Sign conventions make lens problems consistent. By assigning positive and negative directions to focal points, object positions, and image positions, you can interpret lens behavior correctly and avoid common mistakes in AP Physics 2.

Reference points for lens measurements

In lens problems, every location is measured relative to the lens. For a thin lens, the reference position is the center of the lens, sometimes called the lens midline. Distances are measured along the principal axis, the straight line through the center of the lens.

A focal point is an important reference position for a lens.

To assign signs, first choose the direction of incoming light. In many AP problems, light travels from left to right. That lets you identify the incident side, where light arrives, and the transmitted side, where light leaves.

Physicists use a sign convention so that these positions are described consistently.

If the sign is wrong, the physical interpretation of the object, image, or focal point is usually wrong too.

Focal points on both sides of a lens

A lens has two focal points, one on each side of the lens.

Ray diagram for a converging (convex) lens showing the principal axis, lens center, and focal points on both sides (commonly labeled $F$ and $F'$). The three principal rays (parallel ray through the far focal point, focal ray emerging parallel, and central ray) intersect to locate the image. This picture makes the “two focal points” idea concrete while reinforcing how ray-tracing encodes the sign conventions for $f$ and $d_i$. Source

If the lens is surrounded by the same medium on both sides, these focal points are the same distance from the center. That is why a lens is said to have focal points on both sides.

For a converging lens, rays parallel to the principal axis come together on the far side of the lens. If light enters from the left, the relevant focal point is on the right, so the focal length is positive, or $f>0$. The second focal point still exists on the left side and becomes important if light enters from the opposite direction.

For a diverging lens, rays parallel to the axis spread out after passing through the lens and appear to come from a point on the incident side. If light enters from the left, that focal point is on the left, so the focal length is negative, or $f<0$. A second focal point still exists on the right side for light approaching from that side.

The two focal points are geometric features of the lens. The sign of focal length tells you whether the lens converges or diverges for the chosen direction of incoming light.

Sign conventions for object and image locations

The object location is described by the object distance, measured from the lens. In the usual AP setup, the object is on the incident side, so the object distance is positive, or $d_o>0$.

The image location is also measured from the lens. If the image forms on the transmitted side of the lens, the image distance is positive, or $d_i>0$. If the image appears on the same side as the object, the image distance is negative, or $d_i<0$.

Ray diagram for a diverging (concave) lens showing how refracted rays diverge and how backward extensions locate the virtual image on the object side. The focal point on the incident side is used to construct the “parallel ray” rule for diverging lenses. This visual strongly supports why a virtual image corresponds to $d_i<0$ under the usual AP sign convention. Source

These signs carry physical meaning:

• A positive image distance means the image is on the opposite side of the lens from the object.

• A negative image distance means the image is on the same side of the lens as the object.

• A positive focal length identifies a converging lens.

• A negative focal length identifies a diverging lens.

A common mistake is to think that positive always means “to the right.” That is only true if the problem is set up with light entering from the left. The more reliable rule is to think in terms of incident side and transmitted side.

Using the conventions consistently

Before interpreting any lens situation, establish:

• the position of the lens center

• which side is the incident side

• which direction will count as positive

Once that is done, keep the same convention throughout the problem.

Another common error is forgetting that a converging lens still has a focal point on the object side, and a diverging lens still has a focal point on the far side.

Two focal points do not mean two different focal lengths. They are symmetric reference points on opposite sides of the lens, while the sign of $f$ identifies the lens type for the chosen light direction.

In many AP lens problems, correct signs help you decide whether a description makes physical sense. If an image is on the same side as the object, the image distance should be negative. If parallel rays are brought together after passing through the lens, the focal length should be positive. The first step in any lens description is to identify which side of the lens each point lies on.