Problem 54 - Physics

Question

Deep ultrasonic heating is used to promote healing of torn tendons. It is produced by applying ultrasonic sound over the affected area of the body. The sound transducer (generator) is circular with a radius of \(1.8 \mathrm{~cm}\), and it produces a sound intensity of \(5.9 \times 10^{3} \mathrm{~W} / \mathrm{m}^{2}\). How much time is required for the transducer to emit \(4800 \mathrm{~J}\) of sound energy?

Step-by-Step Solution

Answer

The transducer emits 4800 J of sound energy in approximately 2.37 seconds.

Step 1: Understand the Problem

We need to find the time it takes for a circular sound transducer emitting ultrasound with a given intensity to deliver a specific amount of energy.

Step 2: Calculate the Area of the Transducer

The transducer is circular with a radius of \(1.8\, \text{cm}\), which is \(0.018\, \text{m}\). The area \(A\) of the circle can be calculated using the formula:\[A = \pi r^2 = \pi \times (0.018)^2\, \text{m}^2\]

Step 3: Find the Power of the Transducer

The power \(P\) is the product of the area of the transducer and the sound intensity. \(I = 5.9 \times 10^{3}\, \text{W/m}^2\). Thus, the power is:\[P = I \times A = 5.9 \times 10^{3} \times \pi \times (0.018)^2\, \text{W}\]

Step 4: Using Power to Find Time

We use the equation for power \(P = \frac{E}{t}\), where \(E = 4800\, \text{J}\) is the energy and \(t\) is the time. Solve for \(t\):\[t = \frac{E}{P}\]

Step 5: Calculate the Time Required

Substitute the values into the equation:\[t = \frac{4800}{5.9 \times 10^{3} \times \pi \times (0.018)^2}\, \text{s}\]Compute this value to find the required time.

Key Concepts

Sound Intensity

When discussing ultrasonic heating, a key factor is sound intensity. Sound intensity is a measure of the power per unit area carried by a sound wave. It is usually given in watts per square meter (W/m²). Sound intensity is what determines how much energy is being delivered over a certain area.For example, in medical applications like healing torn tendons with ultrasound, different intensities could result in varying levels of heating. The sound intensity in the original exercise is given as \(5.9 \times 10^{3} \, \mathrm{W/m^{2}}\), which is quite significant, reflecting the focused energy needed to promote healing.

Energy Transmission

Energy transmission in the context of ultrasonic heating refers to how sound energy is carried through the air (or other mediums) and eventually absorbed by the body tissues. The efficiency of energy transmission depends on several factors, including:

• The medium through which the sound travels – air, water, or human tissue can all have different transmission characteristics.
• The frequency and intensity of the sound waves – higher frequencies might transmit slightly better through certain tissues.
• The duration of the sound application – longer exposure might mean more energy transmission.

Understanding how energy is transmitted through mediums is crucial for optimizing ultrasound treatments. In medical applications, effective transmission ensures the targeted tissues receive enough energy to aid in healing.

Power Calculation

Power calculation is essential when working with ultrasonic heating devices such as transducers. In physics, power is defined as the rate at which energy is moved or converted. Mathematically, it's the product of intensity (sound intensity, in this context) and the area the sound waves are covering:\[ P = I \times A \]In the case of the circular transducer from the exercise, its power output is determined by multiplying the given sound intensity by the area of the circle the sound waves cover. This calculation helps us understand how much energy is released over time, which can be critical when aiming for precise treatments in medical scenarios.

Ultrasound Transducer

An ultrasound transducer is a device that converts electrical energy into sound waves, which are then transmitted into the body for various medical purposes. The transducer's ability to generate sound waves depends on its size, shape, and the frequency it operates at.The original exercise involves a circular transducer with a radius of 1.8 cm. For such a transducer, the area can be calculated using the formula\[ A = \pi r^2 \]Once the area is known, this is used to calculate the total power output based on the intensity of the soundwaves it produces. These transducers are critical in medical ultrasound applications, helping perform tasks like imaging internal organs or promoting healing by sending focused sound waves to specific body areas.