Numford
Formula

Mirror formula: 1/v + 1/u = 1/f

For a spherical mirror, 1/v + 1/u = 1/f and f = R/2. A concave mirror with f = -15 cm and an object 10 cm in front gives v = +30 cm and m = +3: a virtual, erect, enlarged image.

The ray diagram

Drag the object or type values. Signs follow the New Cartesian convention: distances are measured from the pole or optical centre, along the incident light is positive, against it negative, and heights above the axis are positive. So u is negative for a real object.

Image distance v+30 cmbehind the mirror
Magnification m+3erect, 3 times
Image height h′+6 cmobject +2 cm
Focal length f-15 cmR = 2f = -30 cm
NatureVirtual, erect, enlargedObject between P and F
Image positionBehind the mirrorEnlarged

Concave mirror of focal length -15 cm. An object +2 cm tall at u = -10 cm: the image is at v = +30 cm, virtual, erect, enlarged.

Unit
cm
Show the working
  1. f = R/2 = -30/2 = -15 cm, u = -10 cm (the object is in front, so u is negative), h = +2 cm.
  2. 1/v = 1/f - 1/u = 1/(-15) - 1/(-10) = 5/150 = 1/30
  3. v = +30 cm: positive, so the image is behind the mirror and virtual.
  4. m = -v/u = -(+30)/(-10) = +3: positive, so erect, and enlarged.
  5. h′ = m × h = +3 × +2 = +6 cm
Solved example

Object 10 cm from a concave mirror, f = 15 cm

  1. An object is placed 10 cm in front of a concave mirror of focal length 15 cm. Find the image.
  2. f = R/2 = -30/2 = -15 cm, u = -10 cm (the object is in front, so u is negative), h = +2 cm.
  3. 1/v = 1/f - 1/u = 1/(-15) - 1/(-10) = 5/150 = 1/30
  4. v = +30 cm: positive, so the image is behind the mirror and virtual.
  5. m = -v/u = -(+30)/(-10) = +3: positive, so erect, and enlarged.
  6. h′ = m × h = +3 × +2 = +6 cm
Derivation outline

Where the formula comes from

  • Take an object AB in front of a concave mirror and draw two rays from A: one parallel to the axis, which reflects through F, and one through C, which reflects back on itself. They meet at A′, the top of the image A′B′.
  • Triangles A′B′C and ABC are similar, so A′B′/AB = B′C/BC.
  • For small apertures the reflection point of the parallel ray is close to the pole, so triangles A′B′F and the one made at the mirror give A′B′/AB = B′F/FP.
  • Equate the two: B′C/BC = B′F/FP.
  • Measured as lengths, PB = -u, PB′ = -v, PF = -f and PC = -2f, because all four lie in front of the mirror. So B′C = v - 2f, BC = 2f - u, B′F = f - v and FP = -f.
  • Then (v - 2f)/(2f - u) = (f - v)/(-f). Cross multiply and cancel to get uv = fv + fu.
  • Divide by uvf: 1/v + 1/u = 1/f.
Sign convention

New Cartesian signs

  • The object is always placed to the left of the mirror or lens, so light falls on it from the left.
  • All distances parallel to the principal axis are measured from the pole of the mirror or the optical centre of the lens.
  • Distances measured to the right of the origin, along the direction of the incident light, are positive (+x).
  • Distances measured to the left of the origin, against the incident light, are negative (-x).
  • Distances measured upward and perpendicular to the principal axis, such as the height of an erect object or image, are positive (+y).
  • Distances measured downward, such as the height of an inverted image, are negative (-y).
Questions

Quick answers

What is the mirror formula?

1/v + 1/u = 1/f, relating the object distance u, the image distance v and the focal length f, all measured from the pole P. The focal length is half the radius of curvature, f = R/2.

Does the same formula work for convex mirrors?

Yes. Put f in with its sign: negative for a concave mirror, positive for a convex mirror. The sign of v then tells you where the image is.

What does a positive v mean for a mirror?

The image is behind the mirror, so it is virtual and erect. A negative v means the image is in front of the mirror, real and inverted.

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