Problem 44 - College Physics

Question

Electricity through the body, I. A person with a body resistance of 10 \(\mathrm{k} \Omega\) between his hands accidentally grasps the terminals of a 14 \(\mathrm{kV}\) power supply. (a) If the internal resistance of the power supply is \(2000 \Omega,\) what is the current through the person's body? (b) What is the power dissipated in his body? (c) If the power supply is to be made safe by increasing its internal resistance, what should the internal resistance be for the maximum current in the situation just described to be 1.00 \(\mathrm{mA}\) or less?

Step-by-Step Solution

Answer

(a) 1.17 A, (b) 13,641 W, (c) 13.99 MΩ or higher.

Step 1: Understand the Circuit

The person is in contact with a power supply of voltage 14 kV and has a body resistance of 10 kΩ. Additionally, the power supply has an internal resistance of 2000 Ω. This creates a series circuit where the total resistance is the sum of the internal resistance of the power supply and the body resistance. Calculate the total resistance: \[ R_{\text{total}} = 10\, \text{k}\Omega + 2000\, \Omega = 12\, \text{k}\Omega \]

Step 2: Calculate the Current through the Body

Using Ohm's Law, the current through the series circuit with total resistance is given by: \[ I = \frac{V}{R_{\text{total}}} = \frac{14\, \text{kV}}{12\, \text{k}\Omega} \] Convert the voltage to volts before calculating: \[ I = \frac{14,000\, \text{V}}{12,000\, \Omega} = 1.167\, \text{A} \] Therefore, the current through the body is approximately 1.17 A.

Step 3: Calculate the Power Dissipated in the Body

The power dissipated in the body can be calculated using the formula: \[ P = I^2 \cdot R_{\text{body}} \] Substitute the known values:\[ P = (1.167\, \text{A})^2 \times 10\, \text{k}\Omega = 13,640.89\, \text{W} \] Therefore, the power dissipated in the body is approximately 13,641 W.

Step 4: Determine Required Resistance for Safety

To find the internal resistance that makes the circuit safe, limit the current to a maximum of 1 mA or less. Using 1 mA, rewrite Ohm's Law: \[ 0.001\, \text{A} = \frac{14,000\, \text{V}}{R_{\text{body}} + R_{\text{internal}}} \] Rearrange to solve for \( R_{\text{internal}} \):\[ R_{\text{internal}} = \frac{14,000\, \text{V}}{0.001\, \text{A}} - 10,000\, \Omega \]\[ R_{\text{internal}} = 14,000,000\, \Omega - 10,000\, \Omega = 13,990,000\, \Omega \]Therefore, the internal resistance should be 13.99 MΩ or higher.

Key Concepts

Ohm's Law

Ohm's Law is a fundamental principle in the study of electricity. It states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance between them. The mathematical expression for Ohm's Law is \[ I = \frac{V}{R} \]where:

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

This law is essential for understanding how electrical circuits operate. In the context of the problem, we used Ohm's Law to calculate the current passing through the person's body by identifying the total series resistance of the circuit. Understanding the relationship between voltage, current, and resistance helps explain why even small voltages can cause dangerous currents under certain conditions.

Body resistance

The resistance of the human body plays a critical role in determining how much current flows through it in case of accidental contact with electrical sources. Body resistance can vary widely, depending on factors like moisture, skin condition, and the area of contact. For the illustrated example, a body resistance of 10 kΩ was assumed. This resistance can combine with other circuit components, influencing the total impedance and subsequently the current.
The overall concept here is that our bodies are not perfect insulators, and electricity can pass through us, potentially causing serious harm. Safety standards set limits on exposure, considering the body's resistance. The combination with external resistances—like that of the power supply in our scenario—can help mitigate the risk by reducing the current reaching lethal levels.

Power dissipation

Power dissipation in electrical circuits is the process of converting electrical energy into other forms, such as heat. In human and machine terms, power dissipation occurs when electric current flows through a resistance, creating a voltage drop and releasing energy.
This can be explained with the formula:\[ P = I^2 \cdot R \]where:

• \( P \) is the power in watts (W).
• \( I \) is the current in amperes (A).
• \( R \) is the resistance in ohms (Ω).

In our exercise, the power dissipated in the person's body is calculated using this formula, demonstrating why large amounts of current can lead to severe heating effects. High levels of power dissipation can heat body tissues, potentially causing burns or even electrocution. It's crucial to understand power dissipation both for planning safe circuits and for implementing precautionary measures against accidental contact with live electrical components.

Circuit safety

Circuit safety is vital when designing systems that humans will interact with. Ensuring that electrical equipment is safe involves controlling the amount of current that can flow through a circuit, particularly in situations where human contact is possible.
In the exercise, increasing the power supply's internal resistance is a method to make the system safer. By increasing this resistance, the total resistance of the circuit goes up, which reduces current according to Ohm's Law. To calculate a safer setup, we determined the necessary internal resistance to keep the body current below 1 mA, which is considered a safer threshold for humans.

• Measures such as circuit breakers, fuses, and insulation are also commonly employed to protect people from electrical hazards.
• Understanding and implementing adequate resistance levels can prevent dangers like fires, electrical shocks, or even electrocution.

By designing circuits with safety as a priority, it becomes possible to prevent mishaps and ensure a safer interaction with electricity.