Question

An object is \(6.0 \mathrm{~cm}\) from a thin lens along the axis of the lens. If the lens has a focal length of \(9.0 \mathrm{~cm},\) determine the image distance.

Short answer

Answer: The image distance is -18.0 cm.

Step-by-step solution

Write down the known values

We are given the object distance \(d_o = 6.0 ~cm\) and the focal length \(f = 9.0 ~cm\). We want to find the image distance \(d_i\).

Apply the lens formula

Using the lens formula, we will plug in the known values and solve for \(d_i\): $$ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} $$ $$ \frac{1}{9.0} = \frac{1}{6.0} + \frac{1}{d_i} $$

Solve for the image distance

We need to find the value of \(\frac{1}{d_i}\) by isolating it: $$ \frac{1}{d_i} = \frac{1}{9.0} - \frac{1}{6.0} $$ Now, find the common denominator for the two fractions on the right side of the equation, which is 18: $$ \frac{1}{d_i} = \frac{2}{18} - \frac{3}{18} $$ Combine the two fractions: $$ \frac{1}{d_i} = -\frac{1}{18} $$ Finally, take the reciprocal of both sides to get the image distance: $$ d_i = -18 \mathrm{~cm} $$ Therefore, the image distance is \(-18.0 \mathrm{~cm}\). The negative sign indicates that the image is on the same side of the lens as the object, meaning it is a virtual image.