Angular magnification in normal adjustment.

The idea

Angular magnification compares the angle subtended at the eye by the final image with the angle subtended by the object when viewed directly. In normal adjustment for a two-lens astronomical telescope, the final image is at infinity, so the eye is relaxed. The intermediate image formed by the objective lies at the principal focus of both the objective and the eyepiece, so the lens separation equals f₀ + fₑ. For small angles, the angular magnification is M = β/α, where β is the angle subtended by the image at the eye and α is the angle subtended by the object at the unaided eye. Ray geometry gives M = f₀/fₑ. For example, an objective of focal length 1.20 m with an eyepiece of focal length 0.030 m gives M = 40. The image is inverted, and the magnification is a pure number with no unit.

What earns the marks

• Defines angular magnification as the ratio of the angle subtended at the eye by the final image to the angle subtended at the unaided eye by the object, M = β/α.
• States that in normal adjustment the final image is at infinity and the eye is relaxed.
• Explains that the intermediate image is formed at the common principal focus of the objective and eyepiece, so the separation of the two lenses is f₀ + fₑ.
• Derives or applies M = f₀/fₑ for an astronomical telescope in normal adjustment.
• Uses the small-angle approximation tan θ ≈ θ in radians when relating angles to focal lengths.
• Recognises that the final image is inverted and that angular magnification is a dimensionless ratio.

Common mistakes and how to put them right

• Using M = fₑ/f₀ instead of M = f₀/fₑ.

  Put it right by checking that a long objective focal length and short eyepiece focal length give a large magnification.

• Treating the lens separation as f₀ − fₑ.

  Put it right by recalling that in normal adjustment the intermediate image is at the common focal point, so the separation is f₀ + fₑ.

• Forgetting to convert focal lengths to the same unit before dividing.

  Put it right by converting both to metres, for example 1.20 m and 0.030 m.

• Describing the final image as real and inverted.

  Put it right by stating that in normal adjustment the final image is virtual, inverted and at infinity.

Exam tips

• Sketch a ray diagram with the intermediate image at the common principal focus and mark the angles α and β clearly before substituting into M = β/α.
• When calculating M, convert all focal lengths to metres first, then divide; the result has no unit.
• If asked to explain normal adjustment, state that the final image is at infinity and the eye is relaxed, then link this to the lens separation f₀ + fₑ.