Mirror and Magnification Formulas

1. Setting the Stage: From Geometry to Algebra

Let's begin with the familiar geometric setup from our ray tracing exercises. We'll consider a concave mirror forming a real, inverted image. The principles we derive, however, will be universally applicable to convex mirrors and virtual images, provided we use a consistent sign convention.

Our goal is to find a mathematical relationship between the object distance $(u)$, the image distance $(v)$, and the focal length $(f)$. The derivation hinges on identifying pairs of similar triangles within the ray diagram.

2. Deriving the Mirror Formula

The most rigorous derivation starts by considering specific rays and applying geometric principles. We will use a derivation that formally identifies similar triangles based on the law of reflection.

Let's break down the logic of that derivation to ensure every step is clear. The entire proof rests on finding two different expressions for the ratio of image height to object height ($h_i/h_o$) and then equating them.

Using the standard notation of $u$, $v$, and $f$, and referring to a diagram similar to the one in the resource:

1. First Pair of Similar Triangles: Consider the ray that strikes the pole (vertex) P. The principal axis acts as the normal. By the law of reflection, the angle of incidence equals the angle of reflection. This creates two similar right-angled triangles, one with the object and one with the image.

   • From these triangles, we get the relation: $\frac{h_i}{h_o}=\frac{v}{u}$. If we account for the fact that the real image is inverted ($h_i$ is negative), we write: $\frac{-h_i}{h_o}=\frac{v}{u}$, or $\frac{h_i}{h_o}=-\frac{v}{u}$.

2. Second Pair of Similar Triangles: Now consider the ray that passes through the center of curvature, C. This ray reflects back on itself. This creates another pair of similar right-angled triangles, this time with vertices at C.

   • The base of the object triangle is $u-R$.
   • The base of the image triangle is $R-v$.
   • From these triangles, we get: $\frac{-h_i}{h_o}=\frac{R-v}{u-R}$.

3. Combining and Simplifying: We now have two expressions for the ratio of heights. Equating them gives:

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

   Cross-multiplying gives:

   $$\begin{gathered} v(u-R)=u(R-v) \\ uv-vR=uR-uv \\ 2uv=uR+vR \end{gathered}$$

   Now, we divide the entire equation by $uvR$:

   $$\begin{gathered} \frac{2uv}{uvR}=\frac{uR}{uvR}+\frac{vR}{uvR} \\ \frac{2}{R}=\frac{1}{v}+\frac{1}{u} \end{gathered}$$

4. The Small-Angle Approximation: Finally, we use the relationship you derived in a previous lesson, $f=R/2$, which is based on the paraxial ray approximation (small angles). Substituting $R=2f$ into our equation gives the celebrated Mirror Formula:

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

3. The Cartesian Sign Convention

The derivation above was for a specific case (concave mirror, real image). For the mirror formula to be universally applicable to all situations (including convex mirrors and virtual images), we must adhere to a strict set of rules known as the Cartesian Sign Convention.

• Origin: The pole (P) of the mirror is the origin (0,0).
• Principal Axis: The principal axis is the x-axis.
• Direction of Incident Light: Light is always assumed to travel from left to right.
• Distances:
  • All distances measured along the direction of incident light (to the right of the pole) are positive.
  • All distances measured opposite to the direction of incident light (to the left of the pole) are negative.
• Heights:
  • Heights measured upwards and perpendicular to the principal axis are positive.
  • Heights measured downwards and perpendicular to the principal axis are negative.

Applying this convention:

• Object distance (u): Since the object is always placed to the left of the mirror, $u$ is always negative.
• Focal length (f):
  • For a concave mirror, the focus is on the left, so $f$ is negative.
  • For a convex mirror, the focus is on the right, so $f$ is positive.
• Image distance (v):
  • If the image is real (formed on the left), $v$ is negative.
  • If the image is virtual (formed on the right), $v$ is positive.

Note: Your textbook and some resources may use a different convention. The Cartesian convention described here is standard in modern physics and is the most consistent. While the derivation was done using magnitudes, applying this sign convention allows the final formula to work for all cases. For example, if we substitute $u\to -u$, $v\to -v$, and $f\to -f$ (for a real image from a concave mirror) into our derived formula, we get $\frac{1}{-f}=\frac{1}{-u}+\frac{1}{-v}$, which is algebraically identical.

4. Deriving the Magnification Formula

Magnification tells us about the size and orientation of the image relative to the object.

Linear Magnification ($m$) is defined as the ratio of the height of the image ($h_i$) to the height of the object ($h_o$):

Now, let's find a more useful expression for $m$ in terms of $u$ and $v$.

As you saw in the resource and our breakdown in step 2.1, the similar triangles formed by the ray striking the pole give us:

Here we explicitly include the negative sign for the inverted image height $h_i$.

Rearranging this to match the definition of magnification:

Therefore, the Magnification Formula is:

The sign of $m$ tells you the image orientation:

• If $m<0$, the image is real and inverted.
• If $m>0$, the image is virtual and upright.

The magnitude of $m$ tells you the size:

• If $|m|>1$, the image is magnified.
• If $|m|<1$, the image is diminished.
• If $|m|=1$, the image is the same size as the object.

Conclusion

In this lesson, we have moved from geometric constructions to algebraic formulas, providing you with powerful tools for quantitative analysis of spherical mirrors.

• Key Takeaways:
  • The Mirror Formula, $\frac{1}{f}=\frac{1}{u}+\frac{1}{v}$, relates the object distance, image distance, and focal length.
  • The Magnification Formula, $m=-\frac{v}{u}$, relates magnification to the object and image distances.
  • These formulas are universally applicable if you consistently use the Cartesian Sign Convention, where distances are measured from the pole and are negative to the left and positive to the right.
  • The derived relationship $m=\frac{f}{f-u}$ shows how these fundamental equations can be manipulated to solve specific problems efficiently.

Preview of the Next Lesson:
You have now derived the necessary theoretical tools. The next step is to apply them. In our next lesson, we will focus on solving a variety of numerical problems involving spherical mirrors. This will solidify your understanding of the formulas and the crucial sign convention, preparing you for complex problem-solving scenarios.