Lens and Mirror Equation Calculator

What is the Lens and Mirror Equation?

The lens and mirror equation (also known as the thin lens equation or Gaussian lens formula) is a fundamental relationship in optics that connects the focal length of a lens or curved mirror to the distances of the object and its image. The equation is expressed as 1/f = 1/dₒ + 1/dᵢ, where f is the focal length, dₒ is the object distance, and dᵢ is the image distance. This calculator also computes the linear magnification m = -dᵢ / dₒ, which tells you both the size ratio and orientation of the image.

The thin-lens equation and the spherical-mirror equation share the same algebraic form. What changes between the two is only the physical interpretation of each sign. A positive focal length corresponds to a converging optic (convex lens or concave mirror); a negative focal length corresponds to a diverging optic (concave lens or convex mirror). A positive image distance means a real image where rays actually converge, while a negative image distance means a virtual image where rays only appear to diverge from a point.

Thin Lens / Mirror Equation Formula

The fundamental relationship is:

1/f = 1/dₒ + 1/dᵢ

Where:

• f is the focal length of the lens or mirror (positive for converging, negative for diverging)
• dₒ is the distance from the optic to the object (positive for a real object in front of the optic)
• dᵢ is the distance from the optic to the image (positive for a real image, negative for a virtual image)

The linear magnification is given by:

m = -dᵢ / dₒ

Where m > 0 means the image is upright, m < 0 means inverted, and |m| is the size ratio (|m| > 1 enlarges, |m| < 1 reduces).

How to Use This Calculator

The calculator offers three solve modes:

• Solve for Image Distance: Enter the focal length and object distance to find where the image forms. Positive dᵢ means a real image on the opposite side; negative means a virtual image on the same side.
• Solve for Object Distance: Enter the focal length and desired image distance to find where to place the object.
• Solve for Focal Length: Enter known object and image distances to characterize an unknown lens or mirror.

Types of Images

• Real Image: Light rays actually converge at the image point. Can be projected onto a screen. dᵢ is positive for lenses (opposite side from object) and for mirrors (same side as object).
• Virtual Image: Rays only appear to diverge from the image point. Cannot be projected. dᵢ is negative for lenses (same side as object) and for mirrors (opposite side from object).
• Inverted Image: The image is flipped upside down relative to the object (negative magnification).
• Upright Image: The image has the same orientation as the object (positive magnification).

Sign Conventions

The sign convention is critical for correct use of the formula:

• Focal Length (f): Positive for converging optics (convex lens, concave mirror); negative for diverging optics (concave lens, convex mirror)
• Object Distance (dₒ): Always positive for a real object placed in front of the optic
• Image Distance (dᵢ): Positive for real images (formed on the opposite side for lenses, same side for mirrors); negative for virtual images

Applications of the Lens and Mirror Equation

• Camera Lenses: Predicting where the sensor must sit relative to the lens for a chosen subject distance
• Eyeglass Prescriptions: A +2 diopter lens has a focal length of 0.5 m and corrects farsightedness
• Telescopes: Long focal lengths produce small, sharp real images at the focal plane
• Projectors: Solving for object distance to project a sharp image onto a wall
• Magnifying Glasses: Placing an object inside the focal length produces an upright, enlarged virtual image
• Concave Shaving Mirrors: dₒ less than f gives an upright, enlarged virtual image of the face

Example Calculation

A converging lens has a focal length of 10 cm. An object is placed 30 cm in front of the lens. Find the image distance and magnification.

1. Identify: f = 10 cm (positive for converging lens), dₒ = 30 cm
2. Use dᵢ = (f x dₒ) / (dₒ - f) = (10 x 30) / (30 - 10) = 300 / 20 = 15 cm
3. Positive dᵢ means the image is real and on the opposite side of the lens
4. Magnification: m = -dᵢ / dₒ = -15 / 30 = -0.5
5. |m| = 0.5 means the image is half the height; negative means it is inverted