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How to Find Object Distance in a Concave Mirror?

Published in Mirror Calculations 3 mins read

To find the object distance in a concave mirror, you typically use the mirror equation and magnification formula, along with given values. Let's explore the process using an example from the provided reference.

Understanding the Concepts

  • Object Distance (u): The distance between the object and the mirror (always negative).
  • Image Distance (v): The distance between the image and the mirror (can be positive or negative).
  • Focal Length (f): The distance between the focal point and the mirror (negative for concave mirrors).
  • Magnification (m): The ratio of image height to object height, also related to object and image distances.

Solving for Object Distance

Here's a step-by-step approach, using information from the reference provided:

  1. Identify Given Values: In the given example, we have:

    • Focal length (f) = -10 cm (negative for a concave mirror)
    • Magnification (m) = -2
    • Height of the object = h (this does not directly affect object distance calculation)

  2. Use the Magnification Formula:

    • The magnification is given by: m = -v/u, which can be rewritten as v = -mu (Reference point 2 & 3)
    • In this case, m = -2. So, v = -(-2)u = 2u

  3. Apply the Mirror Equation:

    • The mirror equation is: 1/v + 1/u = 1/f (Reference point 4).

  4. Substitute:

    • Replace 'v' with '2u' in the mirror equation: 1/(2u) + 1/u = 1/f

  5. Simplify:

    • To solve this equation, find the common denominator: (1 + 2) / 2u = 1/f
    • Simplify: 3/2u = 1/f
    • Rearrange: 2u = 3f

  6. Solve for 'u':

    • Substitute the focal length 'f'=-10 cm from Step 1: 2u = 3(-10)
    • Simplify: 2u = -30
    • Divide both sides by 2: u = -15 cm

  7. Alternative approach:

    • The reference also provides a direct magnification formula related to f and u, m=f/(f-u) (Reference point 6)
    • Using the given values, -2 = -10/(-10-u) or -2(-10-u)=-10
    • 20+2u=-10
    • 2u=-30 or u=-15 cm

Example Application

Using the example from the reference we found that:

  • The object distance (u) is -15 cm. This means the object is located 15 cm in front of the mirror.
  • The image distance (v) would be 2u, which is 2 * -15 = -30 cm (real, inverted image)

Key Takeaways

  • The mirror equation (1/v + 1/u = 1/f) and magnification formulas (m = -v/u or m = f/(f-u)) are crucial for solving these problems.
  • Always remember the sign conventions for concave mirrors:
    • Object distance (u) is always negative.
    • Focal length (f) is negative.
    • Image distance (v) is negative for real images and positive for virtual images.

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