Question

Where do you have to place an object in front of a concave mirror with focal length \(f\) for the image to be the same size as the object? a) at \(d_{\mathrm{o}}=0.5 f\) b) at \(d_{\mathrm{o}}=f\) c) at \(d_{\mathrm{o}}=2 f\) d) at \(d_{\mathrm{o}}=2.5 f\) e) none of the above

Short answer

Answer: b) at \(d_{\mathrm{o}}=f\).

Step-by-step solution

Recall the mirror formula

The mirror formula is given by \(\frac{1}{f} = \frac{1}{d_{\mathrm{o}}} + \frac{1}{d_{\mathrm{i}}}\).

Set \(d_{\mathrm{o}}\) equal to \(d_{\mathrm{i}}\)

As the image size is the same as the object size, \(d_{\mathrm{i}}\) must be equal to \(d_{\mathrm{o}}\). So we will replace \(d_{\mathrm{i}}\) with \(d_{\mathrm{o}}\) in the mirror formula: \(\frac{1}{f} = \frac{1}{d_{\mathrm{o}}} + \frac{1}{d_{\mathrm{o}}}\).

Solve for \(d_{\mathrm{o}}\)

We first combine the terms on the right side of the equation: \(\frac{1}{f} = 2 \times \frac{1}{d_{\mathrm{o}}}\). Next, we multiply both sides of the equation by d_o and by f: \(d_{\mathrm{o}}(f) = 2(f)\). Finally, we divide both sides by 2: \(d_{\mathrm{o}} = f\).

Check the answer choices

The object must be placed at d_o = f, which is answer choice (b). Therefore, the correct answer is: b) at \(d_{\mathrm{o}}=f\).