krit.club logo

Waves - Imaging and Applications of Waves

Grade 9IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

Reflection: The bouncing of waves off a surface. The Law of Reflection states that the angle of incidence θi\theta_i is equal to the angle of reflection θr\theta_r measured from the normal.

•

Refraction: The change in direction of a wave as it passes from one medium to another due to a change in speed. The refractive index nn is the ratio of the speed of light in a vacuum cc to the speed of light in the medium vv.

•

Converging (Convex) Lenses: These lenses bring parallel light rays to a single point called the principal focus FF. They can form both real and virtual images depending on the object's distance from the lens.

•

Diverging (Concave) Lenses: These lenses spread out parallel light rays. They always form virtual, upright, and diminished images.

•

Total Internal Reflection (TIR): Occurs when light travels from a more dense medium to a less dense medium at an angle of incidence greater than the critical angle θc\theta_c. This is the basis for fiber optics.

•

Ultrasound Imaging: Uses sound waves with frequencies higher than 20,000 Hz20,000 \text{ Hz}. Pulses are reflected at boundaries between different tissues, and the time taken is used to calculate the depth of organs or a fetus.

•

Electromagnetic Spectrum in Imaging: X-rays are used for bone imaging because they are absorbed by dense materials (bones) but pass through soft tissue. Gamma rays are used in PET scans to detect tracers in the body.

📐Formulae

Refractive Index (n)=cv\text{Refractive Index (n)} = \frac{c}{v}

n1sin⁡(θ1)=n2sin⁡(θ2)n_1 \sin(\theta_1) = n_2 \sin(\theta_2)

sin⁡(θc)=1n\sin(\theta_c) = \frac{1}{n}

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

Magnification (M)=hiho=∣vu∣\text{Magnification (M)} = \frac{h_i}{h_o} = \left| \frac{v}{u} \right|

Distance (d)=v×t2\text{Distance (d)} = \frac{v \times t}{2}

💡Examples

Problem 1:

An object is placed 12 cm12 \text{ cm} away from a converging lens with a focal length of 8 cm8 \text{ cm}. Calculate the image distance vv and determine if the image is real or virtual.

Solution:

Given: u=12 cmu = 12 \text{ cm}, f=8 cmf = 8 \text{ cm}. Using the lens equation: 1f=1u+1v\frac{1}{f} = \frac{1}{u} + \frac{1}{v} 18=112+1v\frac{1}{8} = \frac{1}{12} + \frac{1}{v} 1v=18−112\frac{1}{v} = \frac{1}{8} - \frac{1}{12} 1v=324−224\frac{1}{v} = \frac{3}{24} - \frac{2}{24} 1v=124\frac{1}{v} = \frac{1}{24} v=24 cmv = 24 \text{ cm}

Explanation:

Since the image distance vv is positive, the image is real and formed on the opposite side of the lens from the object.

Problem 2:

An ultrasound pulse is sent into the body and the echo is received after 0.0004 s0.0004 \text{ s}. If the speed of sound in human tissue is 1540 m/s1540 \text{ m/s}, how deep is the organ reflecting the signal?

Solution:

Given: v=1540 m/sv = 1540 \text{ m/s}, t=0.0004 st = 0.0004 \text{ s}. Since the sound travels to the organ and back, the distance dd is: d=v×t2d = \frac{v \times t}{2} d=1540×0.00042d = \frac{1540 \times 0.0004}{2} d=0.6162d = \frac{0.616}{2} d=0.308 md = 0.308 \text{ m}

Explanation:

The total distance traveled by the wave is v×tv \times t. We divide by 22 because the wave travels to the object and reflects back to the transducer.

Problem 3:

Calculate the critical angle θc\theta_c for a glass block with a refractive index of 1.521.52 surrounded by air (n≈1n \approx 1).

Solution:

Given: n=1.52n = 1.52. Using the critical angle formula: sin⁡(θc)=1n\sin(\theta_c) = \frac{1}{n} sin⁡(θc)=11.52\sin(\theta_c) = \frac{1}{1.52} sin⁡(θc)≈0.6579\sin(\theta_c) \approx 0.6579 θc=sin⁡−1(0.6579)\theta_c = \sin^{-1}(0.6579) θc≈41.14∘\theta_c \approx 41.14^{\circ}

Explanation:

The critical angle is the angle of incidence that results in an angle of refraction of 90∘90^{\circ}. Any light hitting the boundary at an angle greater than 41.14∘41.14^{\circ} will undergo total internal reflection.