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Step-by-Step Solution
Step 1: Write the mirror formula
For a spherical mirror, the mirror formula is given by
$ \frac{1}{v} + \frac{1}{u} = \frac{1}{f} $,
where:
$u$ is the object distance (distance of the object from the mirror).
$v$ is the image distance (distance of the image from the mirror).
$f$ is the focal length of the mirror.
Step 2: Express $v$ in terms of $u$ and $f$
Rearranging the mirror formula,
$ \frac{1}{v} = \frac{1}{f} - \frac{1}{u} = \frac{u - f}{u f} $
Thus,
$ v = \frac{u f}{u - f} $.
Step 3: Write the expression for linear magnification
The linear magnification $m$ of a concave mirror is given by
$ m = -\frac{v}{u}. $
Substituting $v = \frac{uf}{\,u - f\,}$ into this, we get
$ m = - \frac{\frac{uf}{\,u - f\,}}{u} = - \frac{f}{\,u - f\,}. $
Hence,
$ m = \frac{f}{f - u} $
(note the negative sign indicates image inversion, but we are interested in the magnitude $|m|$ here).
Step 4: Analyze magnification near $u = f$
When the object approaches the focal point of the concave mirror ($u \rightarrow f$), the denominator $f - u$ tends to zero. As a result,
$ m \to \infty. $
In the magnitude sense, $|m|$ becomes very large near $u = f$.
Step 5: Analyze magnification at $u = 2f$
When the object is at $u = 2f$, substituting in the expression for magnification gives:
$ m = \frac{f}{f - 2f} = \frac{f}{-f} = -1. $
So, $|m| = 1$ at $u = 2f$.
Step 6: Sketch and identify the correct graph
1. As $u$ moves from values just greater than $f$ to values around $2f$ and beyond, $|m|$ decreases from a very large value down to $1$ at $u=2f$ and then continues decreasing towards zero as $u$ becomes very large.
2. Therefore, the graph of $|m|$ versus distance $u$ starts very high near $u = f$, intersects $|m| = 1$ at $u = 2f$, and then further decreases.
3. The correct option must show $|m|$ tending to infinity near $u = f$ and approaching 1 when $u = 2f.$
Conclusion
From these observations, the correct graphical representation is the one that shoots up to a very large value near $u = f$ and crosses $|m| = 1$ at $u = 2f$. That corresponds to the given correct option (Option 3).
