Problem 64 - University Physics with Modern Physics

Question

A person with body resistance between his hands of 10 k\(\Omega\) accidentally grasps the terminals of a 14-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 above situation to be 1.00 mA or less?

Step-by-Step Solution

Answer

(a) 1.167 A; (b) 13,640.89 W; (c) 13.99 MΩ.

Step 1: Calculate the total resistance

The total resistance in the circuit can be calculated by adding the resistance of the person's body and the internal resistance of the power supply. The resistance of the person's body is \(10\, k\Omega\) and the power supply's internal resistance is \(2000\, \Omega\). Therefore, the total resistance \( R_{total} \) is:\[ R_{total} = 10\,000\, \Omega + 2,000\, \Omega = 12,000\, \Omega \]

Step 2: Calculate the current through the body

Using Ohm's Law, \( V = IR \), we can find the current \( I \) flowing through the person's body. The voltage \( V \) is supplied by the power source, which is 14 kV (or 14,000 V). So, the current \( I \) is:\[ I = \frac{V}{R_{total}} = \frac{14,000\, V}{12,000\, \Omega} = 1.167\, A \]

Step 3: Calculate the power dissipated in the body

The power dissipated in the person's body can be found using the formula \( P = I^2R \), where \( I \) is the current through the body and \( R \) is the body resistance. Using the current calculated in the previous step:\[ P = (1.167)^2 \times 10,000 = 13,640.89\, W \]

Step 4: Determine the safe internal resistance for reduced current

To make the power supply safe, we need to ensure the current does not exceed 1.00 mA (0.001 A). The equation \( I = \frac{V}{R_{total}} \) gives us:\[ 0.001 = \frac{14,000}{R_{total}} \]Rearranging the equation for \( R_{total} \):\[ R_{total} = \frac{14,000}{0.001} = 14,000,000\, \Omega \]Since the person's resistance is 10,000 \( \Omega \), the internal resistance \( R \) of the power supply should be:\[ R = 14,000,000\, \Omega - 10,000\, \Omega = 13,990,000\, \Omega \]

Key Concepts

Resistor

A resistor is a fundamental component used in electrical circuits to limit the flow of electric current. It is characterized by its resistance, usually measured in ohms (\(\Omega\)), and is used to control the current and divide voltages within a circuit. Resistors can be made of various materials, like carbon or metal film, and come in many forms and sizes.
In the context of this problem, the human body acts as a resistor. When someone touches an electrical source with a known resistance (10 \,k\(\Omega\)), it functions much like a resistor in a circuit. Adding the body's resistance to any circuit influences the total resistance and hence the current flowing through it.
Understanding how to calculate the total resistance in a circuit is crucial in determining how much current will flow. This requires adding together the resistance of all components, including the internal resistance of any power supply.

Body Resistance

Body resistance refers to the electrical resistance offered by the human body when an electrical current passes through it. It plays a critical role in determining how much current can flow through a person's body when exposed to an electrical source.
Several factors affect body resistance, such as moisture content, skin condition, and the cross-sectional area through which voltage is applied. Typically, dry skin has higher resistance, while wet skin offers much lower resistance.
In this exercise, the body resistance is considered to be 10 \,k\(\Omega\), which is relatively high, implying that the body will allow only a limited amount of electrical current to pass through under normal conditions. However, if the applied voltage is very high, like 14 \,kV in the exercise, even a high resistance will allow considerable current to flow through the body, potentially causing harm.

Power Dissipation

Power dissipation is the process by which an electrical system or circuit converts electrical energy into heat. It is quantified by the power formula \( P = I^2 R \), where \( I \) is the current in amperes and \( R \) is the resistance in ohms.
In the exercise scenario, the power dissipated in the human body is calculated using the current flowing through it and the body's resistance. With a current of 1.167 \,A and a resistance of 10 \,k\(\Omega\), the power dissipated in the body is extremely high at 13,640.89 \,W (watts). This high power dissipation can cause severe injuries or even be fatal.
Understanding power dissipation is crucial for creating safe electrical systems, as excessive heat can damage circuit components or pose safety hazards.

Electric Current

Electric current is the flow of electric charge through a conductor, measured in amperes (A). It is analogous to the flow of water in a pipe, where more water means a stronger flow. Current can be direct (DC) or alternating (AC) depending on the flow type.
In this problem, we calculate the current through a person's body, knowing the total resistance and applying Ohm's Law: \( V = IR \). By rearranging the formula to \( I = \frac{V}{R} \), and using the supply voltage of 14 \,kV and the total resistance of 12 \,k\(\Omega\), we find the current to be about 1.167 \,A.
This demonstrates how with a high enough voltage, significant current can flow through even high resistances, emphasizing the importance of protecting circuit users from accidental exposure to high voltages.

Power Supply

A power supply is an electrical device that provides power to an electrical load. It supplies electric current to allow different components in a circuit to function. It maintains the necessary voltage and current as per the circuit's requirements.
In the exercise, the power supply is described as having a 14-kV potential, with an internal resistance that affects the overall current. Internal resistance is inherent in all power supplies and is usually minimized to ensure efficient power delivery.
When designing circuits, it's important to consider not only the voltage but also the internal resistance of power supplies, as it can impact current flow and lead to potential safety concerns.

Internal Resistance

Internal resistance is the resistance within a power supply that impedes the flow of electric current. It is an inherent part of any power supply and affects how efficiently the power is delivered to the external circuit.
In practice, internal resistance causes some of the electric power to be lost as heat within the power supply itself, reducing the power available to the load.
In this exercise, we initially measure the internal resistance at 2000 \,\( \Omega \), which combines with the body resistance to affect the total resistance and current through the person. To ensure safety, internal resistance can be adjusted to limit current flow to non-dangerous levels by increasing to 13,990,000 \,\( \Omega \), cutting down the available current to safer levels like 1.00 \,mA.
Understanding internal resistance helps in designing safer electrical equipment and managing energy efficiency in electrical supply systems.