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During normal beating, the heart creates a maximum \(4.00-\mathrm{mV}\) potential across \(0.300 \mathrm{~m}\) of a person's chest, creating a \(1.00\) \(\mathrm{Hz}\) electromagnetic wave. (a) What is the maximum electric field strength created? (b) What is the corresponding maximum magnetic field strength in the electromagnetic wave? (c) What is the wavelength of the electromagnetic wave?

Short Answer

Expert verified
The maximum electric field strength is \text{13.3 kV/m}, the maximum magnetic field strength is \text{44.4 pT}, and the wavelength of the electromagnetic wave is \text{3.00 x 10^8 m}.

Step by step solution

01

Calculate the maximum electric field strength

To find the maximum electric field strength (E), we use the relationship between potential difference (V) and electric field (E) in a uniform field, which is given by the equation: \( E = \frac{V}{d} \), where V is the potential difference and d is the distance over which the potential difference is measured. In this exercise, V is given as 4.00 mV (which is also 0.004 V) and d is 0.300 m. Plug these values into the equation to find E.
02

Calculate the corresponding maximum magnetic field strength

The maximum magnetic field strength (B) in an electromagnetic wave can be related to the electric field (E) using the equation \( B = \frac{E}{c} \), where c is the speed of light (\(3.00 \times 10^8 \mathrm{m/s}\)). We already calculated E in Step 1, so now substitute that value and the speed of light into the equation to find B.
03

Determine the wavelength of the electromagnetic wave

The frequency (\(f\)) of the wave is given as 1.00 Hz. The relationship between wavelength (\(\lambda\)), frequency (f), and the speed of light (c) is given by the equation \( c = \lambda f \). We are given f and we know the value of c, so we can rearrange the equation to solve for the wavelength: \( \lambda = \frac{c}{f} \). Substitute the values for c and f to find the wavelength.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Electric Field Strength
Electric field strength is a fundamental concept in electromagnetism, representing the force exerted by an electric field on a charged particle. In the context of the heart's activity, the electric field is generated by the potential difference across a distance in the human chest.

When we talk about the maximum electric field strength, it refers to the peak value of this force per unit charge. To calculate it, we divide the potential difference by the distance over which this potential is applied. Hence, the equation used is: \[ E = \frac{V}{d} \].

With the provided values, the potential difference of 4.00 mV (or 0.004 V) and the distance of 0.300 m, we find the electric field strength to be substantial even with such a small voltage because the distance is relatively short. This calculation underscores the sensitivity of the human body to electrical activities, such as those produced by the heart, and how they translate into electromagnetic phenomena.

Since electric fields are vectors, the direction of this field strength would be in the direction in which a positive charge would move, which in this case is from one side of the chest to the other, aligned with the potential difference created by the heart's activity.
Exploring Magnetic Field Strength in Electromagnetic Waves
Magnetic field strength is related to how much force a magnetic field can exert on a moving charged particle or a current. Within an electromagnetic wave, such as the one produced by the heart, the electric and magnetic fields are interconnected.

The formula linking the electric field strength (E) to the magnetic field strength (B) is given by: \[ B = \frac{E}{c} \], where 'c' denotes the speed of light. In our case, the electric field strength is calculated first, and then this equation is used to find the corresponding magnetic field strength.

This relationship reveals how changes in one field can induce changes in the other and is a cornerstone of how electromagnetic waves propagate. These waves are essentially fluctuations of electric and magnetic fields that move through space at the speed of light, allowing the wireless transmission of energy and information, such as the electrical signals from the heart.

The calculated magnetic field strength is pivotal for understanding how the heart's electrical signals can influence other bodily functions and sensitive devices like pacemakers.
Determining the Wavelength of Electromagnetic Waves
The wavelength of an electromagnetic wave is the distance over which the wave's shape repeats. It is inversely proportional to the frequency, the number of times the wave oscillates per second. When it comes to the heartbeat and the biology of the human body, even a low frequency like 1.00 Hz can generate an electromagnetic wave with a significant wavelength.

The essential equation that connects the speed of light (c), frequency (f), and wavelength (\(\lambda\)) is: \[ c = \lambda f \]. By rearranging this for wavelength, we get: \[ \lambda = \frac{c}{f} \].

Substituting the known values into this formula, we reveal that a frequency typical of a human heartbeat can create electromagnetic waves with wavelengths comparable to the dimensions found in everyday life. Such waves can travel through various mediums, interacting with other biological systems and electronic devices, a fact that is leveraged in medical imaging and diagnostic procedures.

The understanding of wavelength in biological electromagnetic phenomena enhances our awareness of the scale at which our body operates and interacts with the environment, reaffirming the incredible precision of physiological functions.

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Most popular questions from this chapter

Laser vision correction often uses an excimer laser that produces \(193-\mathrm{nm}\) electromagnetic radiation. This wavelength is extremely strongly absorbed by the cornea and ablates it in a manner that reshapes the cornea to correct vision defects. Explain how the strong absorption helps concentrate the energy in a thin layer and thus give greater accuracy in shaping the cornea. Also explain how this strong absorption limits damage to the lens and retina of the eye.

Assume the mostly infrared radiation from a heat lamp acts like a continuous wave with wavelength \(1.50 \mu \mathrm{m}\). (a) If the lamp's \(200-W\) output is focused on a person's shoulder, over a circular area \(25.0 \mathrm{~cm}\) in diameter, what is the intensity in \(\mathrm{W} / \mathrm{m}^{2} ?\) (b) What is the peak electric field strength? (c) Find the peak magnetic field strength. (d) How long will it take to increase the temperature of the \(4.00\) -kg shoulder by \(2.00^{\circ} \mathrm{C}\), assuming no other heat transfer and given that its specific heat is \(3.47 \times 10^{3} \mathrm{~J} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C} ?\)

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The rate at which information can be transmitted on an electromagnetic wave is proportional to the frequency of the wave. Is this consistent with the fact that laser telephone transmission at visible frequencies carries far more conversations per optical fiber than conventional electronic transmission in a wire? What is the implication for ELF radio communication with submarines?

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