Reflection of Light: Spherical Mirrors

The image is virtual, erect, and diminished

Step 1: m = +0.5. The sign of magnification tells us about orientation. Step 2: A positive magnification means the image is erect (same side as object with respect to the principal axis). Step 3: A negative magnification means the image is inverted (real image). Step 4: |m| = 0.5 < 1, which means the image is smaller (diminished) than the object. Step 5: So m = +0.5 → virtual, erect, and diminished. m = -0.5 would be real, inverted, and diminished. m = +2 would be virtual, erect, and magnified.

Negative (-)

Step 1: In the New Cartesian Sign Convention, the pole P is the origin and the principal axis is the x-axis. Step 2: All distances are measured from the pole. Distances to the left (where the object is placed) are negative. Step 3: For a concave mirror, the focus F lies in front of the mirror, i.e., on the same side as the object. Step 4: Since the focus is to the left of the pole, PF is measured in the negative x-direction. Step 5: Therefore, the focal length f of a concave mirror is negative. A convex mirror's focus is behind the mirror (positive side), so its f is positive.

Virtual, erect, and magnified

Step 1: When the object is between the pole (P) and focus (F) of a concave mirror, the object distance u satisfies f < u < 0 (in magnitude terms, u < f). Step 2: Applying the mirror formula, 1/v = 1/f - 1/u. With f < 0 and |u| < |f|, 1/v comes out positive. Step 3: Positive v means the image is formed behind the mirror, so it is virtual. Step 4: A virtual image formed by a mirror is always erect. Step 5: Since |v| > |u| in this case, the image is magnified. This is the principle used in shaving mirrors and makeup mirrors.

30 cm in front of the mirror

Step 1: For a convex mirror, f = +15 cm. The image is 1/3 the size, so m = +1/3 (positive since convex mirror always gives erect image). Step 2: Use m = f/(f - u): 1/3 = 15/(15 - u). Step 3: Cross multiply: 15 - u = 45, so u = 15 - 45 = -30 cm. Step 4: u = -30 cm means the object is 30 cm in front of the mirror (negative sign = in front). Step 5: Verify: 1/v = 1/15 - 1/(-30) = 1/15 + 1/30 = 3/30, v = 10 cm; m = -10/(-30) = +1/3. Correct! '10 cm' is the image distance, not the object distance.