Problem 21 - College Physics

Question

Current in the body. The resistance of the body varies from approximately 500 \(\mathrm{k} \Omega\) (when it is very dry) to about 1 \(\mathrm{k} \Omega\) (when it is wet). The maximum safe current is about 5.0 \(\mathrm{mA} .\) At 10 \(\mathrm{mA}\) or above, muscle contractions can occur that may be fatal. What is the largest potential difference that a person can safely touch if his body is wet? Is this result within the range of common household voltages?

Step-by-Step Solution

Answer

The largest safe voltage is 5 V, which is below common household voltages.

Step 1: Understanding the Problem

We need to find the maximum potential difference (voltage) that a person can safely endure given a wet body resistance and maximum safe current. The wet resistance is 1 kΩ, and the maximum safe current is 5 mA.

Step 2: Using Ohm's Law

Ohm's Law states that the potential difference (voltage) across a resistor is the product of the current through it and its resistance: \( V = I \times R \). Here, \( I = 5 \text{ mA} = 5 \times 10^{-3} \text{ A} \) and \( R = 1 \text{ kΩ} = 1000 \text{ Ω} \). Substituting these into the formula gives us the maximum voltage.

Step 3: Calculating the Maximum Voltage

Plug the values into the formula: \( V = (5 \times 10^{-3}\, \text{A}) \times (1000\, \text{Ω}) = 5 \text{ V} \). Thus, the maximum safe potential difference is 5 volts.

Step 4: Comparing with Household Voltage

Common household voltages are usually 110 or 220 volts. 5 volts is significantly less than these values, indicating the calculated voltage is not within the range of typical household voltages.

Key Concepts

Ohm's Law

Ohm's Law is one of the foundational principles in the study of electricity. It explains how voltage (potential difference), current, and resistance are related in an electrical circuit. Simply put, Ohm's Law is described by the formula:

• \( V = I \times R \)

where:

• \(V\) is the voltage in volts \((V)\),
• \(I\) is the current in amperes \((A)\),
• \(R\) is the resistance in ohms \((Ω)\).

Understanding Ohm's Law allows us to predict how changing one of these quantities affects the others. For example, if resistance remains constant, increasing the current will increase the voltage across an electrical component. In many practical situations, like calculating the safe voltage one can be subjected to in the presence of body resistance, Ohm's Law comes in handy to evaluate the potential risks.

Body Resistance

Body resistance is a crucial factor when considering electrical safety. The resistance of the human body can vary widely.

• In dry conditions, body resistance can be as high as 500 \( kΩ \).
• In wet conditions, resistance drops significantly, to as low as 1 \( kΩ \).

Higher resistance in dry conditions provides more protection against electric current. However, when the skin is wet, resistance lowers drastically, allowing more current to pass through. This is why situations involving water increase the risk of electrical shock.
Body resistance acts as a natural barrier to current flow, influencing how much voltage can be safely tolerated. However, decreased body resistance in wet conditions heightens vulnerability, making it essential to understand this variation for assessing potential electrical hazards.

Potential Difference

Potential difference, or voltage, is the driving force that pushes electric current through a circuit. It is measured in volts \((V)\) and represents the energy difference per unit charge. When considering human safety, the potential difference is crucial because it determines how much energy is delivered to the body if an electrical contact occurs.

• The larger the potential difference, the higher the risk, as more energy is transferred.
• With a lower body resistance, such as when it is wet, less voltage is needed to reach dangerous current levels.

Understanding potential difference and how it interacts with body resistance helps determine safe and unsafe levels of exposure to electricity.
For instance, in the given exercise, a wet body's 1 \( kΩ \) resistance limits the safe potential difference to 5 volts. This calculation clarifies that typical household voltages, which start at around 110 volts, are far above safe exposure levels.

Safe Current Levels

Current levels are a critical aspect in determining the safety of electrical exposure. The basic idea is that the amount of current flowing through the body can have various effects on health.

• Safe current levels are considered to be up to 5 milliamperes \((mA)\).
• Currents above this can lead to involuntary muscle contractions or even fatal outcomes when reaching 10 \( mA \) or higher.

Electrical appliances and systems are designed to minimize the risk of exposing people to dangerous current levels. If a person comes into contact with a voltage source, the current passing through them is dictated by Ohm's Law, accounting for the body's resistance.
This understanding is pivotal in assessing electrical risks much like wearing insulated shoes or using a dry towel by the pool to enhance safety measures. It's about knowing where to draw the line when it comes to handling electricity safely.