Understanding Optical Sign Conventions for Lenses

When dealing with lenses in physics, we use specific sign conventions to measure distances. The most common system is the Cartesian sign convention, which is similar to coordinate geometry. Understanding this convention is crucial for applying lens formulas correctly and determining the nature and position of images formed by lenses.

Standard Cartesian Sign Convention

Here are the key rules of the standard Cartesian sign convention:

• All distances are measured from the optical centre of the lens.
• The incident light is assumed to travel from left to right.
• Distances measured in the direction of the incident light (to the right of the optical centre) are taken as positive.
• Distances measured in the direction opposite to the incident light (to the left of the optical centre) are taken as negative.
• Heights measured upwards perpendicular to the principal axis are taken as positive.
• Heights measured downwards perpendicular to the principal axis are taken as negative.

Focal Length of Lenses

The focal length of a lens also follows specific sign conventions:

• For a convex lens (converging lens), the principal focus is located on the right side of the lens, where parallel rays converge after refraction. According to the sign convention, distances to the right are positive. Therefore, the focal length of a convex lens is considered positive.
• For a concave lens (diverging lens), parallel rays appear to diverge from a point on the left side of the lens after refraction. This point is the principal focus. According to the sign convention, distances to the left are negative. Therefore, the focal length of a concave lens is considered negative.

Analyzing the Given Statements

Let's analyze each statement provided in the question based on the standard Cartesian sign convention:

1. Focal length of a convex lens is positive.

   As explained above, for a convex lens, the focal point is on the right side, and distances to the right are positive. So, this statement is TRUE.

2. Focal length of a concave lens is negative.

   For a concave lens, the focal point is on the left side, and distances to the left are negative. So, this statement is also TRUE.

3. All measurements to the right of the optic centre are positive.

   According to the standard Cartesian sign convention, distances measured in the direction of incident light (which is typically rightward) from the optical centre are positive. So, this statement is TRUE.

4. All measurements to the left of the optic centre are positive.

   According to the standard Cartesian sign convention, distances measured opposite to the direction of incident light (which is typically leftward) from the optical centre are negative. Therefore, this statement is FALSE.

Based on the analysis, the false statement is Statement 4.

Conclusion on Sign Conventions

The standard sign convention dictates that measurements to the left of the optical centre are negative, not positive. This is a fundamental rule when applying lens formulas like the lens maker's formula or the thin lens formula: $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$, where $f$ is focal length, $v$ is image distance, and $u$ is object distance. Object distance $u$ is often negative because objects are usually placed to the left of the lens.

Revision Table: Optical Sign Conventions

Additional Information on Lenses and Sign Conventions

Understanding sign conventions is vital for predicting the characteristics of images formed by lenses. For example:

• A positive image distance ($v$) usually indicates a real image formed on the right side of the lens (where the refracted rays actually meet).
• A negative image distance ($v$) usually indicates a virtual image formed on the left side of the lens (where the refracted rays appear to meet).
• The magnification ($m = \frac{h_i}{h_o} = \frac{v}{u}$) sign tells us about the image orientation:
  • Positive magnification ($m > 0$) means the image is erect.
  • Negative magnification ($m < 0$) means the image is inverted.

Always apply the sign convention consistently when solving problems involving lenses and mirrors.