Key Takeaways
- The mirror equation 1/f = 1/do + 1/di works for both concave and convex mirrors
- Concave mirrors have positive focal lengths; convex mirrors have negative focal lengths
- Positive image distance = real image; negative image distance = virtual image
- Magnification m = -di/do tells you image size and orientation
- The radius of curvature is always twice the focal length (R = 2f)
Understanding the Mirror Equation
The mirror equation (also called the mirror formula) is a fundamental relationship in geometric optics that connects three essential quantities: focal length, object distance, and image distance. This equation allows physicists, engineers, and students to predict exactly where an image will form when light reflects off a curved mirror and what properties that image will have.
Unlike flat mirrors which simply reverse images, curved mirrors (both concave and convex) can create images that are larger, smaller, upright, inverted, real, or virtual depending on the object's position relative to the mirror's focal point and center of curvature. Understanding this behavior is essential for applications ranging from telescopes and car mirrors to dental mirrors and security systems.
1/f = 1/do + 1/di
How to Use the Mirror Equation Calculator
Step-by-Step Guide
Enter the Object Distance
Input the distance from your object to the mirror surface. This value is always positive because the object is in front of the mirror. Measure from the object's position to the mirror vertex (center of the mirror surface).
Enter the Focal Length
Input the focal length of your mirror. Use a positive value for concave mirrors (converging) and a negative value for convex mirrors (diverging). The focal length equals half the radius of curvature.
Calculate and Interpret Results
Click Calculate to find the image distance and magnification. A positive image distance means a real image forms in front of the mirror; a negative value indicates a virtual image behind the mirror.
Analyze Image Characteristics
Review the image type (real/virtual), orientation (upright/inverted), and relative size (enlarged/diminished/same size) to fully understand the optical properties of your mirror system.
Concave vs. Convex Mirrors: Complete Comparison
Understanding the differences between concave and convex mirrors is crucial for correctly applying the mirror equation. Each type has distinct properties and applications based on how they focus or diverge light rays.
| Property | Concave Mirror | Convex Mirror |
|---|---|---|
| Shape | Curves inward (like a cave) | Curves outward (bulges out) |
| Focal Length Sign | Positive (+f) | Negative (-f) |
| Light Behavior | Converges parallel rays | Diverges parallel rays |
| Image Types | Real or virtual (depends on position) | Always virtual |
| Image Orientation | Upright or inverted | Always upright |
| Image Size | Enlarged, same, or diminished | Always diminished |
| Applications | Telescopes, makeup mirrors, headlights, solar furnaces | Car side mirrors, security mirrors, ATM cameras |
Example Calculations
Example 1: Concave Mirror (Object Beyond Center)
Result: Real, inverted, diminished image formed 15 cm in front of the mirror.
Example 2: Convex Mirror (Any Position)
Result: Virtual, upright, diminished image formed 6.67 cm behind the mirror.
Understanding Sign Conventions
Proper use of sign conventions is essential for accurate calculations with the mirror equation. Here are the standard conventions used in physics:
- Object Distance (do): Always positive (objects are always in front of mirrors)
- Focal Length (f): Positive for concave mirrors, negative for convex mirrors
- Image Distance (di): Positive for real images (in front), negative for virtual images (behind)
- Magnification (m): Positive for upright images, negative for inverted images
- Heights: Positive above the principal axis, negative below
Pro Tip: The Radius of Curvature Shortcut
If you know the radius of curvature (R) of your mirror, you can quickly find the focal length using f = R/2. This is particularly useful when measuring mirrors directly, as the radius is often easier to determine than the focal point.
Real-World Applications of Mirror Optics
Understanding mirror equations has practical applications across many fields:
Concave Mirror Applications
- Reflecting Telescopes: Large concave mirrors collect and focus light from distant stars
- Makeup and Shaving Mirrors: When positioned close, create magnified upright images
- Car Headlights: Concave reflectors focus light into parallel beams
- Solar Furnaces: Concentrate sunlight to generate extreme heat
- Dental Mirrors: Provide magnified views of teeth when positioned correctly
- Satellite Dishes: Focus electromagnetic waves to a receiver
Convex Mirror Applications
- Car Side Mirrors: Provide wider field of view for safer driving
- Store Security Mirrors: Monitor large areas with a single mirror
- ATM Cameras: Hidden convex mirrors allow surveillance
- Road Intersection Mirrors: Help drivers see around blind corners
Common Mistake: Forgetting Sign Conventions
The most frequent error when using the mirror equation is forgetting to apply proper sign conventions. Remember: convex mirrors ALWAYS have negative focal lengths, and virtual images ALWAYS have negative image distances. Double-check your signs before calculating!
Understanding Magnification
Magnification tells us how the image size compares to the object size and whether the image is upright or inverted. The magnification formula for mirrors is:
m = -di/do = hi/ho
Interpreting magnification values:
- |m| > 1: Image is enlarged (larger than object)
- |m| = 1: Image is same size as object
- |m| < 1: Image is diminished (smaller than object)
- m > 0: Image is upright (same orientation as object)
- m < 0: Image is inverted (upside down)
Special Cases and Interesting Positions
Certain object positions produce notable results with concave mirrors:
- Object at infinity: Image forms at the focal point (used in telescopes)
- Object at center of curvature (C): Image forms at C, same size, real, inverted
- Object at focal point (F): No image forms (rays become parallel)
- Object between F and mirror: Virtual, upright, enlarged image (makeup mirror mode)
Pro Tip: Quick Image Location Rules
For concave mirrors: object beyond C gives diminished real image between F and C; object at C gives same-size image at C; object between C and F gives enlarged real image beyond C; object inside F gives enlarged virtual image behind mirror.
Frequently Asked Questions
The mirror equation is 1/f = 1/do + 1/di, where f is focal length, do is object distance, and di is image distance. This formula works for both concave and convex mirrors using proper sign conventions. For concave mirrors, f is positive; for convex mirrors, f is negative.
A positive image distance indicates a real image (formed in front of the mirror), while a negative image distance indicates a virtual image (formed behind the mirror). Real images can be projected onto a screen and are formed where light rays actually converge, while virtual images cannot be projected and appear to be behind the mirror.
Concave mirrors have a positive focal length and curve inward (like the inside of a spoon). They can form real or virtual images depending on object position. Convex mirrors have a negative focal length and curve outward (like the back of a spoon). They always form virtual, upright, diminished images regardless of object position.
Magnification (m) is calculated as m = -di/do, where di is image distance and do is object distance. Positive magnification means an upright image, negative means inverted. The absolute value tells you size: |m| > 1 means enlarged, |m| < 1 means diminished, and |m| = 1 means same size as the object.
Convex mirrors are used in car side mirrors because they always produce upright, diminished images with a wider field of view. This allows drivers to see more of the road behind them, improving safety. The tradeoff is that objects appear smaller and farther than they actually are, which is why these mirrors carry the warning "Objects in mirror are closer than they appear."
When an object is placed at the focal point of a concave mirror, reflected rays become parallel and no image is formed (mathematically, the image is at infinity). This is why the mirror equation gives undefined results when object distance equals focal length. This property is used in flashlights and headlights to create parallel light beams.
The radius of curvature (R) is twice the focal length (f). So R = 2f. For example, if a concave mirror has a focal length of 10 cm, its radius of curvature is 20 cm. This relationship holds for both concave and convex mirrors and is derived from the geometry of spherical surfaces.
While the thin lens equation has the same mathematical form (1/f = 1/do + 1/di), lenses have different sign conventions than mirrors. For lenses, converging lenses have positive focal lengths, and the image forms on the opposite side from the object. We recommend using a dedicated lens calculator for lens problems to ensure proper sign conventions are applied.