physics

Herschel's condition

Herschel's condition is a physics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Herschel's condition rather than just read about it. In short: In optics, the Herschel's condition is a condition for an optical system to produce sharp images for objects over an extended axial range, i.e. for objects displaced along the optical axis. It was formulated by John Herschel.

Key takeaways

Herschel's condition belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Herschel's condition to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Herschel's condition from memory before moving on to harder problems.

Reference excerpt

In optics, the Herschel's condition is a condition for an optical system to produce sharp images for objects over an extended axial range, i.e. for objects displaced along the optical axis. It was formulated by John Herschel.

Mathematical formulation The Herschel's condition in mathematical form is

1 − cos ⁡ α o 1 − cos ⁡ α i = 1 − cos ⁡ β o 1 − cos ⁡ β i = n i 2 n o 2 | M T | 2 {\displaystyle {\frac {1-\cos \alpha _{\mathrm {o} }}{1-\cos \alpha _{\mathrm {i} }}}={\frac {1-\cos \beta _{\mathrm {o} }}{1-\cos \beta _{\mathrm {i} }}}={\frac {n_{i}^{2}}{n_{o}^{2}}}|M_{T}|^{2}}

where α o , β o {\displaystyle \alpha _{o},\beta _{o}} are the object side ray angles, α i , β i {\displaystyle \alpha _{i},\beta _{i}} are the image side ray angle. n o , n i {\displaystyle n_{o},n_{i}} are the object and image side refractive index, and M T {\displaystyle M_{T}} is the transverse magnification. This condition can be derived by the Fermat's principle. This condition can also be expressed as

M L = n o sin 2 ⁡ ( α o / 2 ) n i sin 2 ⁡ ( α i / 2 ) = n o ( 1 − cos ⁡ α o ) n i ( 1 − cos ⁡ α i ) and M T = n o sin ⁡ ( α o / 2 ) n i sin ⁡ ( α i / 2 ) {\displaystyle M_{L}={\frac {n_{o}\sin ^{2}(\alpha _{\mathrm {o} }/2)}{n_{i}\sin ^{2}(\alpha _{\mathrm {i} }/2)}}={\frac {n_{o}(1-\cos \alpha _{\mathrm {o} })}{n_{i}(1-\cos \alpha _{\mathrm {i} })}}{\text{ and }}M_{T}={\frac {n_{o}\sin(\alpha _{\mathrm {o} }/2)}{n_{i}\sin(\alpha _{\mathrm {i} }/2)}}}

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Herschel's condition

Start with the simplest possible case. Write down what Herschel's condition claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Herschel's condition before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Herschel's condition ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Herschel's condition

In research: Herschel's condition appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Herschel's condition in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Herschel's condition is common in secondary-school and first-year university syllabi. It links to neighbouring topics Geometrical optics, Glass physics, Microscopes, so understanding it makes those chapters shorter.
In everyday life: Look for Herschel's condition outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Herschel's condition in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Herschel's condition means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Herschel's condition out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Herschel's condition in simple terms?

Why does Herschel's condition matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Herschel's condition?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Herschel's condition.