A rod is placed along the principal axis as shown. Find the length of the image. The focal length of the mirror is \(f = 10\,\text{cm}\).
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If an object is placed along the principal axis, find the image of both ends separately using the mirror formula.
The difference of their image positions gives the \textbf{image length}.
To find the length of the image of the rod placed along the principal axis of the concave mirror, we will employ the mirror formula and magnification concepts.
The mirror formula is given by:
\(\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\)
Where:
\(f\) is the focal length of the mirror,
\(v\) is the image distance,
\(u\) is the object distance.
Step 1: Identify object distances from the figure.} Point \(A\) is \(20\,\text{cm}\) from the mirror. \[ u_A = -20\,\text{cm} \] Point \(B\) is \(10\,\text{cm}\) further left. \[ u_B = -30\,\text{cm} \] Step 2: Find the image position of point \(A\).} \[ \frac{1}{f}=\frac{1}{v_A}+\frac{1}{u_A} \] \[ \frac{1}{10}=\frac{1}{v_A}-\frac{1}{20} \] \[ \frac{1}{v_A}=\frac{1}{10}+\frac{1}{20} \] \[ \frac{1}{v_A}=\frac{3}{20} \] \[ v_A=\frac{20}{3}\,\text{cm} \] Step 3: Find the image position of point \(B\).} \[ \frac{1}{10}=\frac{1}{v_B}-\frac{1}{30} \] \[ \frac{1}{v_B}=\frac{1}{10}+\frac{1}{30} \] \[ \frac{1}{v_B}=\frac{4}{30} \] \[ v_B=7.5\,\text{cm} \] Step 4: Calculate the image length.} \[ \text{Image length} = |v_B - v_A| \] \[ = \left|7.5 - \frac{20}{3}\right| \] \[ = \frac{45 - 40}{6} \] \[ = \frac{5}{6}\times12 \] \[ = 10\,\text{cm} \]
