Problem Statement
Solve the optics problem: An object is in water ($n_1 = 1.33$) 30 cm from a convex glass surface ($n_2 = 1.5$) with radius $R = 10$ cm. Find the image position. Refraction at a single spherical surface: $$\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}$$ With $u = -30$ cm, $R = +10$ cm: $$\frac{1.5}{v}-\frac{1.33}{-30}=\frac{0
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.
- See the step-by-step solution for the specific equations applied.
- All quantities are in SI units unless otherwise stated.
Step-by-Step Solution
Step 1 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 2 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 3 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Worked Calculation
$$\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}$$
$$\frac{1.5}{v}-\frac{1.33}{-30}=\frac{0
Given Information
- Refractive index $n$ or focal length $f$ as given
- Object distance $u$ (negative for real objects in Cartesian convention)
- Radius of curvature $R$ or lens/mirror parameters as given
Physical Concepts & Formulas
Geometric optics is governed by Snell’s Law ($n_1 \sin\theta_1 = n_2 \sin\theta_2$) at each interface and the mirror/lens formulas in the paraxial limit. The Cartesian sign convention assigns the incident direction as positive: distances measured opposite to light are negative. For mirrors: $1/v + 1/u = 2/R = 1/f$. For thin lenses: $1/v – 1/u = 1/f$. Magnification $m = -v/u$ for mirrors and $m = v/u$ for lenses. A real image has $v > 0$ for lenses; a virtual image has $v < 0$.
- $\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}$ — mirror formula
- $\dfrac{1}{v} – \dfrac{1}{u} = \dfrac{1}{f}$ — thin lens formula (Cartesian)
- $n_1 \sin\theta_1 = n_2 \sin\theta_2$ — Snell’s Law
- $m = -v/u$ (mirror) or $m = v/u$ (lens) — linear magnification
Step-by-Step Solution
Step 1 — Apply correct sign convention: Real object: $u < 0$. Concave mirror/converging lens: $f > 0$.
Step 2 — Use the appropriate formula:
$$
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Step 3 — Solve for image distance $v$ and compute magnification.
Worked Calculation
Substituting all values with units:
Substitute given values of $u$, $f$ into the formula and solve for $v$.
Answer
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Answer
$$\boxed{\frac{1}{v} – \frac{1}{u} = \frac{1}{f}}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
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