AP Unit 13 Solutions

Draw horizontal rays pointing right, perpendicular to the vertical fronts. Large optical surfaces compared with wavelength allow the geometric ray approximation.

The laser can still be coherent and approximately monochromatic, but the observed diffraction/interference needs phase and wave superposition. Straight rays alone do not predict the spreading and alternating bands.

θᵢ = 90.0° − 40.0° = 50.0°. θᵣ = 50.0°. Draw incoming toward the point and outgoing away on the same side of the surface.

Smooth flat surface: approximately parallel normals, hence a common reflected direction (specular). Rough sheet: varying normals, hence many reflected directions (diffuse). Each patch obeys equal local angles.

f = R/2 = +20.0 cm. 1/d_i = 1/20.0 − 1/30.0 = 1/60.0 cm⁻¹, so d_i = +60.0 cm.

m = −60.0/30.0 = −2.00, h_i = −4.00 cm.

From the tip draw a parallel ray reflecting through F; a ray to the vertex reflecting at equal angles to the axis; and a ray through F reflecting parallel. Their reflected portions meet below the axis 60.0 cm in front.

The vertex ray is not generally normal to the mirror and does not generally retrace its path. A ray through the center of curvature is a separate useful normal-incidence ray.

Plane: 1/f = 0, d_i = −6.00 cm and m = +1. Convex: 1/d_i = −1/12.0 − 1/6.00 = −1/4.00, so d_i = −4.00 cm and m = +2/3. Concave: 1/d_i = 1/12.0 − 1/6.00 = −1/12.0, so d_i = −12.0 cm and m = +2.00.

All are virtual and upright in these cases. Plane is same size, convex reduced, concave enlarged.

A parallel incident ray reflects so its backward extension runs through the virtual focus behind the mirror. The vertex ray’s backward extension intersects it at the virtual tip.

A screen behind the mirror cannot receive this image because no reflected light rays actually converge there.

θ₁ = 35.0°. sin θ₂ = sin35.0°/1.60 = 0.3585, so θ₂ = 21.0°.

v = c/1.60 = 1.875 × 10⁸ m/s and λ = 540/1.60 = 337.5 nm. Frequency is unchanged.

Normal incidence gives θ₂ = 0, so the ray does not bend. The same index gives the same material speed and wavelength as before.

At the critical angle θ₂ = 90°, so n₁sinθ_c = n₂. Thus θ_c = sin⁻¹(n₂/n₁) = sin⁻¹(1.00/1.50) = 41.8°.

At 30.0°, θ₂ = sin⁻¹(1.50sin30.0°) = 48.6°, away from the normal; some reflection may also occur. At θ_c the transmitted limiting ray is along the surface. At 50.0° there is no transmitted ray and all light reflects at 50.0° to the normal.

Air-to-block has n₁ < n₂, so n₂/n₁ > 1 and no critical angle for total internal reflection exists. The ray bends toward the normal instead.

1/d_i = 1/10.0 − 1/25.0 = 0.0600 cm⁻¹, so d_i = +16.7 cm. m = −16.7/25.0 = −0.667 and h_i = −1.00 cm.

Mark F at 10.0 cm on each side. Parallel incoming ray goes through the far/right focus. Center ray continues straight. A ray through the near/left focus exits parallel; all transmitted rays meet at the image tip.

Diverging: 1/d_i = −1/12.0 − 1/8.00 = −5/24.0, so d_i = −4.80 cm and m = +0.600. Converging: 1/d_i = 1/12.0 − 1/8.00 = −1/24.0, so d_i = −24.0 cm and m = +3.00.

Both have negative d_i and positive m: virtual and upright. The diverging image is reduced; the converging lens with object inside f gives an enlarged image.

The parallel incident ray leaves as if from the near/incident focus. The center ray proceeds straight. Their backward extensions meet on the object side at 4.80 cm. A ray aimed at the far focus exits parallel.

Trace the block outline and normal; align a narrow ray through the curved-face center so it meets that curved surface normally. Measure the paired directions at the flat face, vary angle, and repeat markings without moving the outline. Hold wavelength/source and block fixed.

At normal incidence on the curved face, Snell gives n₁sin0° = n₂sinθ₂, so θ₂ = 0°. The ray therefore passes straight there; the measured angle change comes from the flat interface.

For reflection measure incident and reflected angles relative to the local flat-face normal and compare them. Use a smooth face so the specular direction is well defined.

Plot sinθ_air vertically against sinθ_block horizontally. Snell gives y = (n_block/n_air)x; with n_air ≈ 1, the slope is n_block. The three ratios are approximately 1.50, 1.50, and 1.50.

A fit constrained through the origin, as the ideal Snell model predicts, has slope Σxy/Σx² ≈ 1.4986. The rounded angle data therefore give n_block ≈ 1.50. With real uncertainties, also inspect whether an unconstrained fit’s intercept is consistent with zero rather than forcing away a systematic offset. The completed graph below shows the plotted points and line.

A fixed angular reading error is a larger fraction of a very small angle; use several well-separated angles and a fit with uncertainty rather than one ratio.

Send light within the block toward the flat block-to-air interface while entering the curved face normally. For n ≈ 1.50, θ_c ≈ 41.8°. Near it the transmitted ray approaches the surface; above it the transmitted ray disappears and reflection remains.

Both positive-f systems give d_i = +30.0 cm and m = −1: real, inverted, same size. The mirror image is in front on the reflected side; the lens image is on the transmitted side.

Both negative-f systems give d_i = −10.0 cm and m = +1/3: virtual, upright, reduced. The mirror image is behind the mirror; the lens image is on the incident/object side.

The mirror vertex ray reflects with equal angles to the axis; the lens center ray proceeds approximately straight. A parallel mirror ray reflects through the front focus; a parallel lens ray transmits through the far focus.

$1/d_o+1/d_i=1/f\quad\Rightarrow\quad1/d_i=-1/d_o+1/f$. Let x = 1/d_o and y = 1/d_i. This is y = −x + 1/f, predicting dimensionless slope −1 and vertical intercept 1/f in cm⁻¹.

All three reciprocal sums are 0.08333 cm⁻¹, so the intercept gives f = 1/0.08333 cm⁻¹ = +12.0 cm. Actual experimental data would require a slope/intercept uncertainty, rather than an exact line imposed on rounded values. The completed graph below shows the decreasing fit and all three measurements.

The images can be projected on the outgoing-side screen, indicating actual ray intersections.

Keep object, lens, and screen centered on one axis; measure from the lens midline, adjust the screen for best focus, bracket the range of equally acceptable focus, and repeat each setting. Record ruler resolution, alignment, and focus range.

At d_o = 8.00 cm, 1/d_i = 1/12.0 − 1/8.00 = −1/24.0, so d_i = −24.0 cm and m = +3.00. The screen cannot capture it; look through the lens from the transmitted side to see an upright enlarged virtual image.

The two surface refractions combine so a near-axis central ray is approximately undeviated in the thin-lens model. A thick lens or rays far from the axis require more detailed geometry and can have aberrations.