The Thin Lens Equation

Uses the thin lens equation 1/a + 1/b = 1/f to find where the image forms and how large it is. A negative image distance means a virtual image on the same side as the object. For a diverging lens, enter a negative focal length.

Explanation

Where a lens forms its image follows from a single equation.

\dfrac{1}{a} + \dfrac{1}{b} = \dfrac{1}{f} \qquad \text{magnification} = \left| \dfrac{b}{a} \right|

Here $a$ is the object distance, $b$ the image distance and $f$ the focal length.

Reading the signs

This is the heart of it.

$b > 0$ — a real image. It forms on the far side of the lens, can be caught on a screen, and is inverted
$b < 0$ — a virtual image. It appears on the same side as the object, cannot be projected, and is upright
$f < 0$ — a diverging (concave) lens

Example

Place an object 30 cm from a converging lens of focal length 10 cm.

\dfrac{1}{30} + \dfrac{1}{b} = \dfrac{1}{10} \quad \Longrightarrow \quad b = 15\ \text{cm}

The magnification is $15 \div 30 = 0.5$ . Fifteen centimetres beyond the lens sits an inverted image at half size. This is the geometry of a camera and of a projector.

A magnifying glass makes a virtual image

Take the same lens but move the object inside the focal length, to $a = 5$ cm.

b = \dfrac{5 \times 10}{5 - 10} = -10\ \text{cm}

The negative $b$ means a virtual image, magnified twofold. Looking through the lens you see an upright image at twice the size, apparently 10 cm behind the object. That is a magnifying glass.

The equation is telling you why a magnifier only magnifies when the object is inside the focal point.

Watch out

Put the object exactly at the focal point ( $a = f$ ) and the refracted rays leave perfectly parallel: no image forms anywhere. This calculator returns an error for that case.