step in solving optics problems involving mirrors. The focal length \( f \) is a measure of how a concave mirror converges light to a focal point. In our exercise, we used the mirror formula:\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]By substituting the known values:

• \( \frac{1}{v} = \frac{1}{500} \)
• \( \frac{1}{u} = \frac{1}{-200} \)

The right-hand side of the equation simplifies to:\[ \frac{1}{500} - \frac{1}{200} = \frac{-3}{1000} \]From this, we solve for \( f \) by inverting the result:\[ f = -\frac{1000}{3} \approx -333.33 \text{ cm} \]This negative value indicates that the focal point is on the same side of the mirror as the object, which is typical for concave mirrors. Understanding focal length calculation is vital in determining how mirrors affect the path of light.