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Chapter 3: Problem 15

Find the current through a person and identify the likely effect on her if she touches a \(120-\mathrm{V}\) AC source: \((\mathrm{a})\) if she is standing on a rubber mat and offers a total resistance of \(300 \mathrm{k} \Omega ;(\mathrm{b})\) if she is standing barefoot on wet grass and has a resistance of only \(4000 \mathrm{k} \Omega\).

Short Answer

Expert verified
(a) Current is 0.4 mA, effect is negligible. (b) Current is 30 mA, effect is dangerous.

Step by step solution

01

Understanding Ohm's Law

Ohm's Law states that the current \( I \) through a conductor between two points is directly proportional to the voltage \( V \) across the two points and inversely proportional to the resistance \( R \). This can be expressed as \( I = \frac{V}{R} \). We'll use this formula to determine the current for both scenarios.
02

Step 2a: Calculate Current with Rubber Mat

For part (a), the voltage \( V = 120 \, \text{V} \) and the resistance \( R = 300 \, \text{k}\Omega = 300,000 \, \Omega \). By using the formula \( I = \frac{V}{R} \), we get:\[ I = \frac{120}{300,000} = 0.0004 \, \text{A} \]Thus, the current is \( 0.4 \, \text{mA} \).
03

Step 3a: Determine Effect with Rubber Mat

When the current is \( 0.4 \, \text{mA} \), it is generally considered to be below the threshold for sensation or harm to the human body. Therefore, the likely effect is negligible and non-harmful.
04

Step 2b: Calculate Current Barefoot on Wet Grass

For part (b), the resistance is \( R = 4000 \, \Omega \). Using the formula \( I = \frac{V}{R} \), calculate the current:\[ I = \frac{120}{4000} = 0.03 \, \text{A} \]Thus, the current is \( 30 \, \text{mA} \).
05

Step 3b: Determine Effect Barefoot on Wet Grass

With a current of \( 30 \, \text{mA} \), this is a dangerous level of current through the human body, capable of causing pain, muscle contractions, or even possibly more severe effects depending on the duration and path through the body.

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Key Concepts

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

Current Calculation
Calculating the current flowing through an object involves the relationship defined by Ohm's Law. According to Ohm鈥檚 Law, current \( I \) is determined by dividing the voltage (\( V \)) by the resistance (\( R \)). This relationship can be succinctly expressed with the formula: \\[ I = \frac{V}{R} \\]In scenarios where a person comes into contact with an AC source, this calculation helps predict the current that flows through them. This is critical since the amount of current can indicate potential dangers. For practical examples:
  • Standing on a rubber mat, the total resistance is high, resulting in a low current.
  • Standing barefoot on wet grass reduces the resistance significantly, leading to a higher current flow.
Understanding this calculation principle aids in assessing potential electrical risks.
Electrical Resistance
Resistance is a measure of how much a material or object resists the flow of electric current. It鈥檚 measured in Ohms (\( \Omega \)). The higher the resistance, the less current will flow for a given voltage. This is why materials with high resistance, like rubber, are excellent insulators.Different factors affect resistance:
  • Material type 鈥� Insulating materials like rubber significantly increase resistance, reducing current flow.
  • Temperature 鈥� Generally, as a material's temperature increases, so does its resistance.
  • Contact quality 鈥� Better contact can reduce resistance, increasing current.
In the exercise, a barefoot person standing on wet grass has lower resistance compared to standing on a rubber mat. This difference dramatically affects the current flow and highlights the importance of resistance in electrical safety.
Effects of Electric Current
The effects of electric current on the human body vary dramatically with the magnitude and duration of the current flow. Current is often measured in milliamperes (mA), and understanding these levels is crucial for safety:
  • Below 1 mA: Generally not felt, with little to no physiological effects.
  • 1 - 10 mA: A tingling sensation might be felt. This is typically not harmful but can startle.
  • 10 - 30 mA: Might cause muscle contractions or a painful shock. This level increases the risk of injury due to involuntary movement.
  • Above 30 mA: Dangerously high, causing severe pain and possible muscle paralysis, making it difficult to let go of the live conductor.
In our scenario, standing barefoot on wet grass results in a current of 30 mA, which poses a risk of harmful effects, unlike when insulated by a rubber mat.
AC Voltage
Alternating Current (AC) voltage is the type of voltage that comes from wall outlets in most households and businesses. It differs from Direct Current (DC) since the current direction changes periodically. AC voltage is represented by the root mean square (RMS) value, typically measuring the effective voltage level used for power estimation in circuits. Key points about AC voltage:
  • Frequency: The frequency of AC in most homes is 50 or 60 Hz, which refers to the number of cycles per second.
  • Phase: In three-phase systems, multiple AC voltages are used together for more efficient power distribution.
  • Safety: Handling AC voltage incorrectly can be dangerous due to its ability to cause significant harm if in contact with low resistance paths through the body.
In the exercise, the source of 120 V AC is used to explore practical impacts on safety, emphasizing the need to understand and respect household electricity.

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Most popular questions from this chapter

(a) To what temperature must you raise a resistor made of constantan to double its resistance, assuming a constant temperature coefficient of resistivity? (b) To cut it in half? (c) What is unreasonable about these results? (d) Which assumptions are unreasonable, or which premises are inconsistent?

Electron guns are used in X-ray tubes. The electrons are accelerated through a relatively large voltage and directed onto a metal target, producing X-rays. (a) How many electrons per second strike the target if the current is \(0.500 \mathrm{~mA} ?\) (b) What charge strikes the target in \(0.750\) s?

(a) Find the voltage drop in an extension cord having a \(0.0600-\Omega\) resistance and through which \(5.00 \mathrm{~A}\) is flowing. (b) \(\mathrm{A}\) cheaper cord utilizes thinner wire and has a resistance of \(0.300 \Omega\). What is the voltage drop in it when \(5.00\) A flows? (c) Why is the voltage to whatever appliance is being used reduced by this amount? What is the effect on the appliance?

What is the current when a typical static charge of \(0.250 \mu \mathrm{C}\) moves from your finger to a metal doorknob in \(1.00 \mu\) s?

Some surgery is performed with high-voltage electricity passing from a metal scalpel through the tissue being cut. Considering the nature of electric fields at the surface of conductors, why would you expect most of the current to flow from the sharp edge of the scalpel? Do you think high- or low- frequency \(\mathrm{AC}\) is used?
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