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The average bulk resistivity of the human body (apart from surface resistance of the skin) is about 5.0Ω m. The conducting path between the hands can be represented approximately as a cylinder 1.6 m long and 0.10 m in diameter. The skin resistance can be made negligible by soaking the hands in salt water. (a) What is the resistance between the hands if the skin resistance is negligible? (b) What potential difference between the hands is needed for a lethal shock current of 100 mA? (Note that your result shows that small potential differences produce dangerous currents when the skin is damp.) (c) With the current in part (b), what power is dissipated in the body?

Short Answer

Expert verified
(a) 1018.34 Ω, (b) 101.834 V, (c) 10.1834 W

Step by step solution

01

Calculate the Cross-Sectional Area

The cross-sectional area of a cylinder with diameter d=0.10m is given by the formula: A=π(d2)2First, calculate the radius: d2=0.05m.Then, substitute the radius into the area formula: A=π(0.05)2=π×0.0025m20.00785m2
02

Calculate the Bulk Resistance

The resistance R of a cylindrical conductor is calculated with the formula: R=ρLAwhere ρ=5.0Ωm, L=1.6m, and A0.00785m2. Substitute these values:R=5.0×1.60.007851018.34Ω
03

Calculate Potential Difference for Lethal Current

To find the potential difference for a current I=100mA=0.1A, we use Ohm's Law: V=IRSubstitute R=1018.34Ω and I=0.1A:V=0.1×1018.34=101.834V
04

Calculate Power Dissipated in the Body

The power P dissipated is given by the formula: P=I2RSubstitute I=0.1A and R=1018.34Ω:P=(0.1)2×1018.34=0.01×1018.34=10.1834W

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Key Concepts

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

Ohm's Law
Ohm's Law is a cornerstone concept in the study of electrical circuits. It states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points, provided the temperature remains constant. Mathematically, it is expressed as:
  • V=IR
Here, V is the potential difference (voltage) in volts, I is the current in amperes, and R is the resistance in ohms.
This fundamental principle helps us understand how electrical circuits operate, making it a crucial element in the analysis of electrical systems.
In the context of our exercise, Ohm's Law was used to determine the voltage necessary to create a specific current flow through the human body, represented as a cylindrical conductor. This shows how even a small voltage can result in a dangerous current if the resistance is sufficiently low.
Bulk resistivity
Bulk resistivity is a material property that quantifies how strongly a material opposes the flow of electric current. It's essentially the resistance of a material regardless of its shape or size. The formula to determine resistance based on bulk resistivity is:
  • R=ρLA
Where ρ is the bulk resistivity in ohm-meters, L is the length of the material in meters, and A is the cross-sectional area in square meters.
Bulk resistivity is particularly useful when you need to calculate the resistance of a material configured into shapes like wires or cylinders, as seen in our exercise. By knowing the resistivity of the human body, and the approximate dimensions of the conducting path, we could calculate the body's resistance. This step not only aids in academic exercises but also has practical implications in safety standards, where understanding human-body resistivity can help prevent electrical accidents.
Power dissipation
Power dissipation refers to the process in which electrical energy is converted into heat energy in a resistive component. The power dissipated in a resistor can be calculated from the formula:
  • P=I2R
Where P is power in watts, I is the current through the resistor in amperes, and R is the resistance in ohms.
This formula shows that the power dissipated is proportional to the square of the current, which means that even small changes in current can lead to significant changes in the thermal energy generated.
In the exercise, this concept is vital for understanding how much power would be lost in the human body when subjected to a dangerous current. The calculated power, 10.1834W, while not inherently large, can be significant when considering the biological impacts on the human body, indicating potential harm when exposed to such conditions.
Cylinder cross-sectional area
The cross-sectional area of a cylinder is essential for various calculations related to the flow of electricity through it. It is particularly important in determining resistance when combined with the length of the cylinder and the material's resistivity. The formula for the cross-sectional area A of a cylinder is:
  • A=π(d2)2
Where d is the diameter of the cylinder.
In our exercise, calculating this area was necessary to find the body's resistance as a cylindrical conductor between two points.
By understanding the geometry and applying this formula, we were able to calculate the cross-sectional area accurately, allowing for further calculations involving bulk resistivity and resulting resistance.
This concept is equally applicable in various engineering fields and physics problems, making it a foundational part of understanding material properties in different shapes.

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Most popular questions from this chapter

A typical small flashlight contains two batteries, each having an emf of 1.5 V, connected in series with a bulb having resistance 17Ω. (a) If the internal resistance of the batteries is negligible, what power is delivered to the bulb? (b) If the batteries last for 5.0 h, what is the total energy delivered to the bulb? (c) The resistance of real batteries increases as they run down. If the initial internal resistance is negligible, what is the combined internal resistance of both batteries when the power to the bulb has decreased to half its initial value? (Assume that the resistance of the bulb is constant. Actually, it will change somewhat when the current through the filament changes, because this changes the temperature of the filament and hence the resistivity of the filament wire.)

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A resistor with resistance R is connected to a battery that has emf 12.0 V and internal resistance r= 0.40Ω. For what two values of R will the power dissipated in the resistor be 80.0 W?

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Compact fluorescent bulbs are much more efficient at producing light than are ordinary incandescent bulbs. They initially cost much more, but they last far longer and use much less electricity. According to one study of these bulbs, a compact bulb that produces as much light as a 100-W incandescent bulb uses only 23 W of power. The compact bulb lasts 10,000 hours, on the average, and costs 11.00,whereastheincandescentbulbcostsonly0.75, but lasts just 750 hours. The study assumed that electricity costs $0.080 per kilowatt-hour and that the bulbs are on for 4.0 h per day. (a) What is the total cost (including the price of the bulbs) to run each bulb for 3.0 years? (b) How much do you save over 3.0 years if you use a compact fluorescent bulb instead of an incandescent bulb? (c) What is the resistance of a "100-W" fluorescent bulb? (Remember, it actually uses only 23 W of power and operates across 120 V.)

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