MCAT Exam > MCAT Notes > Formula Sheets > Formula Sheet: Spherical Mirrors
Formula Sheet: Spherical Mirrors
Spherical Mirrors: Complete Formula Sheet
Mirror Fundamentals
Types of Spherical Mirrors
- Concave Mirror (Converging): Reflects light inward; center of curvature and focal point are in front of the mirror (real side)
- Convex Mirror (Diverging): Reflects light outward; center of curvature and focal point are behind the mirror (virtual side)
Key Definitions
- Center of Curvature (C): Center of the sphere from which the mirror is a section
- Radius of Curvature (r): Distance from mirror surface to center of curvature
- Focal Point (F): Point where parallel rays converge (concave) or appear to diverge from (convex)
- Focal Length (f): Distance from mirror surface to focal point
- Principal Axis: Line passing through center of curvature and center of mirror
- Object Distance (do or so): Distance from object to mirror surface
- Image Distance (di or si): Distance from image to mirror surface
- Object Height (ho): Height of the object
- Image Height (hi): Height of the image
Primary Mirror Equations
Focal Length and Radius of Curvature
\[f = \frac{r}{2}\]- f = focal length
- r = radius of curvature
- The focal point is located halfway between the mirror surface and center of curvature
- Units: meters (m), centimeters (cm)
Mirror Equation (Gaussian Form)
\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]- f = focal length
- do = object distance from mirror
- di = image distance from mirror
- All distances measured from the mirror surface along the principal axis
- Alternative notation: \(\frac{1}{f} = \frac{1}{s_o} + \frac{1}{s_i}\)
Alternative Forms of Mirror Equation
\[d_i = \frac{f \cdot d_o}{d_o - f}\]\[d_o = \frac{f \cdot d_i}{d_i - f}\]
\[f = \frac{d_o \cdot d_i}{d_o + d_i}\]
Sign Conventions
Standard Sign Convention (Real is Positive)
- Focal Length (f):
- Positive (+) for concave mirrors (converging)
- Negative (-) for convex mirrors (diverging)
- Object Distance (do):
- Positive (+) when object is in front of mirror (real object) - standard case
- Negative (-) when object is behind mirror (virtual object) - rare
- Image Distance (di):
- Positive (+) when image is in front of mirror (real image)
- Negative (-) when image is behind mirror (virtual image)
- Heights (ho, hi):
- Positive (+) when upright
- Negative (-) when inverted
Magnification
Linear Magnification Formula
\[m = -\frac{d_i}{d_o} = \frac{h_i}{h_o}\]- m = magnification (dimensionless)
- di = image distance
- do = object distance
- hi = image height
- ho = object height
- The negative sign accounts for image inversion
Magnification Interpretation
- |m| > 1: Image is enlarged (magnified)
- |m| <>: Image is reduced (diminished)
- |m| = 1: Image is same size as object
- m > 0: Image is upright (erect) relative to object
- m <>: Image is inverted relative to object
Image Height Formula
\[h_i = -m \cdot h_o = -\frac{d_i}{d_o} \cdot h_o\]Image Characteristics by Mirror Type
Concave Mirror (f > 0)
- Object beyond C (do > r or do > 2f):
- Real, inverted, diminished image
- Image between F and C
- di positive, |m| < 1,="" m=""><>
- Object at C (do = r or do = 2f):
- Real, inverted, same size image
- Image at C
- di = do = 2f, |m| = 1, m <>
- Object between C and F (2f > do > f):
- Real, inverted, enlarged image
- Image beyond C
- di positive, |m| > 1, m <>
- Object at F (do = f):
- No image formed (rays parallel after reflection)
- di = ∞
- Object between F and mirror (do <>:
- Virtual, upright, enlarged image
- Image behind mirror
- di negative, |m| > 1, m > 0
Convex Mirror (f <>
- Object anywhere in front of mirror:
- Virtual, upright, diminished image
- Image always behind mirror between F and mirror surface
- di negative, |m| < 1,="" m=""> 0
- Convex mirrors always produce virtual, upright, reduced images regardless of object position
Power of a Mirror
Mirror Power Formula
\[P = \frac{1}{f}\]- P = power of the mirror
- f = focal length in meters
- Units: diopters (D) where 1 D = 1 m-1
- Positive power: concave (converging) mirror
- Negative power: convex (diverging) mirror
Ray Tracing Rules
Principal Rays for Concave Mirrors
- Ray 1 (Parallel Ray): Ray parallel to principal axis reflects through focal point F
- Ray 2 (Focal Ray): Ray through focal point F reflects parallel to principal axis
- Ray 3 (Central Ray): Ray through center of curvature C reflects back on itself
- Ray 4 (Vertex Ray): Ray striking mirror vertex reflects symmetrically about principal axis
Principal Rays for Convex Mirrors
- Ray 1 (Parallel Ray): Ray parallel to principal axis reflects as if coming from focal point F (behind mirror)
- Ray 2 (Focal Ray): Ray directed toward focal point F (behind mirror) reflects parallel to principal axis
- Ray 3 (Central Ray): Ray directed toward center of curvature C (behind mirror) reflects back along same path
Special Cases and Conditions
Object at Infinity
\[d_o \to \infty \implies d_i = f\]- Parallel rays from distant object converge at focal point
- Used to locate focal point experimentally
- Magnification approaches zero
Object at Focal Point
\[d_o = f \implies d_i \to \infty\]- Reflected rays are parallel
- No real or virtual image formed
- Used in searchlights and flashlights
Plane Mirror (Special Case)
\[r \to \infty \implies f \to \infty\] \[d_i = -d_o\] \[m = +1\]- Image is virtual, upright, same size, and same distance behind mirror
- Can be considered limiting case of spherical mirror with infinite radius
Multiple Mirror Systems
Two Mirror System
- Image from first mirror becomes object for second mirror
- Total magnification: \(m_{total} = m_1 \times m_2\)
- Apply mirror equation sequentially for each mirror
- Distance to second mirror measured from position of first image
Spherical Aberration
Paraxial Approximation
- Mirror equations valid only for paraxial rays: rays close to and nearly parallel to principal axis
- Assumes small angles: sin θ ≈ tan θ ≈ θ (in radians)
- Spherical aberration: rays far from principal axis focus at different points
- Minimized by using mirrors with small aperture or parabolic mirrors
Important Relationships and Notes
Key Concepts for MCAT
- Real images: formed by actual convergence of light rays; can be projected on screen; always inverted for single mirror
- Virtual images: formed by apparent divergence of light rays; cannot be projected; always upright for single mirror
- Concave mirrors can produce both real and virtual images depending on object position
- Convex mirrors always produce virtual, upright, diminished images
- For real images: light actually passes through image location
- For virtual images: light appears to come from image location but does not actually pass through it
Problem-Solving Strategy
- Identify mirror type (concave or convex) and determine sign of f
- Identify given quantities and assign proper signs
- Choose appropriate equation(s)
- Solve algebraically before substituting numbers
- Check sign of result for physical meaning
- Verify image characteristics make sense for given configuration
Common MCAT Applications
- Makeup/shaving mirrors: concave mirrors with object inside focal length (virtual, enlarged, upright image)
- Security/surveillance mirrors: convex mirrors (wide field of view, virtual, reduced image)
- Telescopes: concave mirrors to collect and focus light
- Dental mirrors: may use either type depending on application
- Headlight reflectors: concave mirrors with light source at focal point
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