A concave mirror for face viewing has focal length of 0.4 m. The distance at which you hold the mirr

Q: Why is the magnification taken as positive?

An upright image is always virtual. By sign convention, virtual images formed above the principal axis have a positive height, making the magnification $m = \frac{h'}{h}$ positive.

Q: Why is the focal length of the concave mirror negative?

According to the Cartesian sign convention, the focus of a concave mirror lies in front of the reflecting surface (against the direction of incident light), so its coordinate is negative.

Q: Can a concave mirror form a real, upright image?

No, a single spherical mirror can never form an image that is both real and upright. Real images formed by mirrors are always inverted.

Q: What does the negative sign in the final answer $u = -0.32\text{ m}$ indicate?

The negative sign simply confirms that the object (your face) is placed in front of the mirror, which aligns perfectly with our sign convention.

Q: Is there a faster way to solve this without finding $v$ first?

Yes! You can use the combined formula $m = \frac{f}{f - u}$. Plugging in $m = 5$ and $f = -0.4\text{ m}$ gives the answer directly in one algebraic step.

Question

A concave mirror for face viewing has focal length of $0.4	ext{ m}$. The distance at which you hold the mirror from your face in order to see your image upright with a magnification of $5$ is

Options:

(A) $0.16	ext{ m}$

(B) $1.60	ext{ m}$

(C) $0.32	ext{ m}$

(D) $0.24	ext{ m}$

Sigma Percentile · Accepted Answer

### The Magic of Concave Mirrors

Have you ever looked into a shaving mirror or a makeup mirror and marveled at how your face appears so much larger? This isn't just magic; it's the elegant physics of spherical mirrors at play. In this problem, we are tasked with finding the exact distance you need to hold a concave mirror to achieve a specific magnification. Let's dive into the mechanics of this fascinating phenomenon.

### Analyzing the Setup

The problem states that the concave mirror has a focal length of $0.4	ext{ m}$. Because it is a concave mirror, we must apply the Cartesian sign convention, which tells us that the focal length is negative: $f = -0.4	ext{ m}$.

We are also told that the image is **upright** and magnified $5$ times. In the realm of spherical mirrors, an upright image is always a **virtual image**. Virtual images are formed behind the mirror, and their heights are measured positively above the principal axis. Therefore, the magnification $m$ is strictly positive: $m = +5$.

### The Master Equation

To find the object distance $u$, we need to link the magnification to the mirror formula. We start with the magnification formula for spherical mirrors:

$$m = -\frac{v}{u}$$

Substituting our known magnification, we get:

$$5 = -\frac{v}{u}$$

Rearranging this gives us a direct relationship between the image distance and the object distance:

$$v = -5u$$

This equation beautifully illustrates that the virtual image is formed behind the mirror at a distance five times greater than the object's distance in front of the mirror.

### Final Calculation

Now, we bring in our heavy artillery—the mirror formula:

$$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$$

Let's carefully substitute our expression for $v$ and our known focal length $f$:

$$\frac{1}{-5u} + \frac{1}{u} = \frac{1}{-0.4}$$

To solve this, we find a common denominator on the left side:

$$\frac{-1 + 5}{5u} = -\frac{1}{0.4}$$

$$\frac{4}{5u} = -2.5$$

Now, it's just a matter of simple algebra to isolate $u$:

$$5u = \frac{4}{-2.5} = -1.6$$

$$u = -0.32	ext{ m}$$

The negative sign is a reassuring confirmation of our sign convention—it simply means the object (your face) must be placed $0.32	ext{ m}$ in front of the mirror.

### Pro-Tip: The Speed Formula

While the step-by-step method is fantastic for building intuition, competitive exams demand speed. You can bypass finding $v$ entirely by using the combined magnification formula:

$$m = \frac{f}{f - u}$$

Let's plug in our values and watch the magic happen in one step:

$$5 = \frac{-0.4}{-0.4 - u}$$

$$-2.0 - 5u = -0.4$$

$$-5u = 1.6 \Rightarrow u = -0.32	ext{ m}$$

Mastering these alternate forms will give you a significant edge in your physics journey!

Animated Solution for Physics - Optics: A concave mirror for face viewing has focal length of $0.4 m$ . The distance at which you hold the mirror from your face in order to see your image upright with a magnification of $5$ is

Select Answer:

Visualized Solution

\text{Visualizing the Setup}

\text{Magnification Sign Convention}

\text{Magnification Formula}

\text{Relation between } v \text{ and } u

\text{The Mirror Formula}

\text{Substituting Values}

\text{Algebraic Computation}

\text{Final Answer}

\text{The Way Forward}

The Sigma Insight: Spherical Mirror

The Magic of Concave Mirrors

Analyzing the Setup

The Master Equation

Final Calculation

Pro-Tip

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